{"http:\/\/dx.doi.org\/10.14288\/1.0367001":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Li, Ka-Fai","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2018-05-24T19:04:29Z","type":"literal","lang":"en"},{"value":"2018","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"We first study the general theory of K\u00e4hler-Ricci flow on non-compact complex\r\nmanifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate,\r\nwe prove a general existence theorem for K\u00e4hler metrics lying in the C^\\infty\r\n_\\loc closure of complete bounded curvature K\u00e4hler metrics that are uniformly equivalent\r\nto a fixed background metric. In particular we do not assume any curvature bounds. Next, we compare the maximal existence time of two complete bounded curvature solutions by using the equivalence of the initial metrics and using this, we also estimate the maximal existence time of a complete bounded curvature solution in terms of the curvature bound of a background metric. We also prove a uniqueness theorem for K\u00e4hler-Ricci flow which slightly improves the result of\r\n[13] in the K\u00e4hler case.\r\nWe apply the above results to study the K\u00e4hler-Ricci flow on some specific non-compact complex manifolds. We first study the K\u00e4hler-Ricci flow on C^n. By applying our general existence theorem and existence time estimate, we show that any complete non-negatively curved U(n)-invariant K\u00e4hler metric admits a longtime U(n)-invariant solution to the K\u00e4hler-Ricci flow, and the solution converges to the standard Euclidean metric after rescaling.\r\nThen we study the K\u00e4hler-Ricci flow on a quasi-projective manifold. By modifying the approximation theorem of [1] and applying a general existence theorem of Lott-Zhang [23], we construct a K\u00e4hler-Ricci flow solution starting from certain smooth K\u00e4hler metrics. In particular, if the metric is the restriction of a smooth K\u00e4hler metric in the ambient space, then the solution instantaneously becomes complete and has cusp singularity at the divisor. We also produce a solution starting from some complete metrics that may not have bounded curvature, and the solution is likewise complete with cusp singularity for positive time. On the other hand, if the initial data has bounded curvature and is asymptotic to the standard cusp model in a certain sense, we find the maximal existence time of the corresponding complete bounded curvature solution to the K\u00e4hler-Ricci flow.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/66082?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"The Ka\u00a8hler-Ricci flow on non-compact manifoldsbyKa-Fai LiA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)May 2018c\u00a9 Ka-Fai Li, 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:The Ka\u00a8hler-Ricci flow on non-compact manifoldssubmitted by Ka-Fai Li in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin MathematicsExamining Committee:Albert ChauSupervisorJingyi ChenSupervisory Committee MemberAilana FraserSupervisory Committee MemberYoung-Heon KimUniversity ExaminerJoanna KarczmarekUniversity ExamineriiAbstractWe first study the general theory of Ka\u00a8hler-Ricci flow on non-compact complexmanifolds. By using a parabolic Schwarz lemma and a local scalar curvature es-timate, we prove a general existence theorem for Ka\u00a8hler metrics lying in the C\u221eloc-closure of complete bounded curvature Ka\u00a8hler metrics that are uniformly equiva-lent to a fixed background metric. In particular we do not assume any curvaturebounds. Next, we compare the maximal existence time of two complete boundedcurvature solutions by using the equivalence of the initial metrics and using this,we also estimate the maximal existence time of a complete bounded curvature so-lution in terms of the curvature bound of a background metric. We also prove auniqueness theorem for Ka\u00a8hler-Ricci flow which slightly improves the result of[13] in the Ka\u00a8hler case.We apply the above results to study the Ka\u00a8hler-Ricci flow on some specificnon-compact complex manifolds. We first study the Ka\u00a8hler-Ricci flow on Cn. Byapplying our general existence theorem and existence time estimate, we show thatany complete non-negatively curved U(n)-invariant Ka\u00a8hler metric admits a long-time U(n)-invariant solution to the Ka\u00a8hler-Ricci flow, and the solution convergesto the standard Euclidean metric after rescaling.Then we study the Ka\u00a8hler-Ricci flow on a quasi-projective manifold M \\D. Bymodifying the approximation theorem of [1] and applying a general existence the-orem of Lott-Zhang [23], we construct a Ka\u00a8hler-Ricci flow solution starting fromcertain smooth Ka\u00a8hler metrics. In particular, if the metric is the restriction of asmooth Ka\u00a8hler metric in the ambient space M, then the solution instantaneouslybecomes complete and has cusp singularity at D. We also produce a solution start-ing from some complete metrics that may not have bounded curvature, and theiiisolution is likewise complete with cusp singularity for positive time. On the otherhand, if the initial data has bounded curvature and is asymptotic to the standardcusp model at D in a certain sense, we find the maximal existence time of thecorresponding complete bounded curvature solution to the Ka\u00a8hler-Ricci flow.ivLay SummaryThe Ricci flow was introduced by Richard Hamilton in 1982 to solve the famousPoincare\u00b4 conjecture in mathematics, and it is a geometric process that deforms theshape of a given space according to how the space is curved. Hamilton provedthat some nicely curved 3 dimensional spaces deform to a sphere along the Ricciflow. Similar results have been proved for the Ricci flow since then. On the otherhand, some spaces cannot be deformed at all due to their roughness, and this thesisdiscusses conditions under which a space can be deformed along the Ricci flow,and when is there a unique way to carry out the deformation. We also apply theRicci flow to spaces with certain symmetries. For example, we prove that certainrotationally symmetric spaces will be flattened along the Ricci flow, and we relatethis to a well known conjecture in geometry.vPrefaceAll of the work presented in this thesis was conducted as I was a PhD student inthe Mathematics Department of University of British Columbia.Materials in Chapter 2 and 3 are from [10], [11] and [12], these are joint workswith Professor Albert Chau and Professor Luen-Fai Tam and they are all published.In [10], I contributed the parabolic Schwarz computations and the construction ofbackground U(n)-invariant metrics. In [11], all authors contributed equally. In[12], I made contributions to the existence of long-time solution to U(n)-invariantKa\u00a8hler-Ricci flow. Some arguments in [12] are replaced by new original argumentsin this thesis.Materials in Chapter 4 are from [9], which is a joint work with Professor AlbertChau and Doctor Liangming Shen, it has been posted on arXiv. In this work, allauthors contributed equally.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Non-negatively curved U(n)-invariant metrics . . . . . . . . . . . 51.3 Ka\u00a8hler-Ricci flow on quasi-projective manifolds . . . . . . . . . . 62 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 C0 estimates in terms of background metrics . . . . . . . . . . . . 112.3 General existence theorems . . . . . . . . . . . . . . . . . . . . . 182.4 A uniqueness theorem and an existence time estimate . . . . . . . 223 U(n)-invariant Ka\u00a8hler metrics . . . . . . . . . . . . . . . . . . . . . 273.1 Background materials . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Short time existence of U(n)-invariant Ka\u00a8hler-Ricci flow . . . . . 313.3 Long time solution . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Bounding the scalar curvature . . . . . . . . . . . . . . . . . . . 43vii3.5 Convergence after rescaling . . . . . . . . . . . . . . . . . . . . . 484 Quasi-projective manifolds . . . . . . . . . . . . . . . . . . . . . . . 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . 554.2.1 Proof of Theorem 4.1.1 when \u03d50 \u2208C\u221e(M) . . . . . . . . 564.2.2 Proof of Theorem 4.1.1 when \u03d50 \u2208 L\u221e(M)\u22c2C\u221e(M) . . . 624.3 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 Approximate solutions \u03c9\u03b1, j(t) . . . . . . . . . . . . . . . 654.3.2 A priori estimates for \u03c9\u03b1, j(t) . . . . . . . . . . . . . . . . 674.3.3 Completion of Proof of Theorem 4.1.2 . . . . . . . . . . . 734.4 Proof of Theorem 4.1.3 . . . . . . . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.1 Interior estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.2 Plurisubharmonic functions . . . . . . . . . . . . . . . . . . . . . 83viiiAcknowledgmentsI would like to thank my supervisor Professor Albert Chau, not only for his guid-ance and motivation during my Ph.D. study, but also for his support during mytough times. I would also like to thank Professor Jingyi Chen and Professor AilanaFraser for serving on my thesis committee. I am also grateful to Liangming Shenand John Ma for many helpful discussions. Last but not least, I would like to thankmy family and my friends outside of UBC for their encouragement and support.ixChapter 1IntroductionIn this thesis, we study the Ka\u00a8hler-Ricci flow on non-compact complex manifolds.Let (M,g) be a n dimensional Ka\u00a8hler manifold, a smooth family of Ka\u00a8hler metricsg(t) is said to satisfy the Ka\u00a8hler-Ricci flow on M\u00d7 [0,T ) starting from g if\uf8f1\uf8f2\uf8f3\u2202\u2202 tgi j\u00af(t) =\u2212Ri j\u00af(g(t))g(0) = g,(1.0.1)on M\u00d7 [0,T ), where Ri j\u00af(g(t)) is the Ricci curvature of g(t). Any Ka\u00a8hler metriccan also be represented by a Ka\u00a8hler form, if \u03c9 is the Ka\u00a8hler form of g then theequation (1.0.1) is equivalent to\uf8f1\uf8f2\uf8f3\u2202\u2202 t\u03c9(t) =\u2212Ric(\u03c9(t))\u03c9(0) = \u03c9,(1.0.2)where Ric(\u03c9(t)) = \u2212\u221a\u22121\u2202 \u2202\u00af log\u03c9n(t) is the Ricci form of \u03c9(t). In the follow-ing content, we will use Greek alphabets to denote a Ka\u00a8hler form while Romanalphabets will be used to denote a Ka\u00a8hler metric.The thesis is divided into three parts. In the first part, we study the generaltheory of Ka\u00a8hler-Ricci flow on non-compact complex manifolds. We prove anexistence theorem for initial metrics that can be approximated by complete Ka\u00a8hlermetrics of bounded curvature satisfying some C0 conditions. We also prove anexistence time estimate and a uniqueness theorem for complete bounded curvature1solutions.In the second part, we study the Ka\u00a8hler-Ricci flow on Cn starting from a com-plete U(n)-invariant Ka\u00a8hler metric. We show that if the U(n)-invariant Ka\u00a8hlermetric has non-negative holomorphic bisectional curvature, then the Ka\u00a8hler-Ricciflow has a corresponding long-time bounded curvature solution to the Ka\u00a8hler-Ricciflow. And the solution we obtain converges to the standard Euclidean metric afterrescaling.In the third part, we study the Ka\u00a8hler-Ricci flow on quasi-projective manifold.A Ka\u00a8hler manifold M is called a quasi-projective manifold if M = M \\D, whereM is compact Ka\u00a8hler and D \u2282 M is a divisor with normal crossings. We willconstruct a solution to Ka\u00a8hler-Ricci flow starting from certain initial metrics on Mand the solution we construct becomes instantaneously complete and with a cuspsingularity at D. We also estimate the existence time of our solution.1.1 General TheoremsOne of the most fundamental questions in the study of Ka\u00a8hler-Ricci flow is theexistence of solutions. Unlike the Ka\u00a8hler-Ricci flow on compact Ka\u00a8hler manifolds,the existence problem on non-compact Ka\u00a8hler manifolds is not a direct conse-quence of general theory. The most classical result which is due to Shi( [26, 27])states that if (M,g) is a complete Ka\u00a8hler manifold with bounded curvature, thenthere is a complete bounded curvature solution to (1.0.1). Shi\u2019s solution has beenused extensively in the application of Ricci flow on non-compact Riemannianmanifolds. Chen-Zhu [13] showed that Shi\u2019s solution is unique among completebounded curvature solutions while Lott-Zhang [23] gave an analytic characteriza-tion of the maximal existence time of the solution. As a result, given a completebounded curvature solution, we can discuss the maximal existence time of the cor-responding complete bounded curvature solution to Ka\u00a8hler-Ricci flow. There areother more general existence results concerning complete Ka\u00a8hler metrics on non-compact manifolds in the literature, but they often require rather specific curvatureassumptions and their uniqueness and existence time are often not known.In case the initial metric has unbounded curvature, we prove that if it can be ap-proximated by certain complete bounded curvature metrics, then we can construct2a solution to 1.0.1:Theorem 1.1.1. Let g0 be a complete continuous Hermitian metric on a non-compact complex manifold Mn. Suppose there exists a sequence {hk,0} of smoothcomplete Ka\u00a8hler metrics with bounded curvature on M converging uniformly oncompact subsets to g0 and another complete Ka\u00a8hler metric g\u02c6 on M with boundedcurvature and holomorphic sectional curvature bounded from above by K \u2265 0 suchthat(i) 1C g\u02c6\u2264 hk,0 \u2264Cg\u02c6 for some C independent of k;(ii) hk has bounded curvature for every k.Let T = 1\/(2nCK) if K > 0, otherwise let T = \u221e. Then the Ka\u00a8hler-Ricci flow(1.0.1) has a smooth solution g(t) on M\u00d7 (0,T ) such that(a) (1\/(nC)\u22122Kt)g\u02c6\u2264 g(t)\u2264 B(t)g\u02c6 on M\u00d7 (0,T ) for some positive continuousfunction B(t) depending only on C, g\u02c6 and n.(b) g(t) has bounded curvature for t > 0. More precisely, for any 0 < T \u2032 < Tand for any l \u2265 0 there exists a constant Cl depending only on C, l, T \u2032, g\u02c6 andthe dimension n such thatsupM|\u2207lRm(g(t))|2g(t) \u2264Clt l+2,(c) g(t) converges uniformly on compact subsets to g0 as t\u2192 0.Moreover, if g0 is smooth and {hk,0} converges smoothly and uniformly oncompact subsets of M, then g(t) extends to a smooth solution on M\u00d7 [0,T ) withg(0) = g0.The proof of this theorem is based on estimating the local equivalence betweenthe evolving metric and the fixed background metric g\u02c6. This is achieved by using aparabolic Schwarz lemma, an Aubin-Yau trace estimate and a local scalar curvatureestimate. After establishing the equivalence, the higher order estimates follow fromEvans-Krylov theory. Furthermore, our proof to this theorem actually shows that ifh is a complete bounded curvature Ka\u00a8hler metric such that 1C g\u02c6\u2264 h\u2264Cg\u02c6 for some3constant C, then the maximal existence time of Shi\u2019s solution satisfies T \u2265 12nCKif K > 0 and T = \u221e otherwise. That is, we can estimate the existence time ofa complete bounded curvature solution by the curvature bound of an equivalentcomplete bounded curvature Ka\u00a8hler metric.On the other hand, we will also prove a theorem which compares the maximalexistence time of two complete bounded curvature metrics using their C0 data.Theorem 1.1.2. If g and h are two equivalent complete Ka\u00a8hler metrics of boundedcurvature and C is a constant such that g \u2264 Ch, then T (g) \u2264 CT (h). Here T (\u00b7)is the maximal existence time of the complete bounded curvature solution to theKa\u00a8hler-Ricci flow starting from the given metric.In particular, our result implies that if g is a complete Ka\u00a8hler metric of boundedcurvature with T (g) = \u221e, then this property is shared by all complete boundedcurvature Ka\u00a8hler metrics equivalent to g.We also prove a theorem about the uniqueness:Theorem 1.1.3. Let (Mn, g\u0302) be a complete non-compact Ka\u00a8hler manifold. Supposethere is an exhaustion function \u03b6 > 0 on (Mn, g\u0302) with limx\u2192\u221e \u03b6 (x) = \u221e such that|\u2202 \u2202\u00af\u03b6 |g\u0302 and |\u2207\u0302\u03b6 |g\u0302 are bounded.Let g1(x, t) and g2(x, t) be two solutions of the Ka\u00a8hler-Ricci flow (1.0.1) onM\u00d7 [0,T ] with the same initial data g0(x) = g1(x,0) = g2(x,0). Suppose there isa positive function \u03c3 with limx\u2192\u221e log\u03c3(x)\/ log\u03b6 (x) = 0 such that the followingconditions hold for all (x, t) \u2208M\u00d7 [0,T ]:(i)g\u0302(x)\u2264 \u03b6 (x)g1(x, t); g\u0302(x)\u2264 \u03b6 (x)g2(x, t),(ii)\u2212\u03c3(x)\u2264 det((g1)i j\u00af(x, t))det((g2)i j\u00af(x, t))\u2264 \u03c3(x).Then g1 \u2261 g2 on M\u00d7 [0,T ].This theorem implies that if g1 and g2 are uniformly equivalent to g\u0302 on M\u00d7[0,T ], then g1 \u2261 g2. This slightly improves the uniqueness result of Chen-Zhu inthe Ka\u00a8hler case.41.2 Non-negatively curved U(n)-invariant metricsThe result in Chapter 3 is related to Yau\u2019s uniformization conjecture for non-compact Ka\u00a8hler manifolds: Let (Mn,g) be a complete non-compact Ka\u00a8hler man-ifold with positive holomorphic bisectional curvature, then M is biholomorphicto Cn. This is a long standing conjecture and there have been several partial re-sults so far, in particular using the Ka\u00a8hler-Ricci flow. For example, Chau-Tam[8] used Ka\u00a8hler-Ricci flow to prove that a complete non-compact Ka\u00a8hler manifoldwith bounded non-negative holomorphic bisectional curvature and maximal vol-ume growth is biholomorphic to Cn. Recently, this result was generalized by Liu[21] by removing the bounded curvature condition. Liu did not use Ka\u00a8hler-Ricciflow in his proof, but his result was re-proved by Lee-Tam [20] using the Ka\u00a8hler-Ricci flow.The U(n)-invariant Ka\u00a8hler metrics on Cn were first studied by Wu-Zheng [33]in an attempt to give examples of positively curved compete Ka\u00a8hler metrics on Cn.Prior to their works, there were only three examples of positively curved completeKa\u00a8hler metrics on Cn and they are all U(n)-invariant ( [6, 7, 19]). In Wu-Zheng\u2019swork, they gave a very convenient parametrization of the complete U(n)-invariantmetrics with non-negative bisectional curvature, in particular illustrating that thesegenerically had unbounded curvature. Motivated by these examples, Yang-Zheng[35] showed that if g(t) is a complete U(n)-invariant solution to the Ka\u00a8hler-Ricciflow with initial metric having non-negative holormophic bisectional curvature,then the solution will also have non-negative holomorphic bisectional curvaturefor all time. Furthermore the asymptotic volume ratio remains constant along theflow. They also used the construction of Cabezas-Rivas and Wilking [2] to producea short time solution for complete U(n)-invariant initial metric with non-negativesectional curvature satisfying some technical assumptions. Using a different ap-proach, we prove that:Theorem 1.2.1. Let g0 be a complete U(n)-invariant Ka\u00a8hler metric on Cn withnon-negative holomorphic bisectional curvature. Then(i) the Ka\u00a8hler-Ricci flow (1.0.1) has a unique smooth long-time U(n)-invariantsolution g(t) which is equivalent to g0 and has bounded non-negative bisec-tional curvature;5(ii) g(t) converges, after rescaling at the origin, to the standard Euclidean metricon Cn.Here g(t) is equivalent to g0 means that for all T < \u221e, there exists constantC such that 1C g0 \u2264 g(t) \u2264 Cg0 for all t \u2208 [0,T ]. Our result does not assume anyvolume growth rate on the initial metrics but yet the solution converges to thestandard Euclidean metric in a certain sense. This showed that the Ka\u00a8hler-Ricciflow may still be used to attack Yau\u2019s conjecture even if we do not assume anyvolume growth conditions.1.3 Ka\u00a8hler-Ricci flow on quasi-projective manifoldsIn Chapter 4, we study the Ka\u00a8hler-Ricci flow on quasi-projective manifolds. AKa\u00a8hler manifold M is called a quasi-projective manifold if M = M \\D, where M isKa\u00a8hler and D\u2282M is a divisor with normal crossings. M is clearly a non-compactmanifold, given any Ka\u00a8hler metric on M, we can define different notions of singu-larity at D. A metric on M is said to have cusp singularities at D if it is equivalent toa Carlson-Griffiths type form \u03b7\u2212 i\u2202 \u2202\u00af log log2 |S|2 where \u03b7 is a Ka\u00a8hler form on Mand S is the holomorphic section of [D] that vanishes on D. The Carlson-Griffithsform is analogous to the Poincar\u00b4e metric on the punctured disk, with the divisorcorresponding to the origin. In particular, cusp metrics are complete on M withpossibly unbounded curvature. In Lott-Zhang [23], they considered bounded cur-vature cusp metrics satisfying certain asymptotics conditions at D, and showed thattheir maximal existence time for (1.0.1) is bounded above byT[\u03c90] := sup{T : [\u03b7 ]+T (c1(KM)+ c1(OD)) \u2208KM} (1.3.1)whereKM is the Ka\u00a8hler cone of M.The main theorems in Chapter 4 are:Theorem 1.3.1. Let \u03c90 = \u03b7+ i\u2202 \u2202\u00af\u03d50 be a smooth Ka\u00a8hler metric on M, and let T[\u03c90]be given by (1.3.1).(a) If \u03d50 \u2208 L\u221e(M)\u22c2C\u221e(M)\u22c2PSH(M,\u03b7) and \u03c90 \u2265 c\u03b7 for some constant c > 0.6Then (1.0.1) has a unique smooth solution \u03c9(t) on M\u00d7 [0,T[\u03c90]) wherec1(t)\u03c9\u02c6 \u2264 \u03c9t \u2264 c2(t)\u03c9\u02c6 (1.3.2)for all t \u2208 (0,T[\u03c90]) and some positive functions ci(t).(b) Let \u03d50 \u2208 C\u221e(M)\u22c2PSH(M,\u03b7) have zero Lelong number and \u03c90 \u2265 c\u03c9\u02c6 forsome c > 0. Then the Ka\u00a8hler -Ricci flow (1.0.1) has a smooth solution \u03c9(t)on M\u00d7 [0,T[\u03c90]) and\u03c9(t)\u2265 (1n\u2212 4K\u02c6tc)\u03c9\u02c6 (1.3.3)for all t \u2264 c4nK\u02c6.As a particular case of part (a), if \u03c90 is the restriction of a Ka\u00a8hler metric onM or has conical singularities at D, the solution instantaneously becomes completeand is equivalent to a cusp metric for t > 0. On the other hand, the initial metricin (b) is complete with possibly unbounded curvature, and there the solution islikewise complete for a uniform amount of time.There have been several earlier works about solutions to the Ka\u00a8hler-Ricci flowwith conical singularities at D (see for example [14], [22], [25]). These worksestablished the characterization of the existence time in terms of the cohomologyof M, and also produced an instantaneously conical solution starting from initialdata that is restriction from smooth metrics on M. Our results can be viewed as acuspidal versions of these results.In addition to Theorem 1.3.1, we will also discuss the maximal existence timeof a complete bounded curvature solution starting from a Ka\u00a8hler metric which hasbounded curvature and is asymptotic to a cusp model in a certain sense.7Chapter 2General Theorems2.1 BackgroundThe Ricci flow was first introduced by Hamilton [18] to prove that a closed 3 di-mensional Riemannian manifold with positive Ricci curvature is diffeomorphic toa spherical space form or its quotient. Since then the Ricci flow has been appliedto many problems in geometry, it was first applied to Ka\u00a8hler manifolds by Cao in[4] who used the flow to re-prove the Calabi-Yau theorem.The behavior of Ricci flow on compact Ka\u00a8hler manifolds has been well studied.Suppose that (M,g) is a closed Ka\u00a8hler manifold, then it is known that a short-timesolution g(t) to the Ricci flow exists and the solution remains Ka\u00a8hler . The exis-tence follows from the fact that on Ka\u00a8hler manifolds the Ricci flow is a parabolicsystem of partial differential equations. In fact by applying the \u2202 \u2202\u00af -lemma, it canbe shown here the Ricci flow is equivalent to a parabolic Monge-Ampe`re equa-tion for a scalar function. With the aid of this equivalence, the uniqueness followsreadily by a maximum principle argument. The maximal existence time of the so-lution was characterized by Tian-Zhang [31] in terms of the cohomology class ofthe initial metric:Theorem 2.1.1 (Tian-Zhang [31]). Let (M,\u03c9) be a closed Ka\u00a8hler manifold, sup-pose that \u03c9(t) solves the Ka\u00a8hler-Ricci flow. Then the maximal existence time of the8solution is given byT[\u03c9] = sup{t > 0 : [\u03c9]\u2212 tc1(M)> 0}.Here we say a cohomology class [\u03b1] > 0 if there is a Ka\u00a8hler form inside the class[\u03b1], and c1(M) is the first Chern class of the manifold, which is equal to the class[Ric(\u03c9)] for arbitrary Ka\u00a8hler metric \u03c9 on M.When the Ka\u00a8hler manifold (M,g) is non-compact, we do not expect the Ricciflow to have a solution unless g satisfies some extra conditions. And even if itexists, the solution g(t) may not be Ka\u00a8hler or even Hermitian for t > 0. In caseg(t) solves the Ricci flow: \uf8f1\uf8f2\uf8f3\u2202\u2202 tg(t) =\u2212Ric(g(t))g(0) = g.(1.0.1)and g(t) is Ka\u00a8hler for all t, we will call it a solution to Ka\u00a8hler-Ricci flow. Theresearch of Ka\u00a8hler-Ricci flow on non-compact manifolds was pioneered by Shi([26, 27]), and he proved the short time existence of Ka\u00a8hler-Ricci flow on completenon-compact Ka\u00a8hler manifolds with bounded curvature:Theorem 2.1.2. Let (Mn,g0) be a complete non-compact Ka\u00a8hler manifold withcurvature bounded by a constant K. Then for some 0 < T \u2264 \u221e depending only onK and the dimension n, there exists a smooth solution g(t) to (1.0.1) on M\u00d7 [0,T )with g(0) = g0 such that(i) g(t) is Ka\u00a8hler and equivalent to g0 for all t \u2208 [0,T );(ii) g(t) has uniformly bounded curvature on M\u00d7 [0,T \u2032] for all 0< T \u2032< T . Moreprecisely, for any l \u2265 0 there exists a constant Cl depending only on l, T \u2032, g0and the dimension n such thatsupM|\u2207lRm(g(t))|2g(t) \u2264Clt l,on M\u00d7 [0,T \u2032].(iv) If T <\u221e and limt\u2192TsupM|Rm(x, t)|<\u221e, then g(t) extends to a smooth solution to9(1.0.1) on M\u00d7 [0,T1) for some T1 > T so that (ii) is still true with T replacedby T1.From now on, given a bounded curvature complete g0, we will refer to a solu-tion g(t) to (1.0.1) satisfying (i), (ii) as Shi\u2019s solution to (1.0.1). In fact, Chen-Zhu[13] showed that Shi\u2019s solution is unique. More precisely,Theorem 2.1.3 (Chen-Zhu [13]). Let (M,g0) be a complete non-compact Ka\u00a8hlermanifold of bounded curvature, if g1(t), g2(t) are solutions to Ka\u00a8hler-Ricci flow onM\u00d7 [0,T ] such that gi(t) (i= 1,2) are complete, have uniformly bounded curvatureand gi(0) = g0, then g1(t) = g2(t).Remark 2.1.1. Chen-Zhu\u2019s uniqueness theorem was originally proved for generalreal Ricci flow, the version we stated here is the restriction to the Ka\u00a8hler case.Lott-Zhang [23] also found a similar existence time characterization for Shi\u2019ssolution like Theorem 2.1.1, although it is no longer described by the cohomologyclass of M:Theorem 2.1.4 (Lott-Zhang [23]). Let g0 be a complete Ka\u00a8hler metric with boundedcurvature on a non-compact manifold M. Then the maximal existence time Tg0 of abounded curvature solution is equal to the supremum of the numbers T for whichthere is a bounded function FT \u2208C\u221e(M) such that(i) g0\u2212T Ric(g0)+\u221a\u22121\u2202 \u2202\u00afFT \u2265 cT g0 for some cT > 0(ii) |FT | and the quantities |\u2207lRm(g0)|g0 , |\u2207l\u2202 \u2202\u00afFT |g0 , for 0 \u2264 l \u2264 2, are uni-formly bounded on M.Moreover if T satisfies (i) and (ii), then for any T \u2032 < T , there is a constant Cdepending T \u2032, cT , and the bound on the quantities in (ii) such thatC\u22121g0 \u2264 g(t)\u2264Cg0on M\u00d7 [0,T \u2032].In the following sections, we will construct a solution to (1.0.1) for a completeKa\u00a8hler metric g0 that can be approximated by a sequence of complete bounded cur-vature Ka\u00a8hler metrics satisfying some C0 assumptions. Each of sequence members10admits a solution to Ka\u00a8hler-Ricci flow by Shi\u2019s Theorem 2.1.2, then by establishingsome C0 estimates, we can use Evans-Krylov theory to conclude that the sequenceof solutions converges to a limit solution starting from g0. We will also prove auniqueness result of Ka\u00a8hler-Ricci flow which can be applied to unbounded curva-ture metrics satisfying certain conditions.2.2 C0 estimates in terms of background metricsBefore we prove the existence theorems, we need to establish some C0 estimatesfor solutions of Ka\u00a8hler-Ricci flow in terms of a background metric, these estimatesallow us to use the Evans-Krylov theory to conclude that our approximating se-quence converges.In the following, \u2207\u02c6 always denotes the covariant derivative of g\u02c6.Lemma 2.2.1. Let h(t) be a solution to (1.0.1) on Mn\u00d7 [0,T0) with h(0) = h0 suchthat h(t) has uniformly bounded curvature on M\u00d7 [0,T \u2032] for all 0 < T \u2032 < T0. Letg\u02c6 be another complete Ka\u00a8hler metric on M with bounded curvature such that theholomorphic bisectional curvature bounded above by K \u2265 0. Let T = 12nK if K > 0,otherwise let T = \u221e.(i) Suppose h0 \u2265 g\u02c6. Then h(t)\u2265(1n \u22122Kt)g\u02c6 on M\u00d7 [0,min{T0,T}).(ii) Suppose in addition to (i) we have h0 \u2264 Cg\u02c6, that is, suppose g\u02c6 \u2264 h0 \u2264 Cg\u02c6,then(1\u2212w(t))g\u02c6\u2264 h(t)\u2264 (1+w(t))g\u02c6on M\u00d7 [0,min{T0,T}),where w(t) =\u221av2(t)(v1(t)+ v2(t)\u22122n),v1(t) =11n \u22122Kt,v2(t) = nCe\u22122\u03bav1(t)tand \u03ba is a lower bound on the bisectional curvature of g\u02c6. In particular, we havew(0) = n\u221aC(C\u22121).Proof. (i) Let \u03c6(t) := trh(t)g\u02c6 = h(t)i j\u00afg\u02c6i j\u00af. Let 2= \u2202\u2202 t \u2212\u2206, where \u2206 is the Laplacianwith respect to h(t). Then as in [28], we can calculate in a normal coordinate11relative to h(t) and use (1.0.1) to get2\u03c6 =((ht)i j\u00afg\u02c6i j\u00af)\u2212hkl\u00af(hi j\u00afg\u02c6i j\u00af)kl\u00af=(Ri j\u00afg\u02c6i j\u00af)\u2212 (Ri j\u00afg\u02c6i j\u00af)+hkl\u00afhi j\u00afR\u0302i j\u00afkl\u00af\u2212 g\u0302pq\u00afhkl\u00afhi j\u00af\u2202kg\u0302iq\u00af\u2202l\u00af g\u0302p j\u00af\u2264 2K\u03c6 2.(2.2.1)Now v1(t) is the positive solution to the ODEdv1(t)dt= 2Kv21(t); v1(0) = nfor t \u2208 [0,T ). Let S \u2208 (0,min{T0,T}) be fixed. Since h(t) has uniformly boundedcurvature on M\u00d7 [0,S] we have h(t) \u2265C1h0 \u2265C1g\u0302 for some C1 > 0 and hence \u03c6is a bounded function on M\u00d7 [0,S]. Moreover, v1 is also a bounded function onM\u00d7 [0,S]. Let A = supM\u00d7[0,S](\u03c6 + v1). Then on M\u00d7 [0,S]2(e\u2212(2AK+1)t(\u03c6 \u2212 v1))\u2264 e\u2212(2AK+1)t [2K(\u03c6 2\u2212 v21)\u2212 (2AK+1)(\u03c6 \u2212 v1)]= e\u2212(2AK+1)t [2K(\u03c6 + v1)\u2212 (2AK+1)] (\u03c6 \u2212 v1)which is nonpositive at the points where \u03c6\u2212v1\u2265 0. Using the fact that h(t) has uni-formly bounded curvature on M\u00d7 [0,S] and the fact that e\u2212(2AK+1)t(\u03c6 \u2212 v1)\u2264 0 att = 0, which is uniformly bounded on M\u00d7 [0,S], we conclude that e\u2212(2AK+1)t(\u03c6 \u2212v1) \u2264 0 and thus (\u03c6 \u2212 v1) \u2264 0 on M\u00d7 [0,S] by the maximum principle, see [24,Theorem 1.2] for example. This proves (i).(ii) Let \u03c8(t) := trg\u02c6h(t). For any fixed S \u2208 [0,min{Th,T}), as in [4] we calculatein a normal coordinate relative to g\u02c6 and use (1.0.1) to get that on M\u00d7 [0,S):122\u03c8 =(g\u02c6i j\u00af(ht)i j\u00af)\u2212hkl\u00af(g\u02c6i j\u00afhi j\u00af)kl\u00af=\u2212 (g\u02c6i j\u00afRi j\u00af)\u2212hkl\u00af(R\u02c6i j\u00afkl\u00afhi j\u00af)+(g\u02c6i j\u00afRi j\u00af)\u2212 g\u0302i j\u00afhpq\u00afhkl\u00af\u2202ihpl\u00af\u2202 j\u00afhkq\u00af=\u2212hkl\u00afhi j\u00afR\u0302 j\u00afikl\u00af \u2212 g\u0302i j\u00afhpq\u00afhkl\u00af\u2202ihpl\u00af\u2202 j\u00afhkq\u00af\u2264\u22122\u03bav1(t)\u03c8\u2264\u22122\u03bav1(S)\u03c8(2.2.2)by (i). Let wS(t) = nCe\u22122cv1(S)t be the solution to the ODEdwS(t)dt=\u22122cv1(S)wS(t); wS(0) = nC.Then arguing as above, we have \u03c8 \u2264 wS on Mn \u00d7 [0,S]. In particular, we get\u03c8(S)\u2264 wS(S) for every S \u2208 [0,min{T0,T}).So far, we have \u03c6(t) \u2264 v1(t), and \u03c8(t) \u2264 v2(t) on M\u00d7 [0,min{T0,T}) wherev1,v2 are as in the statement of the Lemma. Now we follow an idea from [26].At any point in (p, t) \u2208 M\u00d7 [0,min{T0,T}), let \u03bb \u2032i s be the eigenvalues of h withrespect to g\u02c6, and calculate at (p, t)n\u2211i=11\u03bbi(1\u2212\u03bbi)2 =n\u2211i=11\u03bbi+\u03bbi\u22122=\u03c6 +\u03c8\u22122n\u2264v1(t)+ v2(t)\u22122n(2.2.3)and thus for any fixed i we have\u2212w(t)\u2264 \u03bbi\u22121\u2264 w(t) (2.2.4)where w(t) =\u221av2(t)(v1(t)+ v2(t)\u22122n). The conclusion in (ii) then follows.The following lemma basically says that if a local solution h(t) to (1.0.1) isa priori uniformly equivalent to a fixed metric g\u02c6 in space time, and close to g\u02c6 at13time t = 0, then it remains close to g\u02c6 in a uniform space time region. Note that incontrast to Lemma 2.2.1, the a priori assumption here is on h(t) for all t.Lemma 2.2.2. Let h(t) be a smooth solution to (1.0.1) on B(1)\u00d7 [0,T ) with h(0) =h0 where B(1) is the unit Euclidean ball in Cn. Let g\u02c6 be a smooth Ka\u00a8hler metric onB(1). SupposeN\u22121g\u02c6\u2264 h(t)\u2264 Ng\u02c6 (2.2.5)on B(1)\u00d7 [0,T ) for some N > 0, and thatg\u02c6\u2264 h0 \u2264Cg\u02c6 (2.2.6)on B(1). Then there exists a positive continuous function a(t) : [0,T )\u2192R depend-ing only on g\u02c6,N,C and n such that(1\u2212a(t))Ch0 \u2264 h\u2264 (1+a(t))h0 (2.2.7)on B(1\/2)\u00d7 [0,T ), where a(0) = n\u221aC(C\u22121).Proof. As in the previous Lemma, let \u03c6 = trhg\u02c6, \u03c8 = trg\u02c6h on B(1)\u00d7 [0,T0). Choosesome smooth non-negative cutoff function on \u03b7 : B(1)\u2192 R satisfying \u03b7 |B(1\/2) =1, \u03b7 |(B(3\/4))c = 0, |\u2207\u02c6\u03b7 |2 \u2264 C1\u03b7 , |\u2202 \u2202\u00af\u03b7 |g\u02c6 \u2264 C2 on B(1) for some constants C1,C2depending only on g\u02c6. Using the fact that h(t)\u2265 N\u22121g\u02c6, we have|\u2207\u03b7 |2 = hi j\u00af\u03b7i\u03b7 j\u00af \u2264 N|\u2207\u02c6\u03b7 |2 \u2264 NC1,and|\u2206\u03b7 |=\u2223\u2223\u2223hi j\u00af\u03b7i j\u00af\u2223\u2223\u2223\u2264 N|\u2202 \u2202\u00af\u03b7 |g\u02c6 \u2264 NC2.Now we consider the function \u03b7\u03c6 on B(1)\u00d7 [0,T ). Then in B(1)\u00d7 [0,T ) at14the point where \u03b7 > 0, as in the proof of Lemma 2.2.1 (i) we obtain(\u2202t \u2212\u2206)(\u03b7\u03c6) = \u03b7(\u2202t \u2212\u2206)\u03c6 \u22122 < \u2207\u03b7 ,\u2207\u03c6 >\u2212\u03c6\u2206\u03b7= \u03b7(\u2202t \u2212\u2206)\u03c6 \u22122< \u2207\u03b7 ,\u2207(\u03b7\u03c6)>\u03b7 +2|\u2207\u03b7 |2\u03b7\u03c6 \u2212\u03c6\u2206\u03b7\u2264 \u03b7C3\u03c6 2\u22122< \u2207\u03b7 ,\u2207(\u03b7\u03c6)>\u03b7 +2NC1\u03c6 +NC2\u03c6\u2264C4\u22122< \u2207\u03b7 ,\u2207(\u03b7\u03c6)>\u03b7(2.2.8)where the constants C3,C4 depend only on g\u02c6,N,C and n, where we have used theassumption (2.2.7). Since \u03b7\u03c6 is zero outside B(3\/4), applying the maximum prin-ciple to \u03b7\u03c6 \u2212C4t one can conclude that\u03b7\u03c6 \u2264 n+C4t =: v\u02dc1(t)on B(1)\u00d7 [0,T ).Now consider the function \u03b7\u03c8 on B(1)\u00d7 [0,T ). Using the proof of Lemma2.2.1 (ii) and estimating as above we obtain\u03b7\u03c8 \u2264 nC+C5t =: v\u02dc2(t) (2.2.9)on B(1)\u00d7 [0,T ) for som constants C5 depending only on g\u02c6,N,C and n.Now at any point in (p, t) \u2208 B(1\/2)\u00d7 [0,T ), let \u03bb \u2032i s be the eigenvalues of hwith respect to g\u02c6. Then as in the proof of Lemma 2.2.1 (ii) we get that at (p, t)\u2212 w\u02dc(t)\u2264 \u03bbi\u22121\u2264 w\u02dc(t) (2.2.10)where w\u02dc(t) =\u221av\u02dc2(t)(v\u02dc1(t)+ v\u02dc2(t)\u22122n). Since v\u02dc1(0) = n and v\u02dc2(0) = nC, thelemma follows easily from this.In contrast to the previous lemma, in the following lemmas we only assume alower bound on a solution h(x, t) to (1.0.1).Lemma 2.2.3. Let h(x, t) be a smooth solution to (1.0.1) on M\u00d7 [0,T ) with h(0) =h0. Let p \u2208M. Suppose there is a positive continuous function \u03b1(t) : [0,T )\u2192 R15such thath(t)\u2265 \u03b1(t)g\u02c6.where g\u02c6 is a complete Ka\u00a8hler metric with bounded curvature. Then, there exists apositive continuous function \u03b2 (r, t) : [1,\u221e)\u00d7 [0,T )\u2192 R depending only on g\u02c6 theupper bound of trg\u02c6h0 in Bg\u02c6(p,2r), the lower bound of scalar curvature R(0) of h(0)in Bg\u02c6(p,2r), \u03b1(t) and the dimension n such that for r \u2265 1h(t)\u2264 \u03b2 (r, t)g\u02c6.in Bg\u02c6(p,r)\u00d7 [0,T ).Proof. Let d(x) be the distance with respect to g\u02c6 from x to a fixed point p \u2208 M.Since g\u02c6 has bounded curvature, by [27] there exists a smooth positive function\u03c1(x) satisfying d(x)+1\u2264 \u03c1(x)\u2264 d(x)+C on M for some C > 0, with |\u2207\u02c6\u03c1|, |\u2207\u02c62\u03c1|are bounded on M. Hence without loss of generality, we may assume for simplicitythat d(x) is in fact smooth with |\u2207\u02c6d|, |\u2207\u02c62d| bounded on M.Let \u03c6(s) be smooth function on R such that \u03c6 = 1 for s \u2264 1 and is zero fors\u2265 2. Moreover, we assume \u03c6 \u2032 \u2264 0, (\u03c6 \u2032)2\/\u03c6 \u2264C1, |\u03c6 \u2032\u2032| \u2264C2. Let R be the scalarcurvature of h(t). Then (\u2202\u2202 t\u2212\u2206)R\u2265 1nR2. (2.2.11)on M\u00d7 [0,T ). Let \u03d5(x) = \u03c6(d(x)\/r). Then \u03d5(x) = 0 if d(x) \u2265 2r. Fix someT \u2032 < T . Then as in the proof of the previous lemma, we compute|\u2207\u03d5|2 = 1r2(\u03c6 \u2032)2|\u2207d|2=1r2(\u03c6 \u2032)2hi j\u00afdid j\u00af\u2264 1r2\u03b1(t)(\u03c6 \u2032)2g\u02c6i j\u00afdid j\u00af\u2264C3r2(\u03c6 \u2032)216on B(2r)\u00d7 [0,T \u2032] for some constant C3 depending only on T \u2032,\u03b1(t) and g\u02c6. Similarly,|\u2206\u03d5|=|1r\u03c6 \u2032\u2206d+1r2\u03c6 \u2032\u2032|\u2207d|2|\u2264C4(1r +1r2)on B(2r)\u00d7 [0,T \u2032] where C4 depends on C1,C2,T \u2032,\u03b1(t) and g\u02c6.Now (\u2202\u2202 t\u2212\u2206)(\u03d5R) =\u03d5(\u2202\u2202 t\u2212\u2206)R\u2212R\u2206\u03d5\u22122\u3008\u2207R,\u2207\u03d5\u3009\u22651n\u03d5R2\u2212C5|R|\u22122\u3008\u2207R,\u2207\u03d5\u3009(2.2.12)on B(2r)\u00d7 [0,T \u2032] where C5 depends only on C4 and r. Suppose the infimum of \u03d5Ron B(2r)\u00d7 [0,T \u2032] is attained at t = 0, then R \u2265 min{0, infBg\u02c6(p,2r)R(h0)} on Bg\u02c6(r).Suppose instead that \u03d5R attains a negative minimum at some (x, t)\u2208 B(2r)\u00d7 [0,T \u2032]where t > 0. Then at (x, t), \u2207R =\u2212R\u2207\u03d5\u03c6 . Hence at this point,0\u22651n\u03d5R2\u2212C6|R| (2.2.13)where C6 depends only on C6,C3 and r. Hence\u03d52|R| \u2264 nC6.on B(2r)\u00d7 [0,T \u2032] and we conclude that R\u2265\u2212C7 on Bg\u02c6(p,r)\u00d7 [0,T \u2032] for some C7depending only on T \u2032, g\u02c6,r,\u03b1(t). On the other hand,\u2202\u2202 tlog(det(h\u03b1\u03b2\u00af )(t)det(h\u03b1\u03b2\u00af (0)))=\u2212R\u2264C7.Sodet(h\u03b1\u03b2\u00af )(t)det(g\u02c6\u03b1\u03b2\u00af )\u2264 eC7t det(h\u03b1\u03b2\u00af )(0)det(g\u02c6\u03b1\u03b2\u00af ).on Bg\u02c6(p,r)\u00d7 [0,T \u2032]. Let \u03bbi be eigenvalues of h(t)with respect to g\u02c6. By assumption,\u03bbi(x,T \u2032)\u2265 \u03b1(T \u2032) for each i and x \u2208 Bg\u02c6(p,r), and the above inequality then implies17\u03bbi(x,T \u2032)\u2264 \u03b2 (r,T \u2032) for some \u03b2 (r,T \u2032) depending only on the those constants listedin the Lemma. Moreover, it is not hard to see that \u03b2 (r,T \u2032) can be chosen to dependcontinuously on r,T \u2032 as \u03b1(t) is continuous. The Lemma follows as T \u2032 was chosenarbitrarily.Remark 2.2.1. The completeness assumption of g\u02c6 is only used to ensure the exis-tence of an exhaustion function \u03c1 with bounded gradient and Hessian, it could bedropped if such \u03c1 could be constructed independently.Remark 2.2.2. Given only a local solution h(t) to (1.0.1) on B(1)\u00d7 [0,T ) whereB(1) is the unit ball on Cn, it is not hard to see from its proof that the conclusionof Lemma 2.2.3 will hold in B(r)\u00d7 [0,T ) for all r \u2264 1\/2.2.3 General existence theoremsWe will now prove the main general existence Theorems for (1.0.1) using the esti-mates in the previous section. Theorems 2.3.1 and 2.3.2 provide general existenceTheorems for (1.0.1) when the initial Ka\u00a8hler metric is realized as a limit of a se-quence of Ka\u00a8hler metrics satisfying certain properties. In fact, our initial metricmay have unbounded curvature or may even be only Hermitian continuous withcurvature undefined. When the initial metric is only Hermitian, the convergence ofthe solution g(t) at time zero has to be understood in the C0 sense.In the following, we say that a sequence of smooth metrics hk converge uni-formly (resp. smoothly) to a metric g on a set U , if hk converge to g in the C0 (resp.Ck for all k) norm on U .Theorem 2.3.1. Let g0 be a complete continuous Hermitian metric on a noncom-pact complex manifold Mn. Suppose there exists a sequence {hk,0} of smoothcomplete Ka\u00a8hler metrics with bounded curvature on M converging uniformly oncompact subsets to g0 and another complete Ka\u00a8hler metric g\u0302 on M with boundedcurvature and holomorphic bisectional curvature bounded from above by K \u2265 0such that(i) hk,0 \u2265 g\u02c6 for all k;18(ii) for every k, the Ka\u00a8hler-Ricci flow (1.0.1) has smooth solution hk(t) with ini-tial data hk,0 on M\u00d7 [0,T \u2032) for some T \u2032 > 0 independent of k such that thecurvature of hk(t) is uniformly bounded on M\u00d7 [0,T1] for all 0 < T1 < T \u2032;(iii) The scalar curvature Rk of hk,0 satisfies: for any r > 0, there exists a constantCr > 0 such that Rk \u2265\u2212Cr on Bg\u02c6(p,r) for some fixed point p \u2208M and all k.Let T = min{T \u2032, 12nK} if K > 0, otherwise let T = T \u2032. Then the Ka\u00a8hler-Ricci flow(1.0.1) has a complete smooth solution g(t) on M\u00d7 (0,T ) which extends continu-ously to M\u00d7 [0,T ) with g(0) = g0 and satisfies g(t)\u2265 (1\/n\u22122nK)g\u02c6 on M\u00d7(0,T ).Moreover, if g0 is smooth and {hk,0} converges smoothly and uniformly oncompact subsets of M, then g(t) extends to a smooth solution to (1.0.1) on M\u00d7[0,T ) with g(0) = g0.Proof. By Lemma 2.2.1, we havehk(t)\u2265(1n\u22122Kt)g\u02c6 (2.3.1)as long as t < T0 = 1\/(2nK). By Theorem 2.1.2, let g\u02c6(t) be the solution Ka\u00a8hler-Ricci flow in the theorem with initial condition g\u02c6. Then for any 1 > \u03b5 > 0 small,choose 0 < t0 small enough so that (1\u2212 \u03b5)g\u0302(t0)\u2264 g\u0302\u2264 (1+ \u03b5)g\u0302(t0). Then we havehk(t)\u2265(1n\u22122Kt)(1\u2212 \u03b5)g\u02c6(t0) (2.3.2)and g\u02c6(t0) has bounded geometry of infinite order. By Lemma 2.2.3, for there is apositive continuous function \u03b2 (r, t) : [1,\u221e)\u00d7 [0,T0)\u2192 R such that for r \u2265 1hk(t)\u2264 \u03b2 (r, t)g\u02c6(t0). (2.3.3)in B\u02c6(p,r)\u00d7 [0,T )where T =min{T \u2032, 12nK} and p\u2208M is a fixed point. We concludefrom Theorem A.1.1 (i), that passing to some subsequence, the hk(t)\u2019s converge toa solution g(t) of Ka\u00a8hler-Ricci on M\u00d7 (0,T ) so that (2.3.1) is true. Moreover, ifg0 is smooth and {hk} converges smoothly and uniformly to g0 on compact sets,then we see from Theorem A.1.1 (ii) that in fact g(t) extends to a smooth solutionon M\u00d7 [0,T ) such that g(0) = g0.19We now prove g(t) converge uniformly on compact set to g0 as t \u2192 0 wheng0 is only assumed to be continuous. Fix any x \u2208M and a local biholomorphism\u03c6 : B(1)\u2192 M where B(1) is the open unit ball in Cn, and \u03c6(0) = x. Considerthe pullbacks \u03c6 \u2217hk(t), \u03c6 \u2217hk = \u03c6 \u2217hk(0), \u03c6 \u2217g\u02c6, which by abuse of notation we willsimply denote by hk(t), hk, g\u02c6, respectively, for the remainder of proof. In particular,hk(t) solves Ka\u00a8hler-Ricci flow (1.0.1) on B(1)\u00d7 [0,T ).Now by our hypothesis on the convergence of hk, given any \u03b4 > 0 we may findk0 such that |hk0,0\u2212g0|g\u02c6 \u2264 \u03b4 and(1\u2212\u03b4 )hk0,0 \u2264 hk,0 \u2264 (1+\u03b4 )hk0,0 (2.3.4)for all k\u2265 k0. On the other hand, by (2.3.2) and (2.3.3) we can find N > 0 such thatN\u22121hk0,0 \u2264 hk(t)\u2264 Nhk0,0 (2.3.5)in B(1)\u00d7 [0,T\/2) for all k \u2265 k0. Then by Lemma 2.2.2, there exists a continuousfunction a(t) depending on N,hk0 and \u03b4 such that(1\u2212a(t))(1\u2212\u03b4 )2(1+\u03b4 )hk0,0 \u2264 hk(t)\u2264 (1+a(t))(1+\u03b4 )hk0,0in B(12)\u00d7 [0,T\/2) with a(0) = n\u221aC(C\u22121), with C = (1+\u03b4 )\/(1\u2212\u03b4 ). Note thata(t) is independent of k. Letting k\u2192 \u221e gives(1\u2212a(t))(1\u2212\u03b4 )2(1+\u03b4 )hk0,0 \u2264 g(t)\u2264 (1+a(t))(1+\u03b4 )hk0,0 (2.3.6)in B(12)\u00d7 (0,T\/2). We then getlimsupt\u21920|g(t)\u2212g0|g\u02c6\u2264 limsupt\u21920(|g(t)\u2212hk0,0|g\u02c6+ |hk0,0\u2212g0|g\u02c6)\u2264[\u2223\u2223\u2223\u22231\u2212 (1\u2212a(0))(1\u2212\u03b4 )2(1+\u03b4 )\u2223\u2223\u2223\u2223+ |(1+a(0))(1+\u03b4 )\u22121|] |hk0,0|g\u0302+\u03b4 |hk0,0|g\u030220uniformly on B(12). Then letting \u03b4 \u2192 0 above, and using the fact that a(0)\u2192 0 as\u03b4 \u2192 0, and (2.3.2) and (2.3.3) we conclude thatlimsupt\u21920|g(t)\u2212g0|g\u02c6 = 0.uniformly on B(12). Hence g(t) converge to g0 uniformly on compact sets as t\u2192 0.We do not have any bound on the curvature of the solution g(t) in the previ-ous theorem. Also in the previous theorem, we assume that the Ka\u00a8hler-Ricci flow(1.0.1) has solution with initial condition hk,0 on a fixed time interval independentof k. We want to remove this assumption and obtain curvature bound for the solu-tions. In order to do this, we assume hk,0 also has an uniform upper bound.Theorem 2.3.2. Let g0 be a complete continuous Hermitian metric on a noncom-pact complex manifold Mn. Suppose there exists a sequence {hk,0} of smoothcomplete Ka\u00a8hler metrics with bounded curvature on M converging uniformly oncompact subsets to g0 and another complete Ka\u00a8hler metric g\u0302 on M with boundedcurvature and holomorphic sectional curvature bounded from above by K \u2265 0 suchthat(i) C\u22121g\u02c6\u2264 hk,0 \u2264Cg\u02c6 for some C independent of k;(ii) hk has bounded curvature for every k.Let T = 1\/(2CnK) if K > 0, otherwise let T = \u221e. Then the Ka\u00a8hler-Ricci flow(1.0.1) has a smooth solution g(t) on M\u00d7 (0,T ) such that(a) (1\/(nC)\u22122Kt)g\u02c6\u2264 g(t)\u2264 B(t)g\u02c6 on M\u00d7 (0,T ) for some positive continuousfunction B(t) depending only on C, g\u02c6 and n.(b) g(t) has bounded curvature for t > 0. More precisely, for any 0 < T \u2032 < Tand for any l \u2265 0 there exists a constant Cl depending only on C, l, T \u2032, g\u02c6 andthe dimension n such thatsupM|\u2207lRm(g(t))|2g(t) \u2264Clt l+2,21(c) g(t) converges uniformly on compact subsets to g0 as t\u2192 0.Moreover, if g0 is smooth and {hk,0} converges smoothly and uniformly oncompact subsets of M, then g(t) extends to a smooth solution on M\u00d7 [0,T ) withg(0) = g0.Proof. For each k, let hk(t) be the solution to (1.0.1) with initial condition hk fromTheorem 2.1.2 which is defined on M\u00d7 [0,Tk) for some Tk > 0. We first claimthat there is T > 0 such that Tk \u2265 T for all k. By Lemma 2.2.1, there is a positivecontinuous function B(t) : [0,T )\u2192 R independent of k such that(1\/n\u22122CKt)g\u02c6\u2264 hk(t)\u2264 B(t)g\u02c6in M\u00d7 [0,min{Tk,T}) where T = 1\/(2nCK). As before, we may assume that g\u02c6has bounded geometry of infinite order. By Theorem A.1.1, we conclude that ifTk < T , then |Rm(hk(t))|hk(t) are bounded in M\u00d7 [0,Tk). By Theorem 2.1.2, wesee that one can extend hk(t) so that Tk \u2265 T for all k as claimed. Given upper andlower bounds on hk(t) as above, we may conclude from Theorem A.1.1, as in theproof of Theorem 2.3.1, that there is a smooth solution to the Ka\u00a8hler-Ricci flowg(t) on M\u00d7 (0,T ) satisfying condition (a) and (c) from which we conclude, byTheorem A.1.1 (i), that condition (b) is also satisfied.2.4 A uniqueness theorem and an existence time estimateIn this section we will discuss some results on existence time and uniqueness ofthe complete bounded curvature solutions and uniqueness of solutions in general.We first prove a uniqueness theorem which could be applied to solutions that maynot have bounded curvature:Theorem 2.4.1. Let (Mn, g\u0302) be a complete non-compact Ka\u00a8hler manifold. Supposethere is an exhaustion function \u03b6 > 0 on (Mn, g\u0302) with limx\u2192\u221e \u03b6 (x) = \u221e such that|\u2202 \u2202\u00af\u03b6 |g\u0302 and |\u2207\u0302\u03b6 |g\u0302 are bounded.Let g1(x, t) and g2(x, t) be two solutions of the Ka\u00a8hler-Ricci flow (1.0.1) onM\u00d7 [0,T ] with the same initial data g0(x) = g1(x,0) = g2(x,0). Suppose there is22a positive function \u03c3 with limx\u2192\u221e log\u03c3(x)\/ log\u03b6 (x) = 0 such that the followingconditions hold for all (x, t) \u2208M\u00d7 [0,T ]:(i)g\u0302(x)\u2264 \u03b6 (x)g1(x, t); g\u0302(x)\u2264 \u03b6 (x)g2(x, t), (2.4.1)(ii)\u2212\u03c3(x)\u2264 det((g1)i j\u00af(x, t))det((g2)i j\u00af(x, t))\u2264 \u03c3(x).Then g1 \u2261 g2 on M\u00d7 [0,T ]. In particular, if g1 and g2 are uniformly equivalent tog\u0302 on M\u00d7 [0,T ], then g1 \u2261 g2.Proof. By adding a positive constant to \u03b6 we may assume that \u03b7 := log\u03b6 > 1.Then\u03b7i j\u00af =\u03b6i j\u00af\u03b6\u2212 \u03b6i\u03b6 j\u00af\u03b6 2.Since |\u2202 \u2202\u00af\u03b7 |g\u0302 and |\u2207\u0302\u03b7 |g\u0302 are uniformly bounded, there is c1 > 0 such that|\u2202 \u2202\u00af\u03b7 |g\u0302 \u2264 c1\u03b6on M. Let h(s, t)(x) = sg1(x, t) + (1\u2212 s)g2(x, t), 0 \u2264 s \u2264 1. By (i), we haveg\u0302(x) \u2264 \u03b6 (x)h(s, t)(x) for all (x, t) \u2208 M\u00d7 [0,T ] and for all s. Let (x, t) be fixedand diagonalize \u2202 \u2202\u00af\u03b7 with respect to g\u0302 at x. Then |\u03b7ii\u00af| \u2264 c1\u03b6 . On the other hand,\u2206h(s,t)\u03b7 = (h(s, t))i j\u00af\u03b7i j\u00af = (h(s, t))ii\u00af\u03b7ii\u00af \u2264 n\u03b6 \u00b7c1\u03b6= nc1. (2.4.2)Letw(x, t) =\u222b t0(logdet((g1)i j\u00af(x,s))det((g2)i j\u00af(x,s)))ds.Thenwi j\u00af(x, t) =\u222b t0((R1)i j\u00af(x,s)\u2212 (R2)i j\u00af(x,s))ds =\u2212(g1)i j\u00af(x, t)+(g2)i j\u00af(x, t)where (Rk)i j\u00af is the Ricci tensor of gk, k = 1,2. Here we have used the Ka\u00a8hler-Ricciflow and the fact that g1 = g2 at t = 0. Hence in order to prove the proposition, it23is sufficient to prove that w\u2261 0. Now\u2202\u2202 tw(x, t) =\u222b 10\u2202\u2202 slogdet(hi j\u00af(s, t)(x)))ds=\u222b 10\u2206h(t,s)w(x, t)ds.(2.4.3)Let W (x, t) = eAt\u03b7 where A = nc1+1. By (2.4.2),\u2202\u2202 tW (x, t)\u2212\u222b 10\u2206h(t,s)W (x, t)ds\u2265eAt(A\u03b7\u2212nc1)\u2265eAt\u03b7where we have used the fact that \u03b7 > 1. For any \u03b5 > 0,\u2202\u2202 t(\u03b5W \u2212w)(x, t)\u2212\u222b 10\u2206h(t,s) (\u03b5W \u2212w)(x, t)ds\u2265eAt\u03b7By (ii), limx\u2192\u221e(\u03b5W \u2212w)(x, t) = \u221e uniformly in t. By the maximum principle,we conclude that w \u2264 \u03b5W . Letting \u03b5 \u2192 0 gives w \u2264 0. Similarly, one can provethat \u2212w \u2264 0 and hence w \u2261 0 on M\u00d7 [0,T ]. This completes the proof of theproposition.As a corollary, we haveCorollary 2.4.1. Let (Mn, g\u02c6) be a complete non-compact Ka\u00a8hler manifold withbounded curvature. Let g1(x, t) and g2(x, t) be two solutions of the Ka\u00a8hler-Ricciflow (1.0.1) on M\u00d7 [0,T ] with the same initial data g0(x) = g1(x,0) = g2(x,0).Suppose there is a constant C such that:C\u22121g\u02c6\u2264 g1(t),g2(t)\u2264Cg\u02c6on M\u00d7 [0,T ]. Then g1 \u2261 g2 on M\u00d7 [0,T ].By using Theorem 2.1.4, we can compare the maximal existence time of theKa\u00a8hler-Ricci flow solutons by comparing the C0 data of the initial metric. Forconvenience, if g is a complete Ka\u00a8hler metric with bounded curvature, we let Tg bethe maximal existence time of Shi\u2019s solution.24Theorem 2.4.2. Let M be a non-compact complex manifold, g0 and h0 be completeKa\u00a8hler metrics with bounded curvature. If g0 and h0 are equivalent and g0 \u2265 ah0then Tg0 \u2265 aTh0 ,Proof. First observe that if \u03bb > 0 is a constant, then T\u03bbh0 = \u03bbTh0 . Hence withoutloss of generality, we may assume that a = 1. Also, we assume without loss ofgenerality that g0,h0 have bounded geometry of order infinity. For if not, we letg(t),h(t) are the corresponding solutions as in Theorem 2.1.2, then we first provethe Theorem for g(\u03b5),h(\u03b5) for arbitrary small \u03b5 and then let \u03b5\u2192 0 and use the factthat g(\u03b5),h(\u03b5) converge uniformly to g0,h0 (respectively) on M, and Tg(\u03b5) = Tg\u2212\u03b5and Th(\u03b5) = Th\u2212\u03b5 by Theorem 2.4.1. Note that by Theorem A.1.1, all the covariantderivatives of g(\u03b5) with respect to h0 are bounded. So we may assume in additionthat all the covariant derivatives of g0 with respect to h0 are bounded.Now for any 0 < T < Th0 , we haveg0\u2212T Ric(g0) = h0\u2212T Ric(h0)+T (Ric(g0)\u2212Ric(h0))+(g0\u2212h0).By Theorem 2.1.4, there is a smooth bounded function f with bounded covariantderivatives with respect to h0 such thath0\u2212T Ric(h0)+\u221a\u22121\u2202 \u2202\u00af f \u2265C1h0for some C1 > 0. Then letting F = log\u03c9n0\u03b7n where \u03c90 and \u03b7 are the Ka\u00a8hler forms ofg0 and h0 respectively, givesg0\u2212T Ric(g0)+\u221a\u22121\u2202 \u2202\u00af ( f +T F)\u2265C1h0+(g0\u2212h0)\u2265C2g0for some C2 > 0 because g0 \u2265 h0 and g0 is uniformly equivalent to h0. Fromthe facts that g0,h0 are equivalent and that all the covariant derivatives of g0 withrespect to h0 are bounded, we may conclude that all the covariant derivatives of fare bounded with respect to g0 as well. We may also conclude from these facts thatF and |\u2207l\u2202 \u2202\u00afF |g0 are uniformly bounded for 0 \u2264 l \u2264 2 where we have used that\u221a\u22121\u2202 \u2202\u00afF = \u2212Ric(g0)+Ric(h0). By Theorem 2.1.4, we conclude that T \u2264 Tg0 .25From this the result follows.By the theorem, we have the following monotonicity and continuity of Tg.NamelyCorollary 2.4.2. Let Mn be a non-compact complex manifold.(i) Let g0 \u2265 h0 be complete uniformly equivalent Ka\u00a8hler metrics on M withbounded curvature. Then Tg0 \u2265 Th0 . In particular, if Th0 = \u221e, then Tg0 = \u221e.(ii) LetK be the set of complete Ka\u00a8hler metrics on M with bounded curvature.Then Tg is continuous on K with respect to the C0 norm in the followingsense: Let g0 \u2208K . Then for h0 \u2208K , Th0 \u2192 Tg0 as ||h0\u2212g0||g0 \u2192 0.As a corollary, instead of the C0 data, we can also estimate the existence by theupper bound of the Ricci curvature of another equivalent Ka\u00a8hler metric.Corollary 2.4.3. Let (M,h) be a complete Ka\u00a8hler manifold having bounded cur-vature with Ricci curvature bounded above by K. If g is equivalent to h and g\u2265 h.then Tg \u2265 1K .Proof. Since Ric(h)\u2264Kh, so h\u2212 tRic(h) = (1\u2212Kt)h. By Theorem 2.1.4, Th \u2265 1K .By Theorem 2.4.2, Tg \u2265 1K .Remark 2.4.1. As an example of application, let (M,h) be a complete Ka\u00a8hler man-ifold with non-positive bisectional curvature (e.g. an Euclidean space or a Poincare\u00b4disk), assuming that g is equivalent to h, since K = 0 we have Tg = \u221e.26Chapter 3U(n)-invariant Ka\u00a8hler metricsIn this chapter we will study the Ka\u00a8hler-Ricci flow starting from U(n)-invariantKa\u00a8hler metrics, the main theorem isTheorem 3.0.1. Let g0 be a complete U(n)-invariant Ka\u00a8hler metric on Cn withnon-negative bisectional curvature. Then(i) the Ka\u00a8hler-Ricci flow (1.0.1) has a unique smooth longtime U(n)-invariantsolution g(t) which is equivalent to g0 and has bounded non-negative bisec-tional curvature for all t > 0;(ii) g(t) converges, after rescaling at the origin, to the standard Euclidean metricon Cn.Here we say that the solution g(t) converges to g\u02dc after rescaling at a point p iffor some V \u2208 TpM, the metrics 1|V |2t g(t) converges to g\u02dc smoothly and uniformly oncompact subsets on M where |V |2t = gt(V,V ).The content is organized as follows: We first collect some results about U(n)-invariant Ka\u00a8hler metrics. These results correlate U(n)-invariant metrics with non-negative holomorphic bisectional curvature with non-decreasing real valued func-tions. Then we will construct a short-time U(n)-invariant Ka\u00a8hler-Ricci flow solu-tion starting from any non-negatively curved U(n)-invariant metric. After that, wewill show that the short-time solution can be extended to a long-time solution and27the solution is unique in a certain class. Finally we will show that the long-timesolution converges to the standard Euclidean metric after rescaling.3.1 Background materialsLet g be a U(n)-invariant Ka\u00a8hler metric. Then there is a smooth p \u2208 C\u221e[0,\u221e)such that the Ka\u00a8hler form of the metric satisfies \u03c9 = i\u2202 \u2202\u00af p(r) where r = |z|2. Inthe standard coordinate on Cn, gi j\u00af = f (r)\u03b4i j + f \u2032(r)z\u00afiz j where f (r) = p\u2032(r). Leth = (r f )\u2032, then at the point (z1,0, \u00b7 \u00b7 \u00b7 ,0),g11\u00af = h,gii\u00af = f and gi j\u00af = 0 for all i 6= j.Due to U(n)-invariance, this basically describes the metric on the whole of Cn.Using this observation, Wu-Zheng [33] parametrized the U(n)-invariant Ka\u00a8hlermetrics by a constant C and a smooth function \u03be : [0,\u221e)\u2192 R as follows:Theorem 3.1.1. (a) ([WU-ZHENG] [33]) Every smooth U(n) invariant Ka\u00a8hlermetric g is generated by a function \u03be : [0,\u221e)\u2192 R with \u03be (0) = 0 such that ifh\u03be (r) :=Ce\u222b r0 \u2212 \u03be (s)s ds; f\u03be (r) :=1r\u222b r0h\u03be (s)dswhere h\u03be (0) =C > 0 and f\u03be (0) = h\u03be (0), where r = |z|2, thengi j\u00af = f\u03be (r)\u03b4i j + f\u2032\u03be (r)ziz j.where gi j\u00af are the components of g in the standard coordinates z= (z1, . . . ,zn)on Cn. Moreover g is complete if and only if\u222b \u221e0\u221ah\u03be (s)\u221asds = \u221e.(b) ([WU-ZHENG] [33]) Let h = h\u03be , f = f\u03be . At the point z = (z1,0, . . . ,0),relative to the orthonormal frame e1 = 1\u221ah\u2202z1 ,ei =1\u221af \u2202zi , i\u2265 2, with respectto g, the curvature tensor has componentsA = R11\u00af11\u00af =\u03be \u2032h,28B = R11\u00afii\u00af =1(r f (r))2\u222b r0\u03be \u2032(s)(\u222b t0h(s)ds)dt,C = Rii\u00afii\u00af = 2Rii\u00af j j\u00af =2(r f (r))2\u222b r0h(s)\u03be (s)dt,where 2 \u2264 i 6= j \u2264 n and these are the only non-zero components of thecurvature tensor at z except those obtained from A,B or C by the symmetricproperties of the curvature tensor.(c) ([WU-ZHENG] [33], YANG [34]) g has positive (nonnegative) bisectionalcurvature if and only if \u03be \u2032 > 0 (\u03be \u2032 \u2265 0). In particular, if g has nonnegativebisectional curvature and is complete, then \u03be \u2264 1.Remark 3.1.1. If g1 and g2 are two smooth U(n) invariant Ka\u00a8hler metrics on Cngenerated by \u03be1,\u03be2 respectively, and if the corresponding functions h\u03be1 , h\u03be2 satisfyh\u03be1 \u2265 h\u03be2 , then g1 \u2265 g2. Conversely, if g1 \u2265 g2, then h\u03be1 \u2265 h\u03be2 . This can be seen bycomparing the metrics at the points (a,0, . . . ,0).Using Theorem 3.1.1 (a) and (b), we can find a condition for which the curva-ture of g is uniformly bounded.Lemma 3.1.1. Let g be a complete U(n) invariant Ka\u00a8hler metric on Cn generatedby \u03be . If\u2223\u2223\u2223 \u03be \u2032h \u2223\u2223\u2223 is uniformly bounded, then the curvature of g is uniformly bounded.Proof. It is sufficient to prove that the holomorphic bisectional curvature is uni-formly bounded under the assumption that\u2223\u2223\u2223 \u03be \u2032h \u2223\u2223\u2223 is uniformly bounded by c, say.By Theorem 3.1.1, in the notations of the theorem it is sufficient to prove that|A|, |B|, |C| are uniformly bounded. It is obviously |A| \u2264 c. Now|B| \u2264 1r2 f 2\u222b r0ch(t)dt(\u222b t0h(s)ds)dt\u2264 cr2 f 2(\u222b r0h(t)dt)2=cbecause h > 0 and r f (r) =\u222b r0 h(t)dt. Similarly, since|\u03be (r)| \u2264\u222b r0|\u03be \u2032(t)|dt \u2264 c\u222b r0h(t)dt,29we have|C| \u2264 2c.Theorem 3.1.1 (c) showed that any complete U(n)-invariant Ka\u00a8hler metric withnon-negative bisectional curvature corresponds to a non-decreasing \u03be with \u03be (\u221e) =limr\u2192\u221e \u03be (r) \u2264 1. In fact, the volume growth rate can also be described by \u03be .Chen-Zhu [13] proved that if (M,g) is complete non-compact with non-negativebisectional curvature, then for all x0 \u2208M, there exists c > 0 such that1csn \u2264V (s)\u2264 cs2n, \u2200s > 1,where V (s) be the volume of Bg(x0,s).This means that complete non-negatively curved Ka\u00a8hler manifolds have vol-ume growth rate between half-Euclidean and Euclidean. In case the volume growthrate is half-Euclidean (i.e. V (s) is asymptotic to sn), the space is called a cigar; ifthe volume growth rate is Euclidean (i.e. V (s) is asymptotic to s2n), the space iscalled a conoid. In the U(n)-invariant case, conoids and cigars satisfy the followingconditions:Theorem 3.1.2 (Wu-Zheng, [33]). The metric \u03c9 is a conoid if the corresponding\u03be satisfies \u03be (\u221e)< 1. It is a cigar if\u222b \u221e11\u2212\u03ber dr < \u221e.Using the construction of Cabezas-Rivas and Wilking [2], Yang-Zheng [35]proved the short time existence of the Ka\u00a8hler-Ricci flow for complete non-collapsedU(n)-invariant metric with non-negative sectional curvature. Their solution isU(n)-invariant when some technical assumptions on initial data is satisfied, andwhen the solution is U(n) invariant, they proved the following theorem which wewill use later:Theorem 3.1.3 (Yang-Zheng, [35]). Let g(t), t \u2208 [0,T ] be a complete solution ofthe Ka\u00a8hler-Ricci flow on Cn with U(n)-symmetry. If g(0) has non-negative holo-morphic bisectional curvature, so does g(t) for all t \u2208 [0,T ].In the following content, by rescaling the initial metric if necessary, we willassume that the constant C of g0 in Theorem 3.1.1(a) is equal to 1.303.2 Short time existence of U(n)-invariant Ka\u00a8hler-RicciflowWe will use Theorem 2.3.2 to prove short time existence. We first prove a Proposi-tion:Proposition 3.2.1. Assume that g0 is a complete U(n)-invariant metric with non-negative bisectional curvature \u03be , there exists a complete U(n) invariant metric g\u02c6with bounded curvature andc\u22121g\u02c6\u2264 g0 \u2264 cg\u02c6 (3.2.1)on Cn for some constant c > 0.Proof. We will prove the Theorem by constructing a \u03be\u02c6 satisfying certain proper-ties, the constant C of g\u0302 in Theorem 3.1.1(a) is taken to be 1. Let \u03be and \u03be\u02c6 corre-spond to g0 and g\u02c6 respectively, if \u03be \u2261 0, then g0 is Euclidean and so the statementis true by taking g\u0302 = g0.Suppose now \u03be 6= 0, we consider two cases:Case 1: If\u222b r1\u03be\u22121t dt \u2265 \u2212C for some constant C. Let \u03be\u02c6 be a smooth functionon [0,\u221e) such that \u03be\u02c6 (r) = 1 for all sufficiently large r, it is clear that it generates acomplete bounded curvature g\u0302. By the definition of h, for all r \u2265 0, we haveh\u02c6(r)h(r)= exp(\u222b r0\u03be (t)\u2212 \u03be\u02c6 (t)tdt).Because\u222b r1\u03be\u22121t dt \u2265 \u2212C, there exists C\u2032 such that\u222b r0\u03be (t)\u2212\u03be\u02c6 (t)t dt \u2265 \u2212C\u2032 and there-fore h(r) \u2264 C2h\u02c6(r) for some constant C2. On the other hand, because \u03be \u2264 1 and\u03be\u02c6 (r) = 1 for all sufficiently large r, therefore, h(r) \u2265 C1h\u02c6(r) for some C1. SinceC1h\u02c6(r) \u2264 h(r) \u2264 C2h\u02c6(r) for all r, by Remark 3.1.1, we have C1g\u02c6 \u2264 g0 \u2264 C2g\u02c6 asclaimed.Case 2: If\u222b r1\u03be\u22121t dt is not bounded from below. By the assumption \u03be 6= 0 and\u03be is non-decreasing,\u222b r1\u03bet dt is not bounded from above. We want to construct \u03be\u02c6 byoscillating between 0 and 1 at a suitable rate so that the corresponding metric g\u0302 iscomplete with bounded curvature and is equivalent to g. More precisely, we will31find \u03be\u02c6 and 1\u2264 a0 < a1 < a2 \u00b7 \u00b7 \u00b7 \u2192 \u221e such that \u03be\u02c6 generates a complete U(n) metricg\u0302 such that \u222b a2(i+1)a2i\u03be \u2212 \u03be\u02c6tdt = 0 (3.2.2)for all i\u2265 0; \u2223\u2223\u2223\u2223\u2223\u222b ra2i\u03be \u2212 \u03be\u02c6tdt\u2223\u2223\u2223\u2223\u2223\u2264 c1 (3.2.3)for some c1 for all i\u2265 0 and for all r \u2208 [a2i,a2(i+1)); and\u2223\u2223\u2223\u2223\u2223 \u03be\u02c6 \u2032(r)h\u02c6(r)\u2223\u2223\u2223\u2223\u2223\u2264 c2 (3.2.4)for some c2 for all r \u2265 0. Then by Lemma 3.1.1, Theorem 3.1.1 (a), we can con-clude that g\u0302 satisfies the conditions of the Proposition.Fix a smooth function \u03c1 on R, such that\u03c1(t) ={1, if t \u2264 1+ \u03b5;0, if t \u2265 3\u2212 \u03b5 ,and \u03c1 \u2032 \u2264 0, where \u03b5 > 0 is small enough so that 1+ \u03b5 < 3\u2212 \u03b5 . Then 0\u2264 \u03c1 \u2264 1.Let \u03be\u02c6 be a smooth function on [0,1] with \u03be\u02c6 (0) = 0 and \u03be\u02c6 (r) = 1 near r = 1such that 0\u2264 \u03be\u02c6 \u2264 1. We are going to find ai and \u03be\u02c6 (r) on [ai,ai+1] inductively. Leta0 = 1. \u222b 3a0a0\u03be \u2212\u03c1( ta0 )tdt \u2264\u222b 3a0a01\u2212\u03c1( ta0 )tdt \u2264 log3.Since\u222b r3a0\u03bet dt is not bounded from above, there is a first a1 > 3a0 such that\u222b 3a0a0\u03be \u2212\u03c1tdt+\u222b a13a0\u03betdt = c3where c3 = log3+1. On the other hand,\u222b 3a1a1\u03be \u2212 (1\u2212\u03c1( ta1 ))tdt \u2265\u2212 log3.32Since\u222b r3a1\u03be\u22121t dt is not bounded from below, there exists a first a2 > 3a1, such that\u222b 3a1a1\u03be \u2212 (1\u2212\u03c1( ta1 ))tdt+\u222b a23a1\u03be \u22121tdt =\u2212c3Define\u03be\u02c6 (r) =\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3\u03c1( ra0 ), if a0 \u2264 r \u2264 3a0;0, if 3a0 < r \u2264 a1;1\u2212\u03c1( ra1 ), if a1 < r \u2264 3a1;1, if 3a1 < r \u2264 a2.It is easy to see that \u03be\u02c6 is smooth on [0,a2] with \u03be\u02c6 (r) = 1 near a2. Moreover,0\u2264 \u03be\u02c6 \u2264 1 on [1,a2], and \u222b a2a0\u03be \u2212 \u03be\u02c6tdt = 0.so (3.2.2) is true for i = 0. It is easy to see that|\u03be \u2032| \u2264 c4rwhere c4 = 3max |\u03c1 \u2032|.For a0 \u2264 r \u2264 a1, by the definition of a1 we have\u222b ra0\u03be \u2212 \u03be\u02c6tdt \u2264 c3.For a1 < r \u2264 a2,\u222b ra0\u03be \u2212 \u03be\u02c6tdt =(\u222b a1a0+\u222b ra1)\u03be \u2212 \u03be\u02c6tdt\u2264c3+\u222b ra11\u2212 \u03be\u02c6tdt\u2264c3+ log3.Hence for a0 \u2264 r \u2264 a2, \u222b ra0\u03be \u2212 \u03be\u02c6tdt \u2264 2c3.33Similarly, one can prove that\u222b ra0\u03be \u2212 \u03be\u02c6tdt \u2265\u22122c3.To summarize, we have find \u03be\u02c6 (r) and a0 < a1 < a2 such that \u03be\u02c6 is smooth anddefined on [0,a2] with \u2264 \u03be\u02c6 \u2264 1 on [a0,a2], satisfying (3.2.2) with i = 0, (3.2.3)with i = 0, c1 = 2c3, and |\u03be\u02c6 \u2032| \u2264 c4r on [a0,a2]. Moreover, \u03be\u02c6 (r) = 1 near r = a2.From the above construction, it is easy to see that one can continue and finda2 < a3 < a4 \u00b7 \u00b7 \u00b7 \u2192 \u221e and \u03be\u02c6 with 0 \u2264 \u03be\u02c6 (r) \u2264 1 for r \u2265 a0, satisfying (3.2.2) and(3.2.3) with c1 = 2c3, and |\u03be\u02c6 \u2032| \u2264 c4r on [a0,\u221e).Since \u03be\u02c6 \u2264 1,h\u02c6(r)\u2265 c5 exp(\u2212\u222b r11tdt)\u2265 c5rfor some c5 > 0 for all r \u2265 1. Combing with the fact that |\u03be\u02c6 \u2032| \u2264 c4r on [a0,\u221e), weconclude that (3.2.4) is also true. This completes the proof.We also need the following Lemma:Lemma 3.2.1. Let g0 be a complete U(n)-invariant metric with non-negative bi-sectional curvature and let g\u02c6 be a complete bounded curvature U(n)-invariant met-ric equivalent to g0. Then we can find a sequence of complete U(n)-invariant met-ric hk such that1. hk converges to g0 in C\u221eloc-sense;2. there exists a uniform constant C such that 1C g\u0302\u2264 hk \u2264Cg\u0302 and3. hk has bounded curvature for all k,Proof. Choose \u03b4k > 0 and smooth functions \u03b7k : (\u2212\u221e,\u221e)\u2192 R satisfying\u03b7k(r) :\uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3= 1 if \u2212\u221e< r \u2264 k0 < \u03b7k(r)< 1 if k < r < k+\u03b4k= 0 if k+\u03b4k \u2264 r < \u221e.(3.2.5)34and \u222b k+\u03b4kk\u2223\u2223\u2223\u2223\u2223(\u03be \u2212 \u03be\u02c6 )t\u2223\u2223\u2223\u2223\u2223dt \u2264 1 (3.2.6)for all k. Let {\u03bek} : [0,\u221e)\u2192 \u221e be defined by\u03bek(r) = \u03b7k\u03be +(1\u2212\u03b7k)\u03be\u02c6 .Then each \u03bek generates a U(n) invariant Ka\u00a8hler metric hk and for all r \u2265 0,\u222b r0\u03bek(t)\u2212 \u03be\u02c6 (t)tdt =\u222b r0\u03b7k(\u03be \u2212 \u03be\u02c6 )tdt={ \u222b r0\u03be\u2212\u03be\u02c6t dt, if r \u2264 k;\u222b k0\u03be\u2212\u03be\u02c6t dt+\u03b1k, if r > kwhere|\u03b1k| \u2264\u222b k+\u03b4kk\u2223\u2223\u2223\u2223\u2223\u03be \u2212 \u03be\u02c6t\u2223\u2223\u2223\u2223\u2223dt \u2264 1.By Remark 3.1.1, since g and g\u0302 are equivalent, there exists constant C such that|\u222b r0 \u03be\u2212\u03be\u02c6t dt| \u2264C. Combining the above inequalities and use Remark 3.1.1 again, wehavec\u221212 g\u02c6\u2264 hk \u2264 c2g\u02c6 (3.2.7)for some constant c2 > 0, for all k. Since \u03bek = 1(r) for all sufficiently large r, hk iscomplete with bounded curvature. And because \u03bek = \u03be (and hence hk = g0) on theset {r < k}, we have hk converges to g0 in C\u221eloc-sense.Now we are ready to prove a short time existence theorem for U(n)-invariantKa\u00a8hler-Ricci flow:Theorem 3.2.1. Let g0 be a complete U(n)-invariant Ka\u00a8hler metric on Cn withnon-negative bisectional curvature. Then for some T > 0 the Ka\u00a8hler-Ricci flow(1.0.1) has a complete smooth U(n)-invariant solution g(t) on Cn\u00d7 [0,T ) withg(0) = g0. Moreover, for every l \u2265 0 there exists a constant cl depending only on35such thatsupp\u2208Cn\u2016\u2207lRm(p, t)\u20162t \u2264clt l+2(3.2.8)on Cn\u00d7 (0,T ).Proof. Let g\u02c6 be the U(n) invariant Ka\u00a8hler metric with bounded curvature generatedby \u03be\u02c6 defined in Proposition 3.2.1, so thatc\u221211 g\u02c6\u2264 g0 \u2264 c1g\u02c6 (3.2.9)for some c1 > 0 as in Proposition 3.2.1. Also let hk be the U(n) invariant Ka\u00a8hlermetric with bounded curvature generated by \u03bek defined in Lemma 3.2.1. In partic-ular, hk is complete with bounded curvature, converges in C\u221eloc to g0 andc\u221212 g\u02c6\u2264 hk \u2264 c2g\u02c6 (3.2.10)for some constant c2 > 0, for all k. Now recall that the curvature of g\u02c6 is boundedby a constant K as in Proposition 3.2.1, and thus by Theorem 2.1.2 we may assumewithout loss of generality that g\u02c6 has bounded geometry of order infinity. By Theo-rem 2.3.2, there is a solution g(t) of the Ka\u00a8hler-Ricci flow with initial condition g0on M\u00d7 [0,T ) for some T > 0 so that||Rm(g(t))||2g(t) \u2264c3tfor some c3 > 0 and for all 0 < t < T . The estimates for ||\u2207lRm|| for each l \u2265 0then follows from the general results of [26].Recall that in Theorem 2.3.2, g(t) is obtained as the limit of hk(t), since hk(t)is U(n)-invariant, so g(t) is also U(n)-invariant.Combining Theorem 3.2.1 with Theorem 3.1.3, we haveCorollary 3.2.1. Let g0 be a complete U(n)-invariant metric with non-negative bi-sectional curvature, there exists a complete U(n)-invariant metric g1 with boundednon-negative bisectional curvature such that g1 and g0 are equivalent.Proof. As shown in the proof of Theorem 3.2.1, there is a background metric g\u0302which is complete with bounded curvature and a short time U(n)-invariant solution36g(t) on [0,T ] such that 1c2 g\u0302 \u2264 g(t) \u2264 c2g\u0302 for all t \u2208 [0,T ]. It suffices to take g1 tobe g(\u03b5) where 0 < \u03b5 \u2264 T , the curvature bound follows from Theorem 3.2.1 and thenon-negativity of bisectional curvature is a consequence of Theorem 3.1.3.3.3 Long time solutionThe main theorem of this section is :Theorem 3.3.1. Let g0 be a complete U(n)-invariant Ka\u00a8hler metric on Cn withnon-negative holomorphic bisectional curvature. Then the Ka\u00a8hler-Ricci flow (1.0.1)has a unique long time U(n)-invariant solution g(t) which is equivalent to g0 andhas bounded non-negative bisectional curvature for all t > 0.Recall that g(t) and g0 are equivalent if for all T < \u221e, there exists constant Csuch that 1C g0 \u2264 g(t)\u2264Cg0 for all t \u2208 [0,T ]. The uniqueness is an immediate con-sequence. In fact, suppose g1(t) and g2(t) satisfy Theorem 3.3.1, by Proposition3.2.1, there is a complete bounded curvature g\u0302 equivalent to g0, hence g\u0302 is equiv-alent to both g1(t) and g2(t). The uniqueness then follows from Corollary 2.4.1immediately.In Theorem 3.2.1, we already obtained a short-time solution g(t) satisfying theconditions and by the uniqueness argument above, we only need to extend it to along time solution which is equivalent to g0. And replacing g0 by g(\u03b5) if necessary,we can assume g0 has bounded curvature.We will prove the theorem by considering two cases:Case 1: When g0 is a cigar (i.e. g0 has half-Euclidean volume growth).Proof. Let h be Cao\u2019s cigar constructed in [6]. Recall that it is a complete U(n)-invariant metric which has positive bisectional curvature everywhere and the scalarcurvature decays like 1\u03c1 , where \u03c1 is the distance to the origin with respect to h.It is a soliton so that there is a Ricci flow solution h(t) on [0,\u221e) such that h(t) isisometric to h(0) for all t \u2208 [0,\u221e). Since h has positive bisectional curvature andits scalar curvature decays linearly, h(t) is a complete bounded curvature long timesolution to Ricci flow. Suppose \u03be and \u03beh corresponds to g0 and h respectively, by37Theorem 3.1.2 and the fact that \u03be , \u03beh are both non-decreasing, we have\u2223\u2223\u2223\u2223\u222b r0 \u03be \u2212\u03beht dt\u2223\u2223\u2223\u2223\u2264C0+\u222b \u221e1 1\u2212\u03bet dt+\u222b \u221e11\u2212\u03behtdt \u2264C1where C0, C1 are constants independent of r. In particular, g0 and h are equivalent,so by Corollary 2.4.2 (i), g0 admits a long-time complete bounded curvature solu-tion g(t). And since g(t) has bounded curvature, it must be equivalent to g0. Tosee that g(t) is U(n)-invariant, we observe that g(t) coincides with the short timeU(n)-invariant solution obtained in Theorem 3.2.1 due to the uniqueness theorem.Let T be the supremum of time such that g(t) remains U(n)-invariant on [0,T ) andsuppose that T < \u221e. Since g(t) is a long time bounded curvature solution, g(t)converges in C\u221eloc to g(T ) as t \u2192 T\u2212, this forces g(T ) to be U(n)-invariant. ByTheorem 3.1.3, g(t) has non-negative bisectional curvature for all t \u2208 [0,T ]. Byapplying the short time existence theorem and uniqueness theorem to g(T ), weconclude that the U(n) symmetry is also preserved beyond time T , a contradiction.Case 2: When g0 is not a cigar.In this case, either \u03be (\u221e) = \u03b2 < 1 or \u03be (\u221e) = 1 with\u222b \u221e11\u2212\u03bet dt = \u221e. We firstconstruct background metrics for both cases:Proposition 3.3.1. Let g be a smooth U(n)-invariant metric with bounded cur-vature generated by \u03be and g is not a cigar. Then given any \u03b5 > 0 there exists g\u02dcsatisfying(a) The curvature of g\u02dc is bounded by a constant independent of \u03b5 .(b) (1\/\u03b5)g\u02dc\u2264 g\u2264Cg\u02dc for some constant C.Proof. Suppose first that \u03be (\u221e) = \u03b2 < 1. For each k \u2265 1, consider the linear au-tomorphism of Cn given by \u03c6k(z) = z\/\u221ak and consider the U(n)-invariant Ka\u00a8hlermetric gk := \u03c6 \u2217k g on Cn. Consider the functions hk(r),\u03bek(r) and h(r),\u03be (r) etc cor-responding to gk and g. Then for each k \u2265 1 we have1. hk(r) = (1\/k)h(r\/k)2. \u03bek(r) = \u03be (r\/k)383. The curvature of gk is bounded by a constant independent of k because gk isisometric to g0.Nowhk(r)h(r)=1kh( rk )h(r)=1kexp(\u222b rrk\u03be (s)sds) (3.3.1)and thus by 0\u2264 \u03be \u2264 \u03b2 we havek\u22121 \u2264 hk(r)h(r)\u2264 k(\u03b2\u22121). (3.3.2)By Remark 3.1.1, for any k \u2265 1 we havek(1\u2212\u03b2 )gk \u2264 g\u2264 kgk. (3.3.3)Thus the Proposition follows in this case by the fact that \u03b2 < 1.Suppose now\u222b \u221e11\u2212\u03bet dt = \u221e. Let \u03b5 > 0 be given. Let g\u0302 be any U(n)-invariantnon-negative bisectional curvature metric with h\u0302(0) = 1 and generated by some\u03be\u0302 with \u03be\u0302 (r) = 1 for r \u2265 1. Let the curvature of g\u0302 be bounded by K\u0302. For eachk \u2265 1 define the pullbacks g\u0302k := \u03c6 \u2217k (g\u0302) as before. Let h\u0302k(r), \u03be\u0302k(r) and h\u0302(r) etccorresponding to g\u0302k and g\u0302. Then properties (1) and (2) (3) above still hold, butwith h,\u03be ,hk,\u03bek replaced with h\u0302, \u03be\u0302 , h\u0302k, \u03be\u0302k.STEP 1: First note that by (3.3.1) (applied to h\u02c6(r)) and the fact that \u03be\u02c6 \u2264 1, wesee that hk(r) is non-increasing in k for all r. Now fix \u03b5 > 0, by\u222b \u221e11\u2212\u03bet dt =\u221e thereis r0 > 0 such that if r \u2265 r0, then for k \u2265 1h(r) =h\u0302(r)h(r)h\u0302(r)\u2265h\u0302k(r)exp(\u222b r0\u03be\u0302 (s)\u2212\u03be (s)sds)\u22651\u03b5h\u0302k(r)where we have used the fact that \u03be\u0302 (r) = 1 for r \u2265 1. On the other hand, by (3.3.1)one can see this is true for r \u2264 r0 if k is large enough depending on r0. Hence by39Remark 3.1.1 we can find k > 1 depending on \u03b5 such that,1\u03b5g\u02c6k \u2264 gon Cn.STEP 2: Define\u03be\u02dck(r) := \u03be\u02c6k(r)+ok(r)where ok(r) : [0,\u221e)\u2192 R is a non-positive smooth function with |ok| \u2264 1k to bechosen. Let h\u02dck(0) = 1\/k and consider the corresponding metric g\u02dck.Claim 1: There exists a constant Rk > k and a smooth function ok(r) which is0 on [0,Rk] and satisfies:|o\u2032k(r)| \u22644krand|\u222b rRk\u03be\u02dck(s)\u2212\u03be (s)sds|=\u2223\u2223\u2223\u2223\u222b rRk 1+ok(s)\u2212\u03be (s)s ds\u2223\u2223\u2223\u2223\u2264 1+2log2 (3.3.4)for r \u2265 Rk.We may choose Rk > k such that 1\u2212 1\/k \u2264 \u03be (r) \u2264 1 on [Rk,\u221e). The con-struction of ok(r) follows from the construction in the proof of Proposition 3.2.1.We first choose a smooth non-increasing function \u03c1(r) : [0,\u221e)\u2192 R with \u03c1 = 0 on[0,1], \u03c1 = 1\/k on [2,\u221e) and 0\u2264 \u03c1 \u2032 \u2264 2\/k. LetI(r) :=\u222b r2Rk1+ok(r)\u2212\u03be (s)sds.(note that \u03be\u02dck(r) = 1+ ok(r) for r \u2265 Rk). For any positive sequence {ri} such thatr0 := Rk and 2ri < ri+1, define ok(r) := 0 if r \u2208 [0,r0), ok(r) := \u03c1(r\/r0) if r \u2208[r0,2r0), and40ok(r) :\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3= 1\/k if r \u2208 [2r0,r1]= 1\/k\u22122\u03c1(r\/r1) if r \u2208 [r1,2r1]=\u22121\/k if r \u2208 [2r1,r2]=\u22121\/k+2\u03c1(r\/r2) if r \u2208 [r2,2r2]= 1\/k if r \u2208 [2r2,r3]= 1\/k\u22122\u03c1(r\/r3) if r \u2208 [r3,2r3]...(3.3.5)Now for r \u2208 [2r0,\u221e) we have 1\u2212 1\/k \u2264 \u03be (r) \u2264 1, and as long as k \u2265 2 we have|1+ ok(r)\u2212 \u03be (r)| \u2264 1 as well. The definition of I(r) then gives the following forall i\u2265 0I(r) = I(2ri)+\u222b r2ri(1+(\u22121)i\/k)\u2212\u03be (s)sdsfor r \u2208 [2ri,ri+1),I(ri+1)\u2212 log2\u2264 I(r)\u2264 I(ri+1)+ log2for r \u2208 [ri+1,2ri+1).By the fact \u03be (r)\u2192 1, we may choose the ri\u2019s to be the smallest numbers withI(r1) = 1, I(r2) =\u22121, I(r3) = 1,..etc, and the estimates above give\u22121\u2212 log2\u2264 I(r)\u2264 1+ log2 (3.3.6)for all r \u2208 [2Rk,\u221e). The integral bound in the claim follows from (2.7) and the factthat |1+ok(r)\u2212\u03be (r)| \u2264 1 for r \u2208 [Rk,2Rk].Finally, we also have |o\u2032k(r)|= 2ri\u03c1 \u2032( rri )\u2264 4kri \u2264 4kr for all r \u2208 [ri,2ri].Claim 2: Let ok(r) be as in Claim 1. Then (1\/4e\u03b5)g\u02dck \u2264 g\u2264Ckg\u02dck for some Ckand the curvature of g\u02dck is bounded depending only on g\u0302.To prove the first part of the claim, when r \u2264 Rk we have g\u02dck(r) = gk(r), and so41we only have to consider when r \u2265 Rk. In this case, we have Ck \u2265 h(r)\/h\u02dck(r) =(h(Rk)\/h\u02dck(Rk))e\u222b rRk1+ok(s)\u2212\u03be (s)s ds \u2265 12e\u03b5 for some Ck where we have used Step 1 andClaim 1.To prove the second part of the claim, note that for r \u2265 Rk we have |\u03be\u02dc \u2032k(r)| =|o\u2032k(r)| \u2264 4\/kr andh\u02dck(r) = h\u02dck(1)exp(\u2212\u222b r1\u03be\u02dck(s)sds)= h\u02dck(1)exp(\u2212\u222b Rk1\u03be\u02dck(s)sds\u2212\u222b rRk\u03be\u02dck(s)sds)\u2265 h\u02dck(1) 1Rk exp(\u2212\u222b rRk(\u03be\u02dck(s)\u2212\u03be (s)s+\u03be (s)\u22121s+1s)ds)\u2265 h\u02dck(1) 1RkRk4e1+\u03b4 r= hk(1)14e1+\u03b4 r\u2265 h\u0302(1)k14e1+\u03b4 r(3.3.7)where in the third line we have used that 0\u2264 \u03be\u02dck \u2264 1 by definition, and in the fourthline we have used Claim 1 and\u222b \u221e11\u2212\u03bet dt = \u221e, so that\u222b rRk1\u2212\u03be (s)s ds\u2265\u2212\u03b4 .Thus |\u03be\u02dc \u2032k(r)\/h\u02dck(r)| \u2264 16e1+\u03b4\/h\u0302(1) for r \u2265 Rk. Since g\u02dck(r) = gk(r) for r \u2264 Rk,by the previous computations of \u03bek and hk we have |\u03be\u02dc \u2032k(r)\/h\u02dck(r)|= |\u03be \u2032k(r)\/hk(r)|=|\u03be\u02c6 \u2032( rk )\/h\u02c6( rk )|. As \u03be\u02c6 (r) = 1 if r \u2265 1, so |\u03be\u02dc \u2032k(r)\/h\u02dck(r)| is bounded unformly indepen-dent of k for r \u2264 Rk . As a result, we conclude that the curvature of g\u02dck is boundedby a constant independent of k by Lemma 3.1.1.Proof of Case 2. By Proposition 3.3.1, for all \u03b5 > 0, there is a complete U(n)-invariant metric g\u0302\u03b5 with curvature bounded by K with K being independent of \u03b5such that1\u03b5g\u0302\u03b5 \u2264 g0 \u2264 c(\u03b5)g\u0302\u03b5for some constant c(\u03b5) which may depend on \u03b5 . By Theorem 2.3.2, there exists42a Ka\u00a8hler-Ricci flow g(t) on [0,T\u03b5), where T\u03b5 = 12nK\u03b5 , such that g(t) is uniformlyequivalent to g0 for all t \u2208 [0,T \u2032) for any T \u2032 < T\u03b5 and has uniformly boundedcurvature on (\u03b4 ,T \u2032) for all 0 < \u03b4 < T \u2032 < T\u03b5 . Let \u03b5 \u2192 0 and use Theorem 2.4.1,one may conclude the theorem is true. Since g(t) is a bounded curvature solution,the U(n) symmetry and the non-negativity of bisectional curvature can be provedusing the same argument as in Case 1.3.4 Bounding the scalar curvatureStarting from g0 which is U(n)-invariant with non-negative bisectional curvature,we have proved the long time existence of Ka\u00a8hler-Ricci flow in the last section, butbefore we prove the convergence of the flow, we will prove that the scalar curvatureis uniformly bounded on compact subset along the flow, this is an important prop-erty that will be used when proving the convergence. First we need the followinglocalized version of Ni-Tam\u2019s estimate [24, Theorem 2.1].Lemma 3.4.1. Let (Mn,g(t)) be complete non-compact solution of the Ka\u00a8hler-Ricci flow (1.0.1) on M\u00d7 [0,T ) with bounded nonnegative bisectional curvature.LetF(x, t) = log(det(gi j\u00af(x, t))det(gi j\u00af(x,0)))and for any \u03c1 > 0, letm(\u03c1,x, t) = infy\u2208B0(x,\u03c1)F(y, t). Then there is c> 0 dependingonly on n such that for any x0 \u2208M and for all \u03c1, t > 0\u2212F(x0, t)\u2264c[(1+t(1\u2212m(\u03c1,x0, t))\u03c12)\u222b 2\u03c10sk(x0,s)ds\u2212 tm(\u03c1,x0, t))(1\u2212m(\u03c1,x0, t))\u03c12] (3.4.1)where B0(x,\u03c1) is the geodesic ball with respect to g0 = g(0), andk(x,s) :=\u222bB(x,s)R0(y)dV0is the average of the scalar curvature R0 of g0 over B0(x,s).43Proof. By [24, (2.6)], if G\u03c1 is the positive Green\u2019s function on B0(x0,\u03c1) withDirichlet boundary value, then\u222bB0(x0,\u03c1)G\u03c1(x0,y)(1\u2212 eF(y,t))dV0\u2264 t\u222bB0(x0,\u03c1)G\u03c1(x0,y)R0(y)dV0+\u222b t0\u222bB0(x0,\u03c1)G\u03c1(x0,y)\u22060(\u2212F(y,s))dV0ds=: I+ II.(3.4.2)As in [24, p.126],II \u2264\u2212tm(\u03c1,x0, t). (3.4.3)As in [24, (2.8)] there is a constant c1 depending only on n such that\u03c12\u222bB0(x0, 15\u03c1)(\u2212F(y, t))dV0\u2264 c1t(1\u2212m(\u03c1,x0, t))(\u222bB0(x0,\u03c1)G\u03c1(x0,y)R0(y)dV0\u2212m(\u03c1,x0, t)).Using the fact that \u22060(\u2212F)\u2265\u2212R0 and [24, Lemma 2.1], we obtain\u2212F(x0, t)\u2264\u222bB0(x0, 15\u03c1)G\u03c1(x0,y)R0(y)dV0+ c2\u03c1\u22122t(1\u2212m(\u03c1,x0, t))(\u222bB0(x0,\u03c1)G\u03c1(x0,y)R0(y)dV0\u2212m(\u03c1,x0, t)) (3.4.4)for some c2 depending only on n. As in [24, p.127], we get the result.We also need the following:Lemma 3.4.2. Let g(t) be the complete U(n)-invariant solution of the Ka\u00a8hler-Ricciflow (1.0.1) with nonnegative bisectional curvature. LetF(r, t) := F(z, t) = log(det(gi j\u00af(z, t))det(gi j\u00af(z,0)))44where r = |z|2. Then for r \u2265 1 and for all tF(r, t)\u2265\u2212c\u2212n logr+F(1, t)for some constant c > 0 depending only on g(0). If in addition the generatingfunction \u03be of g0 satisfies:limr\u2192\u221e\u222b r11\u2212\u03be (s)sds = b < \u221e, (3.4.5)then for r \u2265 1 and for all tF(r, t)\u2265\u2212c+nF(1, t)for some constant c > 0 depending only on g(0).Proof. Consider the functions \u03be (r, t),h(r, t), f (r, t) corresponding g(r, t). Then 0\u2264\u03be (r, t) \u2264 1 since g(r, t) has non-negative bisectional curvature, and by Theorem3.1.1 we then get 0\u2264 h(r, t), f (r, t)\u2264 1. Thus for r \u2265 1 we havef (r, t) =1r\u222b r0h(s, t)ds\u2265 1r\u222b 10h(s, t)dt =1rf (1, t).h(r, t) = h(1, t)exp(\u222b r1\u2212\u03be (s)sds)\u2265 1rh(1, t),and using the formula det(gi j\u00af(r, t)) = h(r, t) fn\u22121(r, t) we then getdet(gi j\u00af(r, t))det(gi j\u00af(r,0))\u2265det(gi j\u00af(r, t))\u2265 1rnh(1, t) f n\u22121(1, t)=1rndet(gi j\u00af(1, t))=1rndet(gi j\u00af(1, t))det(gi j\u00af(1,0))\u00b7det(gi j\u00af(1,0)).From this, it is easy to see the first result follows. Now suppose the generating45function \u03be of g0 also satisfies (3.4.5). Then for r \u2265 1,h(r, t)h(r,0)=h(1, t)exp(\u2212\u222b r1 \u03be (s,t)s ds)h(1,0)exp(\u2212\u222b r1 \u03be (s,0)s ds)\u2265 h(1, t)h(1,0)exp(\u222b r1\u03be (s,0)\u22121sds)\u2265c1 h(1, t)h(1,0)for some constant c1 > 0 independent of r, t, provided r \u2265 1. Also for r \u2265 1f (r, t)f (r,0)=\u222b r0 h(s, t)ds\u222b r0 h(s,0)ds=\u222b 10 h(s, t)ds+\u222b r1 h(s, t)ds\u222b 10 h(s,0)ds+\u222b r1 h(s,0)ds\u2265h(1, t)+ c1h(1,t)h(1,0)\u222b r1 h(s,0)ds\u222b 10 h(s,0)ds+\u222b r1 h(s,0)ds\u2265c2 h(1, t)h(1,0)for some constant c2 > 0 independent of r, t, provided r \u2265 1.Hence for r \u2265 1, at the point |z|2 = rdet(gi j\u00af(z, t))det(gi j\u00af(z,0))=h(r, t) f n\u22121(r, t)h(r,0) f n\u22121(r,0)\u2265c3(h(1, t)h(1,0))n\u2265c3(h(1, t) f n\u22121(1, t)h(1,0) f n\u22121(1,0))n=c3(det(gi j\u00af(1, t))det(gi j\u00af(1,0)))nfor some c3 > 0, where we have used the fact that f (r, t) \u2264 f (r,0). From this thesecond result follows.And now we prove a lemma about bounding the scalar curvature on compact46subsets.Lemma 3.4.3. Let g(t) be as in Theorem 3.3.1. Then for any \u03b1 > 0 the curvatureof g(t) is uniformly bounded in D(\u03b1)\u00d7 [0,\u221e), where D(\u03b1) = {|z|2 < \u03b1}.Proof. Letk(z,\u03c1) =1V0(z,\u03c1)\u222bB0(z,\u03c1)R(0)be the average of the scalar curvature R(0) of g0 over the geodesic ball B0(z,\u03c1)with respect to g0. Letk(\u03c1) = sup|z|\u22641k(z,\u03c1).By [33, Theorem 7], there is a constant c1 such thatk(\u03c1)\u2264 c11+\u03c1. (3.4.6)Suppose |z|2 = r, then the distance \u03c1(z) = \u03c1(r) from z to the origin satisfies\u03c1(z) = \u03c1(r) =12\u222b r0\u221ah\u221asds\u2265 c2 logr (3.4.7)for some constant c2 > 0 for all r \u2265 1. LetF(z, t) = logdet(gi j\u00af(z, t))det(gi j\u00af(z,0))and let m(\u03c1, t) = infz\u2208Cn,\u03c1(z)\u2264\u03c1 F(z, t). Fix r0 > 1 and let \u03c10 = \u03c1(r0). Denote\u2212m(\u03c10, t) by \u03b7(t). By Lemmas 3.4.1 and 3.4.2, there exist positive constants c3,c4independent of t and \u03c1 such that\u03b7(t)\u2264c4[(1+t(1\u2212m(\u03c1+\u03c10, t))\u03c12)K(\u03c1)\u2212 tm(\u03c1+\u03c10, t)(1\u2212m(\u03c1+\u03c10, t))\u03c12]\u2264c4[(1+t(1+ c1+ log r\u02dc(\u03c1)+\u03b7(t))\u03c12)K(\u03c1)+t(c3+ log r\u02dc(\u03c1)+\u03b7(t))(1+ c1+ log r\u02dc(\u03c1)+\u03b7(t)\u03c12]47whereK(\u03c1) =\u222b 2\u03c10sk(s)dsand r\u02dc(\u03c1) is such that,\u03c1+\u03c10 =12\u222b r\u02dc(\u03c1)0\u221ah\u221asds.By (3.4.6) and (3.4.7), there is a constant c5 independent of \u03c1 and t, such that1+\u03b7(t)\u2264 c5(\u03c1+t(1+\u03b7(t))\u03c1+t(1+\u03b7(t))2\u03c12+ t).Let \u03c12 = 2c5t(1+\u03b7(t)), then1+\u03b7(t)\u2264 c6(t12 (1+\u03b7(t))12 + t)for some constant c6 > 0 independent of t. From this we conclude that \u03b7(t) \u2264c7(1+ t) for some constant c7 independent of t. Hence\u2212F(z, t)\u2264 c7(1+ t)for all z with |z|2 \u2264 r0, which implies\u222b 2ttR(z,s)ds\u2264\u222b 2t0R(z,s)ds =\u2212F(z, t)\u2264 c7(1+ t).On the other hand, by the Li-Yau-Hamilton inequality [5], sR(z,s) \u2265 tR(z, t) fors\u2265 t. Hence we haveR(z, t)\u2264 c8for all t and for all z with |z|2 \u2264 r0. This completes the proof of the lemma.3.5 Convergence after rescalingTheorem 3.5.1. Let g(t) be a complete longtime U(n)-invariant solution of theKa\u00a8hler-Ricci flow (1.0.1) with bounded non-negative bisectional curvature, andassume g(0) also has bounded curvature. Then g(t) converges, after rescaling atthe origin, to the standard Euclidean metric on Cn.48In order to prove the theorem, we need the following lemmas.Lemma 3.5.1. Let g(t) be as in Theorem 3.5.1. Suppose the curvature of g(t) isuniformly bounded by c1, in D(\u03b1)\u00d7 [0,\u221e), where D(\u03b1) = {|z|2 < \u03b1}. Then thereis a constant c2 depending only on c1 and \u03b1 such thath(r, t)\u2264 h(0, t)\u2264 c2h(r, t); f (r, t)\u2264 f (0, t)\u2264 c2 f (r, t) (3.5.1)for all 0 < r < \u03b1 and for all t.Proof. By Remark 3.1.1 and the fact that g(t) has nonnegative bisectional curva-ture, we have h(0, t)\u2265 h(r, t) and f (0, t)\u2265 f (r, t) for all r > 0. On the other hand,we have A(r, t),B(r, t),C(r, t)\u2264 c1 in D(\u03b1)\u00d7 [0,\u221e) by hypothesis. Thus C =\u22122 frf 2gives1f (r, t)\u2212 1f (0, t)\u2264 c1\u03b12in D(\u03b1)\u00d7 [0,\u221e), and by f (0, t)\u2264 f (0,0) = h(0,0) = 1 we getf (0, t)\u2264(c1\u03b12+1)f (r, t).Also, A = \u03ber(r,t)h(r,t) \u2264 c1 gives\u03ber(r, t)\u2264 c1h(r, t)\u2264 c1h(r,0)\u2264 c1h(0,0) = c1in D(\u03b1)\u00d7 [0,\u221e), and thus \u03be (r, t)\u2264 c1r, givingh(r, t) = h(0, t)exp(\u222b r0\u2212\u03be (s, t)sds)\u2265 exp(\u2212c1\u03b1)h(0, t).This completes the proof of the lemma.Proof of Theorem 3.5.1. Let a(t)= h(0, t). We claim that the curvature of 1a(t)g(x, t)converges to 0 uniformly on compact sets. Note that 1a(t)g(x, t) has nonnegativebisectional curvature. Let R(z, t) be the scalar curvature of g(t) at z \u2208 Cn. Sup-pose first that limt\u2192\u221eR(0, t) = 0. Then by the Li-Yau-Hamilton inequality [5], weconclude that limt\u2192\u221eR(z, t) = 0 uniform on compact sets. Since a(t) \u2264 a(0) =h(0,0) = 1, the claim is true in this case.49Suppose on the other hand that there exist k\u2192\u221e and c1 > 0 such that R(0, tk)\u2265c1 for all k. We may assume that tk+1 \u2265 tk +1. By the Li-Yau-Hamilton inequalityagain, there is c2 > 0 such that R(0, tk + s)\u2265 c2 for all k and for all 0\u2264 s\u2264 1. Bythe U(n) symmetry, the Ricci tensor of g(t) at the origin is Ric = Rn g, using theKa\u00a8hler-Ricci flow equation, we haveh(0, tk+1)\u2264 h(0, tk +1)\u2264 e\u2212c3h(0, tk)for some c3 > 0 for all k. Hence h(0, tk)\u2192 0 as k\u2192\u221e. Since h(0, t) is nonincreas-ing, we have limt\u2192\u221e a(t) = limt\u2192\u221e h(0, t) = 0. On the other hand, the curvature ofg(t) is uniformly bounded on compact sets by Lemma 3.4.3. Thus our claim is truein this case as well.Consider any sequence tk\u2192 \u221e. Let ak = h(0, tk) and let g\u02dck(x, t) = 1ak g(x,akt +tk). Then g\u02dck(t) is a U(n)-invariant solution to the Ka\u00a8hler-Ricci flow onCn\u00d7[\u2212 tkak ,\u221e).Note that \u2212tk\/ak \u2264 \u2212tk because ak \u2264 1. By Lemmas 3.5.1, 3.4.3, for any \u03b1 > 0,g\u02dck(x,0) is uniformly equivalent to the standard Euclidean metric ge on D(\u03b1) (withrespect to k). By the claim above and the Li-Yau-Hamilton inequality [5], thecurvature of the metrics g\u02dck(x, t) approach zero uniformly (with respect to k) oncompact subsets of Cn\u00d7 (\u2212\u221e,0].In particular, we conclude that g\u02dck(t) is uniformly equivalent to ge in D(\u03b1)provided \u22121\u2264 t \u2264 0, and thus by Theorem A.1.1, we have for any m\u2265 0, there isa c4 depending on \u03b1 such that|\u2207me g\u02dck(0)| \u2264 c4on D(\u03b12 ), where \u2207e is the derivative with respect to the standard Euclidean metric.From this it is easy to conclude the subsequence convergence of g\u02dck(0) uniformlyand smoothly on compact subsets of Cn to a flat U(n)-invariant Ka\u00a8hler metric g\u221e,generated by some \u03be\u221e say. Since the curvature is zero, we have \u03be \u2032\u221e \u2261 0 and thus\u03be\u221e \u2261 0. Moreover, at the origin (g\u221e)i j\u00af = \u03b4i j. Hence h\u221e(0) = 1 which impliesthat (g\u221e)i j\u00af = \u03b4i j everywhere. From this the Theorem follows as tk was chosenarbitrarily.50Chapter 4Quasi-projective manifolds4.1 IntroductionIn this chapter, we discuss the Ka\u00a8hler-Ricci flow on quasi-projective manifolds. Acomplex manifold M is said to be quasi-projective if M = M\u00af \\D for some compactKa\u00a8hler manifold M\u00af and D \u2282 M\u00af is a divisor with simple normal crossing. Withthis structure, we can define different notions of singularity for Ka\u00a8hler metrics onM. We are interested in cusplike merics on M, which are metrics equivalent to thestandard complete local modelidz1\u2227dz1\u00af|z1|2 log2 |z1|2 + in\u2211j=2dz j \u2227dz j\u00af,where (zi) is a local holomorphic coordinate in which D = {z1 = 0}. As an exam-ple, let \u03b7 be a Ka\u00a8hler form on M\u00af and let S be a holomorphic section of OD thatvanishes precisely on D, a Ka\u00a8hler metric \u03c9\u02c6 on M is said to be a Carlson-Griffithsform if\u03c9\u02c6 = \u03b7\u2212 i\u2202 \u2202\u00af log log2 ||S||2h= \u03b7\u22122i\u2202 \u2202\u00af log ||S||2hlog ||S||2h+2i\u2202 log ||S||2hlog ||S||2h\u2227 \u2202\u00af log ||S||2hlog ||S||2h(4.1.1)51for some Hermitian metric h on OD. They are cusplike metrices inroduced in [3],below are some facts about Carlson-Griffiths forms and we refer to [3] and [17] formore details and explanations:1. \u03c9\u02c6 has bounded geometry of infinite order.2. \u2212 log log2 \u2016S\u20162h is bounded above and in L1(M).3. log \u03c9\u02c6n\u2016S\u20162h log2 \u2016S\u20162h\u2126 is bounded on M where \u2126 is any smooth volume form onM.In particular (2) implies that \u03c9\u02c6 is a well defined current on M. (see for example[23] (\u00a78, example 8.15).In [23] the authors showed that if \u03c90 is cusplike with so called superstandardspatial asymptotics at D, then a bounded curvature cusplike solution \u03c9(t) to (1.0.1)exists on M\u00d7 [0,T[\u03c90]), having the same asymptotics for all t, whereT[\u03c90] := sup{T : [\u03b7 ]+T (c1(KM)+ c1(OD)) \u2208KM} (4.1.2)and KM is the Ka\u00a8hler cone of M. They also showed under a weaker conditioncalled standard spatial asymptotics, a similar bounded curvature solution existson M\u00d7 [0,T ), for some maximal T where T \u2264 T[\u03c90]. A main point here is thatthe maximal existence time T[\u03c90] depends only on the cohomology class of theinitial form \u03c90. One example of metric having superstandard spatial asymptotics is\u03b7\u2212 i\u2202 \u2202\u00af log log2 |S|2+ i\u2202 \u2202\u00af\u03d5 where \u03d5 \u2208C\u221e(M) and these metrics will play a crucialrole in our discussion.In this chapter, we will construct a solution to (1.0.1) when\u03c90 = \u03b7\u2212 i\u2202 \u2202\u00af log log2 |S|2+ i\u2202 \u2202\u00af\u03d5for general class of potentials \u03d5 . Assuming that \u03c90 is only bounded below onM by a cusplike metric, and has zero Lelong number (see the Appendix A.2 forthe definition and the related propostions we are going to use in this chapter), wewill construct a solution to (1.0.1) on M\u00d7 [0,T[\u03c90]) which is bounded below by acusplike metric on some definite positive time subinterval of [0,T[\u03c90]) (see Theorem524.1.2 and below for details). In particular, \u03c90 may have unbounded curvature onM here. On the other hand, in cases when \u03c90 may be incomplete on M, includingwhen \u03c90 is smooth on M, we can still construct solutions on M\u00d7 [0,T[\u03c90]) which iscusplike on M for all positive times (see Theorem 4.1.1). In these cases\u03c90 becomesinstantaneously complete on M under (1.0.1). We now describe our results in moredetails below.We first consider the case \u03c90 \u2265 c\u03b7 on M for some c > 0 where \u03d50 is boundedand smooth on M. In particular, \u03c90 is typically incomplete on M here. Our mainresult here isTheorem 4.1.1. Let \u03d50 \u2208 L\u221e(M)\u22c2C\u221e(M)\u22c2Psh(M,\u03b7) such that\u03c90 = \u03b7+ i\u2202 \u2202\u00af\u03d50 \u2265 c\u03b7 (4.1.3)for some constant c> 0. Let T[\u03c90] be as in (4.1.2). Then (1.0.1) has a unique smoothsolution \u03c9(t) on M\u00d7 [0,T[\u03c90]) wherec1(t)\u03c9\u02c6 \u2264 \u03c9t \u2264 c2(t)\u03c9\u02c6 (4.1.4)for all t \u2208 (0,T[\u03c90]) ,and some positive functions ci(t) and Carlson-Griffiths formon \u03c9\u02c6 on M.Also, for any Hermitian metric h on OD and volume form \u2126 on M, (4.1.6)and (4.1.7) hold on M\u00d7 [0,T[\u03c90]) for some \u03d5(t) which is bounded on M for eacht \u2208 [0,T[\u03c90]).Remark 4.1.1. Theorem 4.1.1 includes as a special case, when \u03c90 has conical sin-gularities at D or is in fact smooth on M.Next we consider when \u03c90 is a complete metric on M in which case \u03d50 may beunbounded on M. Our first result here isTheorem 4.1.2. Let \u03d50 \u2208C\u221e(M)\u22c2Psh(M,\u03b7) have zero Lelong number such that\u03c90 = \u03b7+ i\u2202 \u2202\u00af\u03d50 \u2265 c\u03c9\u02c6for some c > 0 and Carlson-Griffiths form \u03c9\u02c6 on M. Let T[\u03c90] be as in (4.1.2). Then53the Ka\u00a8hler -Ricci flow (1.0.1) has a smooth solution \u03c9(t) on M\u00d7 [0,T[\u03c90]) and\u03c9(t)\u2265 (1n\u2212 4K\u02c6tc)\u03c9\u02c6 (4.1.5)for all t \u2264 c4nK\u02c6where K\u02c6 is a non-negative upper bound on the bisectional curvaturesof \u03c9\u02c6 . Moreover,(1) For any hermitian metric h on OD and volume form \u2126 on M, (4.1.6) and(4.1.7) hold on M\u00d7 [0,T[\u03c90]) where \u03d5(t) \u2264 c(t) on M\u00d7 [0,T[\u03c90]) for somecontinuous function c(t).(2) Suppose further that \u03c90 is cusplike and\u2212C log log2 \u2016S\u20162h\u2264 \u03d50 on M for someconstant C > 0 and Hermitian metric h. Then for any 0 < T < T[\u03c90], (4.1.6)and (4.1.7) hold on M\u00d7 [0,T ] where \u2212c log log2 \u2016S\u20162h\u2032 \u2264 \u03d5(t) \u2264 c on M\u00d7[0,T ] for some constant c > 0 and Hermitian metric h\u2032.Remark 4.1.2. It can be proved that there is a unique solution g(t) satisfying (4.1.5)and (1) on the time interval [0, c4nK\u02c6), though it is not known if our solution is uniqueon the whole time interval [0,T[\u03c90]).Remark 4.1.3. In Theorem 4.1.1 and 4.1.2 above, we only considered the case ofa single smooth divisor D \u2282M. On the other hand, straight forward extensions ofour definitions and techniques allow us also to consider the case of some collectionof simple normal crossing divisors D1, ..,Dk in which case we can have similarstatements.Note also that Theorem 4.1.2 leaves open the possibility that the solution mayexist beyond t = T[\u03c90]. Note also, in Theorem 4.1.2 \u03c90 is complete while the so-lution may not be complete for all positive times. Meanwhile in Theorem 4.1.1,\u03c90 may be incomplete while the solution is complete for all positive times. Thisseems counterintuitive, and is a result of the stronger a priori estimates in the case\u03d50 is bounded. On the other hand, (2) says cupslikeness is preserved at the poten-tial level for all times in some sense. If we assume \u03c90 above in fact has boundedcurvature and is sufficiently asymptotic to the standard model at D in a sense, thefollowing Theorem says the solution is indeed cusplike for all times, and [0,T\u03c9) isindeed a maximal time interval.54Theorem 4.1.3. Let \u03b7 be a smooth Ka\u00a8hler form on M and \u03c9\u02c6 =\u03b7\u2212i\u2202 \u2202\u00af log log2 \u2016S\u20162be a Carlson-Griffiths form on M. Let \u03c90 = \u03c9\u02c6 + i\u2202 \u2202\u00af\u03d5 be a smooth completebounded curvature Ka\u00a8hler metric on M such that \u03d5log log2 \u2016S\u20162 \u2192 0,|d\u03d5|\u03c9\u02c6log log2 \u2016S\u20162 \u2192 0and |\u03c90\u2212\u03c9\u02c6|\u03c9\u02c6\u2192 0 as \u2016S\u2016\u2192 0. Let T[\u03c90] be as in (4.1.2). Then (1.0.1) has a uniquesmooth maximal bounded curvature solution \u03c9(t) on M\u00d7 [0,T[\u03c90]).In the follow discussion, we will study (1.0.1) through an associated parabolicMonge Ampe`re equation set up as follows. For any Hermitian metric h on OD, andvolume form \u2126 on M, consider a solution \u03d5(t) to the parabolic Monge Ampe`reequation\uf8f1\uf8f2\uf8f3 \u2202t\u03d5(t) = log\u2016S\u20162h log2 \u2016S\u20162h(\u03b8t + i\u2202 \u2202\u00af\u03d5(t))n\u2126;\u03d5(0) = \u03d50.(4.1.6)\u03b8t := \u03b7+ t\u03c7; \u03c7 :=\u2212Ric(\u2126)+\u0398h\u2212 i\u2202 \u2202\u00af log log2 \u2016S\u20162hand the associated family of Ka\u00a8hler metrics\u03c9(t) := \u03b8t + i\u2202 \u2202\u00af\u03d5(t) (4.1.7)on M\u00d7 [0,T ) for some T . Here Ric(\u2126) = \u2212i\u2202 \u2202\u00af log\u2126 and \u0398h represents the cur-vature form of the metric h on OD.It follows that \u03c9(t) solves (1.0.1) on M\u00d7 [0,T ), and conversely, if \u03c9(t) solves(1.0.1) on M\u00d7 [0,T ) then (4.1.7) holds for some solution \u03d5(t) to (4.1.6) (see thederivation of (4.2.2)). The equation (4.1.6) is different from the parabolic Monge-Ampe`re equations considered in the earlier works mentioned above in the appear-ance of the \u2016S\u20162h log\u2016S\u20162h term in the numerator of the right hand side. This termwill be useful in establishing the cusplike-ness of our solutions for positive times.4.2 Proof of Theorem 4.1.1Analogous to [16], we will use Theorem A.2.1 to construct approximation solu-tions to Ka\u00a8hler-Ricci flow from which we will take a limit to obtain the desiredsolution. Our study here is similar in ways to [23] where the authors also studied55cusp type Ka\u00a8hler-Ricci flow solutions. On the other hand, our work here is differentfrom these in the following ways. First, our initial metric is not cusplike (or evencomplete) as in the case of [23], yet we are looking solutions which are cusplikefor positive times. This is one reason that the Monge Ampe`re equation (4.1.6) weconsider is somewhat different than the ones studied in other works. Second, thebackground Carlson-Griffith metric has only L1 volume form so we cannot approx-imate \u03c60 using the procedure from [29] which is based on Kolodziej\u2019s Lp estimate.In the following we will introduce approximation procedures to overcome thesedifficulties.We first make the following technical assumptions which we will use through-out the rest of the section.Assumption 1. Let \u03b7 , \u03c9\u02c6 and \u03d50 be as in Theorem 4.1.1 and T[\u03c90] be as in (4.1.2).Fix some T\u02dc < T[\u03c90] and choose a Hermitian metric h\u02c6 on OD and smooth volumeform \u2126 on M so that \u2016S\u20162h\u02c6< 1 on M and1. \u03b7+ t(\u2212Ric(\u2126)+\u0398h\u02c6)> 0 on M\u00d7 [0, T\u02dc ]2. \u03b7+ t(\u2212Ric(\u2126)+\u0398h\u02c6+2\u0398h\u02c6log\u2016S\u20162h\u02c6)> 0 on M\u00d7 [0, T\u02dc ]Finally, we will abbreviate \u2016S\u20162h\u02c6and \u0398h\u02c6 simply by \u2016S\u20162 and \u0398 repsectively.We first choose h\u02c6 so that \u2016S\u20162h\u02c6< 1 on M. By the definition of T[\u03c90] we can thenchoose \u2126 such that (1) holds for t = T\u02dc . Then, by scaling h\u02c6 smaller if necessary, wemay also assume the inequality in (2) also hold at t = T\u02dc (by the smoothness of \u0398on M. The fact that (1) and (2) holds for all t \u2208 [0, T\u02dc ] then follows by interpolationbetween t = 0 and t = T\u02dc .4.2.1 Proof of Theorem 4.1.1 when \u03d50 \u2208C\u221e(M)In this case, \u03c90 = \u03b7 + i\u2202 \u2202\u00af\u03d50 is a Ka\u00a8hler form on M, from Theorem 8.19 in [23]we haveLemma 4.2.1. For \u03b5 > 0 sufficiently small, \u03c9\u03b5,0 = \u03c90 \u2212 \u03b5\u2202 \u2202\u00af log log2 \u2016S\u20162h is aCarlson-Griffiths metric on M and (1.0.1) has a bounded curvature solution \u03c9\u03b5(t)on M\u00d7 [0,T[\u03c90]) with initial data \u03c9\u03b5,0.56For \u03c9\u03b5(t) above, we will now derive estimates on compact subsets of M\u00d7[0,T[\u03c90]) which are uniform with respect to \u03b5 . We will then let \u03b5 \u2192 0 to obtaina limit smooth solution \u03c9(t) to (1.0.1) on M \u00d7 [0,T[\u03c90]) with initial data \u03c90 =\u03b7+ i\u2202 \u2202\u00af\u03d50. We may write\u03c9\u03b5(t) = \u03b8\u03b5,t + i\u2202 \u2202\u00af\u03d5\u03b5(t) (4.2.1)where \u03d5\u03b5(t) solves the parabolic Monge Ampe`re equation on M\u00d7 [0,T[\u03c90]):\uf8f1\uf8f2\uf8f3 \u2202t\u03d5\u03b5(t) = log\u2016S\u20162 log\u2016S\u20162(\u03b8\u03b5,t + i\u2202 \u2202\u00af\u03d5\u03b5(t))n\u2126;\u03d5\u03b5(0) = \u03d50.(4.2.2)\u03b8\u03b5,t := \u03b7+ t\u03c7\u2212 \u03b5\u2202 \u2202\u00af log log2 \u2016S\u20162; \u03c7 :=\u2212Ric(\u2126)+\u0398\u2212 i\u2202 \u2202\u00af log log2 \u2016S\u20162Indeed, letting\u03d5\u03b5(t) = \u03d50+\u222b t0log\u2016S\u20162 log2 \u2016S\u20162(\u03c9\u03b5(t))n\u2126and defining \u03b8\u03b5,t as above we see that (4.2.2) is obviously satisfied. On the otherhand, we have \u03b8\u03b5,0+ i\u2202 \u2202\u00af\u03d5\u03b5(0) = \u03c9\u03b5(0) while on M\u00d7 [0,T[\u03c90])(\u03b8\u03b5,t + i\u2202 \u2202\u00af\u03d5\u03b5(t))\u2032 =\u2212Ric(\u2126)+\u0398\u2212 i\u2202 \u2202\u00af log log2 \u2016S\u20162+ i\u2202 \u2202\u00af (log\u2016S\u20162+ log log2 \u2016S\u20162+ log(\u03c9n\u03b5 (t))\u2212 log\u2126)=i\u2202 \u2202\u00af log(\u03c9n\u03b5 (t))=\u2212Ric(\u03c9\u03b5(t))(4.2.3)and it follows from (1.0.1) that (4.2.1) holds on M\u00d7 [0,T[\u03c90]). Conversely, reversingthe process above shows that the metric in (4.2.1) satisfies (1.0.1) for any givensolution to (4.2.2).57Note by (2) in Assumption 1 and (4.1.1), for \u03b5 > 0 we havec1(t+ \u03b5)\u03c9\u02c6 \u2264 \u03b8t,\u03b5 \u2264 c2\u03c9\u02c6, (4.2.4)for some constants c1,c2 > 0 independent of \u03b5 for all t \u2208 [0, T\u02dc ]. We will nowderive estimates for \u03d5\u03b5(t) on M\u00d7 [0, T\u02dc ] which will be uniform with respect to \u03b5and will yield uniform C\u221eloc estimates for \u03c9\u03b5(t). These will allow us to let \u03b5 \u2192 0and T\u02dc \u2192 T[\u03c90] and obtain a limit solution on M\u00d7 [0,T[\u03c90]) satisfying the conditionsin Theorem 4.1.1.We begin with the following C0-estimates:Lemma 4.2.2. On the time interval (0, T\u02dc ], we have |\u03d5\u03b5 | \u2264C, |\u03d5\u02d9\u03b5 | \u2264 Ct , where C isindependent of \u03b5 sufficiently small.Proof. Fix \u03b5 > 0 such that (4.2.4) holds. Since \u03b8t,\u03b5 is a Carlson-Griffiths form forall t \u2208 [0, T\u02dc ], so the curvature on M =M\\D is uniformly bounded. Let\u03c8 :=\u03d5\u03b5\u2212Ctfor some C > 0 to be chosen. Then since \u03c9\u03b5(t) is a bounded curvature solution, \u03c8is uniformly bounded on M\u00d7 [0, T\u02dc ]. We first suppose that \u03c8 attains a maximumvalue on M\u00d7 [0, T\u02dc ] at some point (x, t). If t\u00af > 0 then using (4.2.2) we have at (x, t)\u2202t\u03c8\u03b5 \u2264 log\u2016S\u20162 log\u2016S\u20162\u03b8 nt\u00af,\u03b5\u2126\u2212C <\u22121 (4.2.5)by choosing C sufficiently large independent of \u03b5 . Indeed, such a choice of C existsfrom the upper bound in (4.2.4) and property (3) of Carlson-Griffiths metrics (seethe paragraph after (4.1.1)). But (4.2.5) contradicts the maximality of \u03c8 unlesst\u00af = 0 in which case \u03c8 \u2264 sup\u03c8(0) = sup\u03d50 on M\u00d7 [0, T\u02dc ] and thus\u03d5\u03b5 \u2264 sup\u03d50+CT\u02dc (4.2.6)on M\u00d7 [0, T\u02dc ]. Now in general, suppose \u03c8 does not attain a maximal value onM\u00d7 [0, T\u02dc ]. Since \u03c9\u03b5(t) is a bounded curvature solution, we have |\u2202t\u03c8\u03b5 | is boundedon M\u00d7 [0, T\u02dc ], and by the Omori-Yau maximum principal we can find a sequencexk \u2208M and t\u00af \u2208 [0, T\u02dc ] such that \u03c8\u03b5(xk, t\u00af)\u2192 supM\u00d7[0,T\u02dc ]\u03c8\u03b5 and i\u2202 \u2202\u00af\u03c8\u03b5(xk, t\u00af)\u2264 \u03bbk\u03b8t\u00af,\u03b5where \u03bbk decreases to 0 as k\u2192 \u221e. Combining these with (4.2.2), we may argueas above that for any \u03b4 we have \u2202t\u03c8\u03b5(xk, t\u00af) \u2264 \u22121 for some C independent of \u03b558and all k sufficeintly large. On the other hand, |\u2202 2t \u03c8\u03b5 |(xk, t\u00af) is uniformly boundedindependent of k using again that \u03c9\u03b5(t) is a bounded curvature solution. These lasttwo facts contradict that \u03c8\u03b5(xk, t\u00af)\u2192 supM\u00d7[0,T\u02dc ]\u03c8\u03b5 unless t\u00af = 0 and we concludeagain as above that (4.2.6) likewise holds on M\u00d7 [0, T\u02dc ] in this case as well.Using again that \u03c9\u03b5(t) is a bounded curvature solution, we also have |\u2202 2t \u03c8\u03b5 | isbounded on M\u00d7 [0, T\u02dc ].By a similar argument, using (4.2.4) and (4.2.2) we may also getinf\u03d50+C\u2032\u222b t0log(s+ \u03b5)ds\u2264 \u03d5\u03b5 (4.2.7)on M\u00d7 [0, T\u02dc ] for some constant C\u2032 > 0 independent of \u03b5 . We thus conclude theestimates for |\u03d5e| in the Lemma hold.Next we will apply the methods in [29] to derive estimates for \u03d5\u02d9\u03b5 . We have(\u2202t \u2212\u2206)\u03d5\u02d9\u03b5 = tr\u03c9\u03b5\u03c7where the \u2206 denotes the Laplacian with respect to \u03c9\u03b5 , and also(\u2202t \u2212\u2206)(t\u03d5\u02d9\u03b5 \u2212\u03d5\u03b5 \u2212nt) =\u2212tr\u03c9\u03b5 (\u03b7\u2212 \u03b5\u2202 \u2202\u00af log log2 \u2016S\u20162)< 0,where the last inequality comes from the fact that \u03b7\u2212 \u03b5\u2202 \u2202\u00af log log2 \u2016S\u20162h > 0 when\u03b5 is small by (4.1.1). Applying the maximum principle in [27], we conclude thesupremum of (t\u03d5\u02d9e\u2212\u03d5\u03b5\u2212nt) on M\u00d7(0, T\u02dc ], which is indeed finite, is attained whent = 0 and thus \u03d5\u02d9\u03b5 \u2264 \u03d5\u03b5\u2212\u03d50t + n on M\u00d7 (0, T\u02dc ]. On the other hand, for sufficientlylarge A independent of \u03b5 we have(\u2202t \u2212\u2206)(\u03d5\u02d9e+A\u03d5\u03b5 \u2212n log t)= tr\u03c9\u03b5 (\u03c7+A\u03b8t,\u03b5)+A log\u03c9n\u03b5 \u2016S\u20162 log2 \u2016S\u20162\u2126\u2212 (An+ nt)\u2265 A2tr\u03c9\u03b5\u03b8t,\u03b5 +A log(t+ \u03b5)n\u03c9n\u03b5\u03b8 nt,\u03b5\u2212Ct\u2265 A4(\u03b8 nt,\u03b5\u03c9n\u03b5)1\/n\u2212Ct.(4.2.8)where in the second inequality we have made use of (4.2.4) and property (3) of59Carlson-Griffiths metrics, and in the third inequality we have again used (4.2.4).Now let \u03c8 = (\u03d5\u02d9e + A\u03d5\u03b5 \u2212 n log t), and assume \u03c8 attains a minimum value onM\u00d7 [0, T\u02dc ] at some point (x, t). It follows that t\u00af > 0, and (4.2.8) then gives\u03c9n\u03b5 (x, t)\u2265c\u03b8 nt,\u03b5(x, t)t\u00afn for some c > 0 independent of \u03b5 . From this, (4.2.4), (4.2.2) and prop-erty (3) of Carlson-Griffiths metrics we may then have \u03c8 \u2265 C log t \u2212C and thus\u03d5\u02d9e \u2265 C log t \u2212C on M\u00d7 (0, T\u02dc ] for some C > 0 where we have used (4.2.6) and(4.2.7). In general, \u03c8 is only bounded but may not attain a minimum value onM\u00d7 [0, T\u02dc ], though we may argue as before, applying the above estimate along anappropriate space-time sequence obtained by the Omori-Yau maximum principle,to conclude \u03d5\u02d9e \u2265C log t\u2212C on M\u00d7 (0, T\u02dc ] for some C > 0 in this case as well. Wethus conclude the estimates for |\u03d5\u02d9e| in the Lemma hold.Next we want to derive a Laplacian estimate for \u03d5e.Lemma 4.2.3. for each t \u2208 (0, T\u02dc ] we have C1(t)\u03c9\u02c6 \u2264\u03c9\u03b5(t)\u2264C2(t)\u03c9\u02c6, for constantsC1(t),C2(t)> 0 independent of \u03b5 sufficiently small.Proof. First recall that \u03c9\u02c6 is complete and has uniformly bounded bisectional cur-vature. Then the parabolic version of the Chern-Lu inequality (see [28]) gives:(\u2202t \u2212\u2206) log tr\u03c9\u03b5 \u03c9\u02c6 \u2264Ctr\u03c9\u03b5 \u03c9\u02c6+Cwhere the constant C depends on the upper bound of the bisectional curvature of \u03c9\u02c6.For the rest of the proof, C will denote a constant, which may change from line toline, and which is independent of \u03b5 . Now by (4.2.4) we may choose A sufficientlylarge independent of \u03b5 so that(\u2202t \u2212\u2206)(t log tr\u03c9\u03b5 \u03c9\u02c6\u2212A\u03d5e)\u2264 tr\u03c9\u03b5 (Ct\u03c9\u02c6\u2212A\u03b8t,\u03b5)+ log tr\u03c9\u03b5 \u03c9\u02c6\u2212A log\u03c9n\u03b5 \u2016S\u20162 log2 \u2016S\u20162\u2126+(An+Ct)\u2264 \u2212 A2tr\u03c9\u03b5\u03b8t,\u03b5 + log tr\u03c9\u03b5\u03b8t,\u03b5 +A log\u03b8 nt,\u03b5\u03c9n\u03b5\u2212CA log(t+ \u03b5)+C\u2264 \u2212 A4tr\u03c9\u03b5\u03b8t,\u03b5 \u2212CA log(t+ \u03b5)+C,(4.2.9)Now suppose t log tr\u03c9\u03b5 \u03c9\u02c6 \u2212A\u03d5e attains a maximum value on M\u00d7 [0, T\u02dc ] at some60point (x, t). Then if t > 0, using (4.2.9) we have at (x, t) that tr\u03c9\u03b5\u03b8t,\u03b5 \u2264\u2212C log t+Cand sot log tr\u03c9\u03b5 \u03c9\u02c6\u2212A\u03d5e \u2264Ct(log log1t+ log1t+ \u03b5)+C \u2264C.In this case it follows thattr\u03c9\u03b5 \u03c9\u02c6 \u2264 eCt (4.2.10)on M\u00d7 [0, T\u02dc ] where we have used estimate for |\u03d5\u03b5 | Lemma 4.2.2. On the otherhand, if t = 0 we also clearly have (4.2.10). In general, In general, t log tr\u03c9\u03b5 \u03c9\u02c6\u2212A\u03d5eis only bounded but may not attain a maximum value on M\u00d7 [0, T\u02dc ], though wemay argue as in the proof of Lemma 4.2.2 and apply the above estimates alongan appropriate sequence in space-time to likewise conclude that (4.2.10) holds onM\u00d7 [0, T\u02dc ] in this case as well.Finally, for any t \u2208 (0, T\u02dc ], (4.2.2) gives\u03c9n\u03b5\u03c9\u02c6n= e\u03d5\u02d9\u03b5\u2126\u03c9\u02c6n\u2016S\u20162 log2 \u2016S\u20162 \u2264Ct (4.2.11)on M for some constant Ct > 0 where we have used the estimate for |\u03d5\u02d9\u03b5 | in Lemma4.2.2 and property (3) of Carlson-Griffiths metrics. The Lemma follows from(4.2.11) and (4.2.10).Completion proof of Theorem 4.1.1 when \u03d50 \u2208C\u221e(M). The previous two lemmasand the Evans-Krylov theory applied to (4.2.2) imply that for any K \u2282\u2282 M ands \u2208 (0, T\u02dc ) we havemaxK\u2016\u2207k\u03b7\u03d5\u03b5(t)\u2016\u03b7 \u2264Ck,s,K,tindependent of \u03b5 and t \u2208 (s, T\u02dc ] where the norm and covariant derivative here arewith respect to \u03b7 . Thus for some subsequence \u03b5i \u2192 0, \u03d5\u03b5i will converge locallyuniformly to a smooth solution \u03d5 to (4.1.6) on M\u00d7 (0, T\u02dc ) which is bounded on Mfor each t. Thus \u03c9\u03b5i(t) converges locally uniformly to a smooth solution \u03c9(t) tothe flow in equation (1.0.1) on M\u00d7(0, T\u02dc ) and as T\u02dc < T[\u03c90] was arbitrary, we may infact assume the convergence to a solution on M\u00d7 (0,T[\u03c90]) satisfying the estimatesin Lemma 4.2.3.We now show the limit solution \u03c9(t) in fact converges smoothly uniformly on61compact subsets of M to \u03c90 as t\u2192 0, and thus can be extended to a smooth solutionto (1.0.1) on M\u00d7 [0,T[\u03c90])with initial data\u03c90. This basically follows from applyingTheorem 2.3.1 to the sequence \u03c9\u03b5i(t), and observing that the completeness of thebackground metric g\u02c6 in that Theorem is in fact not necessary in our case. Wedescribe this in more detail as follows. Consider the family of solutions \u03c9\u03b5i(t) onM\u00d7 [0,T[\u03c90]). For each i we have \u03c9\u03b5i(0)\u2265 c\u03b7 on M for some c > 0 independent ofi and we conclude from the proof of Lemma 2.2.1 that\u03c9\u03b5i(t)\u2265 c\u03b7 (4.2.12)on M\u00d7 [0,T ] for some c,T > 0 independent of i. For this simply observe that inLemma 2.2.1, the completeness of g\u02c6 is never actually used in the proof. Next wechoose any smooth non-negative function \u03c8 : [0,\u221e)\u2192 R which is identically zeroin some neighborhood of 0 and let \u03d5(t) := \u03c8(\u2016S\u20162). Then using (4.2.12), as in theproof of Lemma 2.2.3 we may have \u2016\u2207\u03b5i,t\u03d5(t)\u2016,\u2016\u2206\u03b5i,t\u03d5(t)\u2016 \u2264 C on M where thenorms here are relative to \u03c9\u03b5i(t) and C is independent of i and t \u2208 [0,T ), and by thesame proof there we may conclude that \u03d5(t)R\u03b5i(t) \u2265C on M\u00d7 [0,T ] where R\u03b5(t)is the scalar curvature of \u03c9\u03b5i(t) and the constant C is independent of i. From thisand the fact that \u03c6 was arbitrarily chosen, we can conclude as in Lemma 2.2.3 thatfor any compact K \u2282\u2282M we have the upper bound\u03c9\u03b5i(t)\u2264 c\u03b7 (4.2.13)on K\u00d7 [0,T ] for some c independent of i. The smooth convergence of \u03c9\u03b5i(t) oncompact subsets of M\u00d7 [0,T ] then follows from (4.2.12), (4.2.13) and the Evans-Krylov theory. Thus the limit solution \u03c9(t) extends smoothly to M\u00d7 [0,T[\u03c90]) asclaimed above.To complete the proof of the Theorem in this case, it remains only to prove theuniqueness statement which we do in the next sub-section in Proposition 4.2.1.4.2.2 Proof of Theorem 4.1.1 when \u03d50 \u2208 L\u221e(M)\u22c2C\u221e(M)By (4.1.3) we can choose some \u03b5 > 0 so that in fact we have \u03d50 \u2208Psh(M,(1\u2212\u03b5)\u03b7).Then, using Theorem A.2.1 we may choose a sequence {\u03d5 j}\u2282C\u221e(M)\u22c2Psh(M,(1\u221262\u03b5)\u03b7) so that \u03d5 j \u2193 \u03d50 pointwise on M and locally smoothly on M. In particular, itfollows that1. |\u03d5 j| \u2264C for all j and some C2. \u03c9 j = \u03b7+ i\u2202 \u2202\u00af\u03d5 j \u2265 \u03b5\u03b7 for all j on M.Now for each j we let \u03c9 j(t) be the solution to (1.0.1) on M\u00d7 [0,T[\u03c90]) withinitial data \u03c9 j(0) = \u03b7 + i\u2202 \u2202\u00af\u03d5 j constructed in the previous sub-section. UnderAssumption 1, let \u03d5 j(t) be the corresponding solution to (4.1.6) on M\u00d7 [0,T[\u03c90])also as previously constructed. From Lemma 4.2.2 and (1) it follows that |\u03d5 j|(t)and |\u03d5\u02d9 j(t)| are uniformly bounded on compact subsets of M\u00d7(0, T\u02dc ) independentlyof j where T\u02dc is from Assumption 1. From Lemma 4.2.3 it further follows that\u03c9 j(t) is uniformly equivalent to \u03c9\u02c6 on M, independent of j, on compact intervalsof (0, T\u02dc ). As T\u02dc < T[\u03c90] was arbitrary and by applying the arguments in the lastsub-section separately for each j, we may conclude smooth local estimates for\u03c9 j(t) on compact subsets of M\u00d7 [0,T[\u03c90]) which are independent of j, and thatsome subsequence of \u03c9 j(t) converges to a solution \u03c9(t) to (1.0.1) on M\u00d7 [0,T[\u03c90])satisfying (4.1.4) for all t \u2208 (0,T[\u03c90]).The proof of Theorem 4.1.1 will be complete once we prove uniqueness of sucha solution which we do in the Proposition below. First, we note that by (4.1.3) each\u03c9 j(t) will satisfy the lower bound in (4.2.12) for constants c,T > 0 independentof j and thus the limit solution \u03c9(t) likewise satisfies the lower bound in (4.2.12).We will use this fact in the following proof.Proposition 4.2.1. Let \u03d51(t),\u03d52(t) be two solutions to (4.1.6) on M\u00d7 [0,T[\u03c90])with initial data \u03d51(0) = \u03d52(0) \u2208 L\u221e(M)\u22c2C\u221e(M). Suppose |\u03d51(t)|, |\u03d52(t)| areboth bounded on M\u00d7 [0,T ) for every T < T[\u03c90]. Then \u03d51(t) = \u03d52(t) on M\u00d7 [0,T ].Proof. We assume without loss of generality that the solutions \u03c91(t) and \u03d51(t) areas constructed as in the proof of Theorem 4.1.1 so far. Now for any T < T[\u03c90]we prove that \u03d52 \u2264 \u03d51 on M\u00d7 [0,T ]. Let | \u00b7 |2 be a Hermitian metric such that\u2016S\u20162 < 1 and let \u0398 denotes its curvature form. As noted above, \u03c91(t) satisfies theinequality in (4.2.12) on M\u00d7 [0,\u03b5] for some constant c,\u03b5 > 0. Thus for all a > 0we can find Ca\u2192 0 as a\u2192 0 such that log (\u03c91(t)+a\u0398)n\u03c91(t)n 0, then at (x\u00af, t\u00af) we have0 < \u2202t\u03c8 = log(\u03c91(t)+a\u0398+ i\u2202 \u2202\u00af\u03c8)n\u03c91(t)n\u2212Ca < 0,a contradiction. Thus since \u03c8(x,0) < 0 we have \u03d52 \u2264 \u03d51\u2212 a log\u2016S\u20162 +Cat, andletting a\u2192 0 we get \u03d52 \u2264 \u03d51.To prove \u03d52 \u2265 \u03d51 we argue similarly. Namely, we first choose Ca\u2192 0 as a\u2192 0so that log(\u03c91\u2212 a\u0398)n\/\u03c9n1 \u2265Ca. Then we let \u03c8 = \u03d52\u2212\u03d51\u2212 a log\u2016S\u20162\u2212C1t andargue as before using the maximum principle that \u03c8 \u2265minM\u03c8(0) everywhere thenconclude by letting a\u2192 0 that \u03d52 \u2265 \u03d51 on M.4.3 Proof of Theorem 4.1.2The proof here roughly the same steps as the proof in Theorem 4.1.1. Namely, weconstruct a suitable approximating family for \u03c90, then consider the correspondingfamily of approximate solutions to (1.0.1) and convergence to a limit solution isproved using the parabolic Monge Ampe`re equation. One major difference hereis that \u03c90 is complete on M and we want to preserve this property in our approxi-mation. Another major difference is that \u03c60 is no longer bounded on M in generalwhich again makes the estimates more difficult.We make the following technical assumptions which we will use throughoutthe rest of the section.Assumption 2. Let \u03b7 , \u03c9\u02c6,\u03d50, be as in Theorem 4.1.2. Let 0 < T < T\u02dc < T[\u03c90] bearbitrary. Choose a Hermitian metric h\u02c6 on OD, a smooth volume form \u2126 on M\u00af anda constant \u03b2\u02c6 > 0 such that \u2016S\u20162h\u02c6< 1 on M and1. on M\u00d7 [0, T\u02dc ] we have\u03b7+ t(\u2212Ric(\u2126)+\u0398h\u02c6+2\u0398h\u02c6log\u2016S\u20162h\u02c6)> 0642. on M we have(1\u2212 c\u03b2\u02c6 )\u03b7 \u2264 \u03b7\u2212 \u03b2\u02c6\u0398h\u02c6 \u2264 (1+ c\u03b2\u02c6 )\u03b7for some c\u03b2\u02c6 <12 with T < (1\u2212 c\u03b2\u02c6 )T\u02dc .3. log log2 \u2016S\u20162h\u02c6> 1 on M, and \u03c9\u02c6 = \u03b7+ i\u2202 \u2202\u00af log log2 \u2016S\u20162h\u02c6As before we will abbreviate \u2016S\u20162h\u02c6and \u0398h\u02c6 simply by \u2016S\u20162 and \u0398.We first choose some h\u02c6 so that \u2016S\u20162h\u02c6< 1 on M. As in the case of Assumption1, we can find a smooth volume form \u2126, then scale h\u02c6 smaller if necessary, so thatthe inequality in (1) holds for t = T\u02dc in which case it must also hold for all t \u2208 [0, T\u02dc ]by interpolation. A choice of \u03b2\u02c6 in (2) is justified by the smoothness of \u0398 on M andby scaling h\u02c6 smaller if necessary. Finally, by scaling h\u02c6 smaller still we may assumethe inequality in (3) holds and that \u03b7+ i\u2202 \u2202\u00af log log2 \u2016S\u20162h\u02c6is also a Carlson-Griffithsmetric. Thus without loss of generality we may assume \u03c9\u02c6 = \u03b7+ i\u2202 \u2202\u00af log log2 \u2016S\u20162h\u02c6where \u03c9\u02c6 is from Theorem 4.1.2.4.3.1 Approximate solutions \u03c9\u03b1, j(t)Recall in Theorem 4.1.2 we have \u03d50 \u2208C\u221e(M)\u22c2Psh(M,\u03b7) with zero Lelong num-ber such that\u03c90 = \u03b7+ i\u2202 \u2202\u00af\u03d50 \u2265 2\u03b4\u03c9\u02c6 (4.3.1)on M for some \u03b4 > 0. We begin by construct a two parameter family \u03d5\u03b1, j ap-proximating \u03d50 as \u03b1 \u2192 0 and j\u2192 \u221e so that the metrics \u03c9\u03b1, j(0) = \u03b7 + i\u2202 \u2202\u00af\u03d5\u03b1, jare likewise bounded below for some fixed Carlson-Griffiths metric for all \u03b1 . Thisuniform lower bound will be key for our later proofs, and this is one reason for ourtwo parameters construction as opposed to a single parameter approximation as inTheorem A.2.1.Lemma 4.3.1. There exists \u03b1\u02c6 such that for all 0 < \u03b1 \u2264 \u03b1\u02c6 there exists a sequence\u03d5\u03b1, j \u2208 Psh(M,\u03b7) such that1. \u03d5\u03b1, j decreases to \u03b1 log\u2016S\u20162 + \u03d50 (as j \u2192 \u221e) pointwise and smoothly oncompact subsets of M.652. \u03c8\u03b1, j = \u03d5\u03b1, j +\u03b4 log log2 \u2016S\u20162 \u2208C\u221e(M)\u2229Psh(M,(1\u2212\u03b4 )\u03b7)Proof. Now i\u2202 \u2202\u00af log\u2016S\u20162 coincides with a smooth form on M and thus for \u03b1 > 0sufficiently small we have \u2212\u03b4\u03c9\u02c6 \u2264 \u03b1i\u2202 \u2202\u00af log\u2016S\u20162 \u2264 \u03b4\u03c9\u02c6 on M and it follows from(4.3.1) that(1\u2212\u03b4 )\u03b7+ i\u2202 \u2202\u00af (\u03b1 log\u2016S\u20162+\u03b4 log log2 \u2016S\u20162+\u03d50)\u2265 \u03b4\u03c9\u02c6 (4.3.2)In particular, since \u03d50 has zero Lelong number (see definition A.2.1) the potentialon the LHS of (4.3.2) approaches \u2212\u221e when approaching D giving\u03b1 log\u2016S\u20162+\u03b4 log log2 \u2016S\u20162+\u03d50 \u2208 Psh(M,(1\u2212\u03b4 )\u03b7) (4.3.3)for all sufficiently small \u03b1 > 0. Thus by Theorem A.2.1 there exists\u03c8\u03b1, j \u2208C\u221e(M)\u2229Psh(M,(1\u2212\u03b4 )\u03b7) decreasing to \u03b1 log\u2016S\u20162 +\u03b4 log log2 \u2016S\u20162 +\u03d50 as j\u2192 \u221e point-wise on M and smoothly on compact sets. In particular, we have\u03b7+ i\u2202 \u2202\u00af (\u2212\u03b4 log log2 \u2016S\u20162+\u03c8\u03b1, j) = \u03b4\u03c9\u02c6+(1\u2212\u03b4 )\u03b7+ i\u2202 \u2202\u00af\u03c8\u03b1, j > \u03b4\u03c9\u02c6 (4.3.4)so that \u03d5\u03b1, j :=\u2212\u03b4 log log2 \u2016S\u20162+\u03c8\u03b1, j \u2208 Psh(M,\u03b7) decreases to\u03b1 log\u2016S\u20162+\u03d50 as j\u2192 \u221e pointwise on M and smoothly on compact sets. Thus forall \u03b1 > 0 sufficiently small \u03d5\u03b1, j satisfies the conclusions in the lemma.Lemma 4.3.2. For each \u03d5\u03b1, j in Lemma 4.3.1, \u03c9\u03b1, j(0) = \u03b7+ i\u2202 \u2202\u00af\u03d5\u03b1, j is completewith bounded curvature on M with lower bound\u03c9\u03b1, j(0)\u2265 \u03b4\u03c9\u02c6 (4.3.5)and the Ka\u00a8hler Ricci flow (1.0.1) has a smooth maximal bounded curvature solu-tion \u03c9\u03b1, j(t) on M\u00d7 [0,T[\u03c90]) with initial condition \u03c9\u03b1, j(0) = \u03b7 + i\u2202 \u2202\u00af\u03d5\u03b1, j whereT[\u03c90] is as in definition 4.1.2.Proof. The lower bound in (4.3.5) follows immediately from (4.3.4). On the otherhand we can also write \u03b7 + i\u2202 \u2202\u00af\u03d5\u03b1, j = \u03b7 + i\u2202 \u2202\u00af (\u2212\u03b4 log log2 \u2016S\u20162 +\u03c8\u03b1, j) where66\u03c8\u03b1, j := \u03d5\u03b1, j + \u03b4 log log2 \u2016S\u20162 \u2208 C\u221e(M) as in Lemma 4.3.1 (2), and the Lemmathen follows from Theorem 8.19 in [23] (see also example 8.15).Consider \u03c9\u03b1, j(t) as in the above Lemma. In the following we will derive localestimates for \u03c9\u03b1, j(t) on M\u00d7 (0, T\u02dc ) which will be independent of \u03b1, j. These willensure \u03c9\u03b1, j(t) converges in C\u221eloc to a solution \u03c9(t) on M\u00d7 (0, T\u02dc ) as \u03b1 \u2192 0 andj\u2192 \u221e. Then we will show that \u03c9\u03b1, j(t) in fact converges in C\u221eloc on M\u00d7 [0, T\u02dc ) andthe limit solution is complete for a short time. Since T\u02dc < T[\u03c90] in Assumption 2was arbitrary, a diagonal argument will provide a solution on M\u00d7 [0,T[\u03c90]) as inTheorem 4.1.2. We may derive as in (4.2.2) that\u03c9\u03b1, j(t) = \u03b8t + i\u2202 \u2202\u00af\u03d5\u03b1, j(t) (4.3.6)where \u03d5\u03b1, j(t) solves the parabolic Monge Ampe`re equation:\uf8f1\uf8f2\uf8f3 \u2202t\u03d5\u03b1, j(t) = log\u2016S\u20162 log\u2016S\u20162(\u03b8t + i\u2202 \u2202\u00af\u03d5\u03b1, j(t))n\u2126;\u03d5\u03b1, j(0) = \u03d5\u03b1, j.\u03b8t := \u03b7+ t\u03c7; \u03c7 :=\u2212Ric(\u2126)+\u0398\u2212 i\u2202 \u2202\u00af log log2 \u2016S\u20162(4.3.7)on M\u00d7 [0,T[\u03c90]).We will derive local uniform bounds of |\u03d5\u03b1, j(t)| and |\u2202t\u03d5\u03b1, j(t)| on K\u00d7 [\u03b5, T\u02dc ]where K\u2282M is compact and 0< \u03b5 are arbitrary where the bounds will independentof \u03b1, j. We will use these to derive uniform local trace estimates for \u03c9\u03b1, j(t) whichcombined with the local Evans-Krylov estimates will yield the desired uniform C\u221elocestimates.4.3.2 A priori estimates for \u03c9\u03b1, j(t)Recall the choices for 0 < T < T\u02dc < T[\u03c90] and h\u02c6,\u2126, \u03b2\u02c6 in Assumption 2 and thenotation there. Recall also the definitions of \u03b8t and \u03c7 from (4.3.7). We fix some \u03b1\u02c6from Lemma 4.1 and will always assume that \u03b1 \u2264 \u03b1\u02c6 in the following.67Local C0 estimates of \u03d5\u03b1, j(t).From Lemma 4.3.1 there is a constant C and constant K(\u03b2\u02c6 ) such that\u03b2\u02c62log\u2016S\u20162\u2212K\u03b2\u02c6 \u2264 \u03d5\u03b1, j 0 and any \u03b1 \u2264 \u03b2\u02c6\/2 and jTheorem 4.3.1. There is a bounded continuous function U(t) on [0, T\u02dc ] such that\u03d5\u03b1, j(t)\u2264U(t) on M\u00d7 [0, T\u02dc ] for all \u03b1 and j. There is a continuous function L\u03b2\u02c6 (t)on [0,(1\u2212c\u03b2\u02c6 )T\u02dc ] such that 32 \u03b2\u02c6 log\u2016S\u20162+L\u03b2\u02c6 (t)\u2264 \u03d5\u03b1, j(t) on M\u00d7 [0,(1\u2212c\u03b2\u02c6 )T\u02dc ] forall \u03b1 \u2264 \u03b2\u02c6\/2 and all j.We first proveLemma 4.3.3. We have1. \u2016S\u20162 log2 \u2016S\u20162\u03b8 nt\u2126 \u2264C1(1+ t) for all t \u2208 [0, T\u02dc ];2. \u2016S\u20162 log2 \u2016S\u20162(\u03b8t\u2212\u03b2\u02c6\u0398)n\u2126 \u2265C2t for all t \u2208 [0,(1\u2212 c\u03b2\u02c6 )T\u02dc ],where the constants Ci > 0 depend on \u2126, h\u02c6 and T\u02dc .Proof. Now it suffices to show that C2t \u2264 \u2016S\u20162 log2 \u2016S\u20162\u03b8 nt\u2126 \u2264C1+C1t for all t \u2208 [0, T\u02dc ],where C is a constant depending only on h and T\u02dc , since by (2) Assumption 2 wehave for all t \u2208 [0,(1\u2212 c\u03b2\u02c6 )T\u02dc ]\u2016S\u20162 log2 \u2016S\u20162(\u03b8t \u2212 \u03b2\u02c6\u0398)n\u2126\u2265(1\u2212 c\u03b2\u02c6 )n\u2016S\u20162 log2 \u2016S\u20162\u03b8 n t1\u2212c\u03b2\u02c6\u2126\u2265 12n\u2016S\u20162 log2 \u2016S\u20162\u03b8 n t1\u2212c\u03b2\u02c6\u2126.68Now \u03b8t = \u03b8\u02dct + 2ti\u2202\u2016S\u20162\u2227\u2202\u00af\u2016S\u20162\u2016S\u20164 log2 \u2016S\u20162 where \u03b8\u02dct = \u03b7\u2212 tRic\u2126+ t\u0398+ 2t\u0398log\u2016S\u20162 . Thus \u03b8 nt = \u03b8\u02dc nt +n\u03b8\u02dc n\u22121t \u2227 (2ti\u2202\u2016S\u20162\u2227\u2202\u00af\u2016S\u20162\u2016S\u20164 log2 \u2016S\u20162 ) and\u2016S\u20162 log\u2016S\u20162\u03b8 nt = \u2016S\u20162 log2 \u2016S\u20162\u03b8\u02dc nt +n\u03b8\u02dc n\u22121t \u22272ti\u2202\u2016S\u20162\u2227 \u2202\u00af\u2016S\u20162\u2016S\u20162From this, the fact that 2i\u2202\u2016S\u20162\u2227\u2202\u00af\u2016S\u20162\u2016S\u20162 is a continuous positive (1,1) form on M\u00af, andthe positivity of \u03b8\u02dct on t \u2208 [0, T\u02dc ] we conclude C2t \u2264 \u2016S\u20162 log\u2016S\u20162\u03b8 nt\u2126 \u2264 C1 +C1t onM\u00d7 [0, T\u02dc ] as claimed.Proof of Theorem 4.3.1. For all \u03b1 \u2264 \u03b2\u02c6\/2 and j, considerH\u03b5 = \u03d5\u03b1, j(t)\u2212\u222b t0log[C1(1+ t)]dt\u2212 \u03b5ton M\u00d7 [0, T\u02dc ] for any \u03b5 > 0 and C1 from Lemma 4.3.3. Since H\u03b5(x,0) = \u03d5\u03b1, j isbounded above by (4.3.8) and |\u2202t\u03d5\u03b1, j(t)| and hence |\u2202tH\u03b5 | is bounded on M\u00d7 [0, T\u02dc ](by (4.3.7) and that \u03c9\u03b1, j(t) is a complete bounded curvature solution to (1.0.1)), itfollows H\u03b5 is bounded above on M\u00d7 [0, T\u02dc ]. Now suppose H\u03b5 attains a maximumvalue on M\u00d7 [0, T\u02dc ] at (x\u00af, t\u00af). Then if t\u00af > 0, using (4.3.7) and Lemma 4.3.3 we haveat (x\u00af, t\u00af):\u2202tH\u03b5 \u2264 log \u2016S\u20162 log\u2016S\u20162\u03b8 nt\u00af\u2126\u2212 log[C1(1+ t\u00af)]\u2212 \u03b5 \u2264\u2212\u03b5which contradics the maximality assumption. Thus t\u00af = 0 in which case we maysimply take U(t) = C+\u222b t0 log[C1(1+ t)]dt for some C by (4.3.8). In general, ifH\u03b5 does not attain a maximum value on M\u00d7 [0, T\u02dc ] we may argue as in the proofof Lemma 4.2.2 and apply the above estimates along an appropriate sequence inspace-time (using the Omori-Yau maximum principle) and likewise take U(t) =C+\u222b t0 log[C1(1+ t)]dt for some C in this case as well.For the lower bound we take Q\u03b5(x, t)=\u03d5\u03b1, j(x, t)\u2212 \u03b2\u02c6 log |S(x)|2\u2212\u222b t0 log(C2t)dt+\u03b5t on M\u00d7 [0,(1\u2212 c\u03b2\u02c6 )T\u02dc ] for any \u03b5 > 0 and C2 from Lemma 4.3.3. It follows from(4.3.8), (4.3.7) and that \u03c9\u03b1, j(t) is a bounded curvature solution that Q\u03b5(x, t)\u2192 \u221e69uniformly as x approaches D on M \u00d7 [0, T\u02dc ] and hence Q\u03b5(x, t) attains an inte-rior minimum on M\u00d7 [0,(1\u2212 c\u03b2\u02c6 )T\u02dc ]. Using again Lemma 4.3.3 we may argueas above for the upper bound and conclude that L\u03b2\u02c6 (t) can be taken as \u2212K\u03b2\u02c6 +\u222b t0 log(C2t)dt.Local C0 estimates of \u03d5\u02d9\u03b1, j(t).Theorem 4.3.2. We have \u03d5\u02d9\u03b1, j(t)\u2264 U(t)\u2212\u03b2\u02c6 log\u2016S\u20162+K\u03b2\u02c6t +n on M\u00d7 [0,(1\u2212 c\u03b2\u02c6 )T\u02dc ] forall \u03b1 \u2264 \u03b2\u02c6\/2 and all j.Proof. The proof is the same as Proposition 3.1 in [16]. Let H = t\u03d5\u02d9\u03b1, j(t)\u2212(\u03d5\u03b1, j(t)\u2212\u03d5\u03b1, j)\u2212 nt. Then using (4.3.7) we have (\u2202t \u2212\u2206)H < 0, where \u2206 is theLaplacian with respect to \u03c9\u03b1, j(t). Also, since that \u03c9\u03b1, j(t) is a bounded curvaturesolution it follows H is a bounded function on M\u00d7 [0, T\u02dc ], and thus by the maxi-mum principle in [27] we H \u2264 supx\u2208M H(x,0) = 0 on M\u00d7 [0, T\u02dc ]. Then combiningwith Lemma 4.3.3 and (4.3.8) we obtain the theorem.Theorem 4.3.3. For all A > 0 with 0 < T\u02dc \u2212 1A , there exists a smooth functionF(\u2016S\u20162(x), t) on M\u00d7 (0,(1\u2212 c\u03b2\u02c6 )(T\u02dc \u2212 1A)] such that \u03d5\u02d9\u03b1, j(x, t) \u2265 F(\u2016S\u20162(x), t) onM\u00d7 (0,(1\u2212 c\u03b2\u02c6 )(T\u02dc \u2212 1A)] for all \u03b1 \u2264 2\u03b2\u02c6 and all j.Proof. For all sufficiently small \u03b5 , there exists a constant C > 0 such that\u03b8t \u2212 s\u03b5i\u2202 \u2202\u00af log log2 \u2016S\u20162 \u2265C\u03c9\u02c6 (4.3.9)on M\u00d7 [0, T\u02dc ] for all 1\u2264 s\u2264 2Fix some A > 0 with 0 < T\u02dc \u2212 1A and \u03b5 as above. Let Q = \u03d5\u02d9\u03b1, j(t)+A(\u03d5\u03b1, j(t)\u2212\u03b2\u02c6 log\u2016S\u20162+\u03b5 log log2 \u2016S\u20162)\u2212n log t. By our previous bounds, Q\u2192\u221e on M as t\u21920 or \u2016S\u2016\u2192 0. In fact, from (4.3.7) and that \u03c9\u03b1, j(t) is a bounded curvature solution,Q(x, t)\u2192 \u221e uniformly on M as x approaches D for all t \u2208 [0,(1\u2212 c\u03b2\u02c6 )(T\u02dc \u2212 1A)]. SoQ has a minimum on M\u00d7 (0,(1\u2212 c\u03b2\u02c6 )(T\u02dc \u2212 1A)] at some point (x\u00af, t\u00af) with t\u00af > 0. Let\u2206 be the Laplacian with respect to \u03c9\u03b1, j(t), using (4.3.7), we have70(\u2202t \u2212\u2206)(\u03d5\u03b1, j(t)\u2212 \u03b2\u02c6 log\u2016S\u20162+ \u03b5 log log2 \u2016S\u20162= \u03d5\u02d9\u03b1, j\u2212n+Tr\u03c9\u03b1, j(\u03b8t \u2212 \u03b2\u02c6\u0398\u2212 \u03b5i\u2202 \u2202\u00af log log2 \u2016S\u20162)(\u2202t \u2212\u2206)\u03d5\u02d9\u03b1, j(t) = Tr\u03c9\u03b1, j\u03c7.Then at (x\u00af, t\u00af) we have the following, where we will use C to denote a constantwhich is independent of \u03b1 , j and which may differ from line to line.0 \u2265 (\u2202t \u2212\u2206)Q(x\u00af, t\u00af)= ATr\u03c9\u03b1, j(\u03b8t\u00af+ 1A \u2212 \u03b2\u02c6\u0398\u2212 \u03b5i\u2202 \u2202\u00af log log2 \u2016S\u20162)+A\u03d5\u02d9\u03b1, j\u2212nA\u2212 nt\u00af\u2265 A(1\u2212 c\u03b2\u02c6 )Tr\u03c9\u03b1, j [\u03b8 11\u2212c\u03b2\u02c6 (t\u00af+1A )\u2212 11\u2212 c\u03b2\u02c6\u03b5i\u2202 \u2202\u00af log log2 \u2016S\u20162]+A\u03d5\u02d9\u03b1, j\u2212nA\u2212 nt\u00af.\u2265 AC(1\u2212 c\u03b2\u02c6 )Tr\u03c9\u03b1, j \u03c9\u02c6+A log\u2016S\u20162 log\u2016S\u20162\u03c9n\u03b1, j\u2126\u2212nA\u2212 nt\u00af\u2265 CTr\u03c9\u03b1, j\u03c9\u02c6 +C log\u03c9n\u03b1, j\u03c9\u02c6n\u2212Ct\u00af.\u2265 CTr\u03c9\u03b1, j \u03c9\u02c6\u2212Ct\u00af\u2265 C(\u03c9\u02c6n\u03c9n\u03b1, j) 1n\u2212Ct\u00af.where we have used Assumption 2 in the third line, c\u03b2\u02c6 \u2264 12 and (4.3.9) in the fourthline, property (3) of Carlson-Griffiths metrics, and the fact 1\u03bb +C log\u03bb is boundedbelow by some constant depending on C in the sixth line. Therefore, at (x\u00af, t\u00af),\u03c9n\u03b1, j \u2265Ct\u00afn\u03c9\u02c6n and so \u03d5\u02d9\u03b1, j(x\u00af, t\u00af)\u2265C+n log t\u00af by (4.3.7) and property (3) of Carlson-Griffiths metrics. Since log log2 \u2016S\u20162 > 1 by Assumption 2, we have Q(x\u00af, t\u00af) \u2265C+A(\u03d5\u03b1, j(x\u00af, t\u00af)\u2212 \u03b2\u02c6 log |S(x\u00af)|2). By Theorem 4.3.1, \u03d5\u03b1, j(t)\u2212 \u03b2\u02c6 log\u2016S\u20162 \u2265C andso Q(x\u00af, t\u00af)\u2265C. From this, and the upper bound of \u03d5\u03b1, j(t) from Theorem 4.3.1, weconclude the lower bound for \u03d5\u02d9\u03b1, j(t) in the Theorem.71Local trace estimates for \u03c9\u03b1, j(t)Note for all \u03b1 and j, since \u03c9\u03b1, j(t) is a bounded curvature solution on M\u00d7 [0,T[\u03c90]),so \u03c9\u03b1, j(t) will be uniformly equivalent to \u03c9\u02c6 on any closed subinterval of [0,T[\u03c90]).In particular, Tr\u03c9\u02c6\u03c9\u03b1, j(t) will be a bounded function on M\u00d7 [0,T ]Theorem 4.3.4. There is a smooth function G(\u2016S\u20162(x), t) on M\u00d7 (0,(1\u2212 c\u03b2\u02c6 )T\u02dc ]such that Tr\u03c9\u02c6\u03c9\u03b1, j(x, t)\u2264 G(\u2016S\u20162(x), t) for all 2\u03b1 \u2264 \u03b2\u02c6 and all j.Proof. ConsiderQ(\u00b7, t) = t logTr\u03c9\u02c6\u03c9\u03b1, j(t)\u2212B(\u03d5\u03b1, j(t)\u2212 \u03b2\u02c6 log\u2016S\u20162+ \u03b5 log log2 \u2016S\u20162),where \u03b5 is chosen as in (4.3.9) and B > 0 is a large constant which will be de-termined later, independently of \u03b1, j. Now Q(x,0) \u2192 \u2212\u221e as x approaches Dfrom (4.9), and from (4.3.7) and that \u03c9\u03b1, j(t) is a bounded curvature solution,Q(x, t)\u2192\u2212\u221e uniformly as x approaches D for all t \u2208 [0,(1\u2212c\u03b2\u02c6 )T\u02dc ]. Hence Q(x, t)attains a maximum on M\u00d7 [0,(1\u2212c\u03b2\u02c6 )T\u02dc ] at some point (x\u00af, t\u00af). In the following, Ci\u2019swill denote positive constants independent of \u03b1, j.If t\u00af > 0, then 0 \u2264 (\u2202t \u2212\u2206)Q(x\u00af, t\u00af), where \u2206 is the Laplacian with respect to\u03c9\u03b1, j(t). Also, we have(\u2202t \u2212\u2206) logTr\u03c9\u02c6\u03c9\u03b1, j(t)\u2264C1Tr\u03c9\u03b1, j \u03c9\u02c6 (4.3.10)for some constant C1 depending only on \u03c9\u02c6 (see [28]), so(\u2202t \u2212\u2206)t logTr\u03c9\u02c6\u03c9\u03b1, j(t) = logTr\u03c9\u02c6\u03c9\u03b1, j(t)+ t(\u2202t \u2212\u2206) logTr\u03c9\u02c6\u03c9\u03b1, j(t)\u2264 logTr\u03c9\u02c6\u03c9\u03b1, j(t)+C1tTr\u03c9\u03b1, j(t)\u03c9\u02c6.Using the computations in the proof of Theorem 4.3.3, we have(\u2202t \u2212\u2206)(\u03d5\u03b1, j(t)\u2212 \u03b2\u02c6 log\u2016S\u20162+ \u03b5 log log2 \u2016S\u20162)\u2265 \u03d5\u02d9\u03b1, j\u2212n+(1\u2212 c\u03b2\u02c6 )C2Tr\u03c9\u03b1, j \u03c9\u02c6.72Therefore at (x\u00af, t\u00af), using that c\u03b2 < 1\/2 from Assumption 2, we have0\u2264 (\u2202t \u2212\u2206)Q\u2264 logTr\u03c9\u02c6\u03c9\u03b1, j\u2212B\u03d5\u02d9\u03b1, j +nB+(C1t\u2212 12BC2)Tr\u03c9\u03b1, j \u03c9\u02c6\u2264 logTr\u03c9\u02c6\u03c9\u03b1, j\u2212Tr\u03c9\u03b1, j \u03c9\u02c6\u2212B\u03d5\u02d9\u03b1, j +nBwhere in the second line we have assumed a choice B, independent of \u03b1, j and t\u00af,such that C1T\u02dc \u2212 12 BC2 <\u22121.Since Tr\u03c9\u02c6\u03c9\u03b1, j \u2264 (Tr\u03c9\u03b1, j \u03c9\u02c6)n\u22121\u03c9n\u03b1, j\u03c9\u02c6n and \u03d5\u02d9a, j \u2265C3 log\u03c9n\u03b1, j\u03c9\u02c6n for some C3 depend-ing only on h and \u03c9\u02c6 , putting them into the above expression, we get0\u2264 (n\u22121) logTr\u03c9\u03b1, j \u03c9\u02c6+(1\u2212BC3) log\u03c9n\u03b1, j\u03c9\u02c6n\u2212Tr\u03c9\u03b1, j \u03c9\u02c6+C4.Assume further that BC3 > 2, we have0 \u2264 \u2212Tr\u03c9\u03b1, j \u03c9\u02c6+(n\u22121) logTr\u03c9\u03b1, j \u03c9\u02c6\u2212 log\u03c9n\u03b1, j\u03c9\u02c6n+C4\u2264 \u221212Tr\u03c9\u03b1, j \u03c9\u02c6\u2212 log\u03c9n\u03b1, j\u03c9\u02c6n+C4,we used \u2212x+C logx is bounded above for x > 0 by some constant depending onC. Now let \u03bbi be the eigenvalue of \u03c9\u03b1, j(x\u00af, t\u00af) relative to \u03c9\u02c6(x\u00af, t\u00af) and let C denote apositive constant independent of \u03b1, j which may differ from line to line. Then theprevious equation says\u2211i(12\u03bbi+ log\u03bbi)\u2264Cand from the fact that the function 1\/2x+ logx is bounded below for all x > 0,we get that ( 12\u03bbi + log\u03bbi) \u2264C for each i, thus Tr\u03c9\u02c6\u03c9\u03b1, j(x\u00af, t\u00af) \u2264C. Since \u03d5\u03b1, j(t)\u2212\u03b2\u02c6 log\u2016S\u20162\u2265C and \u03b5 log log2 \u2016S\u20162\u2265 0 we conclude Q(x\u00af, t\u00af)\u2264C. Thus by our earlierobserved upper bound for Q(x,0) we get Q(x, t)\u2264C on M\u00d7 [0,(1\u2212c\u03b2\u02c6 )T\u02dc ] and theTheorem follows from this and the upper bound in Theorem 4.3.1.4.3.3 Completion of Proof of Theorem 4.1.2Now recall our family \u03c9\u03b1, j(t) of solutions to (1.0.1) on M\u00d7 [0,T[\u03c90]) from Lemma4.3.2. Recall that we wrote \u03c9\u03b1, j(t) = \u03b8t + i\u2202 \u2202\u00af\u03d5\u03b1, j(t) where \u03d5\u03b1, j(t) solves (4.3.7)73on M\u00d7 [0,T[\u03c90]). Also recall the choices made in Assumption 2, and in particularthat 0 < T < T\u02dc < T[\u03c90] was arbitrary.From the Theorem 4.3.3, 4.3.4 and (4.3.7), for any \u03b5 > 0 and compact subsetsK1 \u2282\u2282 K2 \u2282\u2282M we may haveC1\u03b7 \u2264 \u03c9\u03b1, j(t)\u2264C2\u03b7 (4.3.11)on K2\u00d7 [\u03b5,T ] for some constants Ci independent over all \u03b1 \u2264 \u03b2\u02c6\/2 and all j. Itfollows from this and the estimates from the Evans-Krylov theory (see also [32] fora maximum principle proof of these for (1.0.1)), that for some \u03b1k\u2192 0, jk\u2192\u221e, wehave \u03c9\u03b1k, jk(t) converges on K1\u00d7 [\u03b5,T ] smoothly to a limit solution \u03c9(t) to the flowin equation (1.0.1). As T < T[\u03c90] was chosen arbitrarily, by a diagonal argumentwe may in fact assume \u03c9\u03b1k, jk(t) converges on M\u00d7 (0,T[\u03c90]), smoothly on compactsubsets, to a limit solution \u03c9(t) to the flow in (1.0.1), while also \u03c9\u03b1k, jk(0)\u2192 \u03c90smoothly on compact subsets of M. By applying Theorem 2.3.1 to the sequence\u03c9\u03b1k, jk(t) on M\u00d7 [0,T[\u03c90]) and observing the uniform lower bound on \u03c9\u03b1k, jk(0) \u2265\u03b4\u03c9\u02c6 in (4.3.5), we see that \u03c9\u03b1k, jk(t) actually converges smoothly on M\u00d7 [0,T[\u03c90])to a limit solution satisfying (4.1.5). In other words the solution \u03c9(t) extendssmoothly on M\u00d7 [0,T[\u03c90]) and satisfies (4.1.5).We now show (1) in the Theorem 4.1.2 is satisfied. Fix any Hermitian metric hon OD and smooth volume form \u2126 on M. Then as in our derivation of (4.3.7)we see that \u03d5(t) := \u03d50 +\u222b t0 log log\u2016S\u20162h log\u2016S\u20162h(\u03c9(t))n\u2126solves (4.1.6) on M \u00d7[0,T[\u03c90]) and (4.1.7). In particular, \u03d5(t) = limk\u2192\u221eu\u03b1k, jk(t) where u\u03b1k, jk(t) = \u03d5\u03b1k, jk +\u222b t0 log log\u2016S\u20162h log\u2016S\u20162h(\u03c9\u03b1k, jk(t))n\u2126and u\u03b1k, jk(t) solves (4.1.6) on M\u00d7[0,T[\u03c90])withinitial data \u03d5\u03b1k, jk . To see that the upper bound in (1) holds, note that the estimatein Lemma 4.3.3 (1) in fact holds for any, and hence our, choice of h for some con-stant C1. Then from the proof of the upper bound in Theorem 4.3.1, there existsa continuous function U(t) such that u\u03b1k, jk(t) \u2264 U(t) and hence \u03d5(t) \u2264 U(t) onM\u00d7 [0,T[\u03c90]). This completes the proof of (1) in the Theorem.Finally, we show that (2) holds. Let \u03d50 be as in (2). For any choice of 0 0 we haveC\u221211 d(p,x)\u2264 \u03b3(x)\u2264C1d(p, \u00b7) whenever d(p,x)>C2, where d(p, \u00b7) is the distancefunction from p on (M,\u03c9). We begin by proving the following Theorem:Theorem 4.4.1. Let (M, \u03c9\u02c6) be a complete Ka\u00a8hler manifold with bounded cur-vature. Let \u03b3 : M \u2192 R be a smooth distance-like function with |\u2207\u02c6\u03b3|\u03c9\u02c6 < C and|i\u2202 \u2202\u00af \u03b3|\u03c9\u02c6 < C for some constant C. Let \u03d5 \u2208 C\u221e(M) such that |\u03d5|\/\u03b3 \u2192 0 and|\u2207\u02c6\u03d5|\u03c9\u02c6\/\u03b3 \u2192 0 as \u03b3 \u2192 0. If \u03c9 = \u03c9\u02c6 + i\u2202 \u2202\u00af\u03d5 is a complete metric with boundedcurvature and satisfies |\u03c9\u2212 \u03c9\u02c6|\u03c9\u02c6 \u2192 0 as \u03b3 \u2192 0, then T (\u03c9) = T (\u03c9\u02c6).75Proof. Let \u03c1 : R\u2192 R be a smooth function such that \u03c1 = 1 on [0,1] and \u03c1 = 0 on[2,\u221e). Define \u03c1R : M\u2192R by \u03c1R = \u03c1(\u03b3\/R) and let \u03c9R = \u03c9\u02c6+ i\u2202 \u2202\u00af (\u03c1R\u03d5). We claimthat if R is sufficiently large then \u03c9R is a complete Ka\u00a8hler metric and there existsCR\u2192 1 as R\u2192 \u221e such that 1CR\u03c9 \u2264 \u03c9R \u2264CR\u03c9 .We have \u03c9R = \u03c1R\u03c9 + (1\u2212 \u03c1R)\u03c9\u02c6 + 2Re(i\u2202\u03c1R \u2227 \u2202\u00af\u03d5) + i\u03d5\u2202 \u2202\u00af\u03c1R. Since |\u03c9 \u2212\u03c9\u02c6|\u03c9\u02c6 \u2192 0 as \u03b3 \u2192 \u221e, we have 1CR\u03c9 \u2264 \u03c1R\u03c9+(1\u2212\u03c1R)\u03c9\u02c6 \u2264CR\u03c9 for some CR\u2192 1 asR\u2192 \u221e. Now it suffices to show that |2Re(i\u2202\u03c1R\u2227 \u2202\u00af\u03d5)|\u03c9\u02c6 \u2192 0 and |i\u03d5\u2202 \u2202\u00af\u03c1R|\u03c9\u02c6 \u2192 0uniformly on M as R\u2192 \u221e.For any point in M, we have|2Re(i\u2202\u03c1R\u2227 \u2202\u00af\u03d5)|\u03c9\u02c6 \u2264 |\u03c1 \u2032( \u03b3R)R\u2202\u03b3 \u2227 \u2202\u00af\u03d5|\u03c9\u02c6\u2264 |\u03c1\u2032( \u03b3R)|R|\u2207\u02c6\u03b3|\u03c9\u02c6 |\u2207\u02c6\u03d5|\u03c9\u02c6\u2264C(maxR|\u03c1 \u2032|)\u03c7\u03b3\u22121[R,2R]R|\u2207\u03d5|\u03c9\u02c6\u2264 2C(maxR|\u03c1 \u2032|)\u03c7\u03b3\u22121[R,2R]|\u2207\u03d5|\u03c9\u02c6\u03b3.Because |\u2207\u02c6\u03d5|\u03c9\u02c6\/\u03b3\u2192 0 as \u03b3\u2192\u221e, the function on the right hand side convergesuniformly to 0 as R\u2192 \u221e. Similar argument works for|i\u03d5\u2202 \u2202\u00af\u03c1R|\u03c9\u02c6 = |\u03d5\u03c1 \u2032( \u03b3R)i\u2202 \u2202\u00af \u03b3R+\u03d5\u03c1 \u2032\u2032(\u03b3R)i\u2202\u03b3 \u2227 \u2202\u00af \u03b3R2|\u03c9\u02c6 .Therefore, we have a family of complete Ka\u00a8hler metrics \u03c9R such that 1CR\u03c9 \u2264\u03c9R \u2264 CR\u03c9 with CR \u2192 1 as R\u2192 \u221e and it is clear that \u03c9R has bounded curvature.Therefore, by Theorem 2.4.2, we have 1CR T (\u03c9R)\u2264 T (\u03c9)\u2264CRT (\u03c9R). On the otherhand, since \u03c1R\u03d5 has compact support, by Theorem 4.1 in [23], we have T (\u03c9R) =T (\u03c9\u02c6) for all R. Therefore, passing the limit R\u2192 \u221e we obtain T (\u03c9) = T (\u03c9\u02c6).Proof of Theorem 4.1.3. The uniqueness of bounded curvature solutions followsfrom [13]. Let p \u2208M and let d\u03c9\u02c6(p, \u00b7) be the distance function to p relative to \u03c9\u02c6 .76Let \u03b3(x) := log log2 |S(x)|2 on M. Then from (4.1.1) we may write\u03c9\u02c6 = \u03b7\u00af\u2212 i\u2202 \u2202\u00af \u03b3(x) = \u03b7\u22122ddc log\u2016S\u20162hlog\u2016S\u20162h+2i\u2202\u03b3 \u2227 \u2202\u00af \u03b3.Noting that \u03b7 as well as the numerator of the second term above are smooth formson M, we see that for all x \u2208M sufficiently close to D, or equivalently when \u03b3(x) issufficiently large, we have C\u22121\u03b3(x)\u2264 d\u03c9\u02c6(p,x)\u2264C\u03b3(x) and \u2016d\u03b3(x)\u2016\u03c9\u02c6 0, and from this and the first equality above we may concludethat \u2016i\u2202 \u2202\u00af \u03b3(x)\u2016\u03c9\u02c6 \u2264C for some C independent of x. In other words, \u03b3 satisfies theassumption in Theorem 4.4.1 relative to \u03c9\u02c6 , and Theorem follows immediately.Remark 4.4.1. In Theorem 4.1.3 we can remove the condition on d\u03d5 if we assume\u03c90 has the same standard spatial asymptotics as that of \u03c9\u02c6 as defined in [23]. 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Weinkove, Interior derivative estimates for the Khler-Ricciflow. Pacific Journal of Mathematics, 257(2), 491-501. (2012) \u2192 pages 74,82, 83[33] H.-H Wu, F. Zheng, Examples of positively curved complete Ka\u00a8hler mani-fold, Geometry and Analysis Volume I, Advanced Lecture in Mathematics 17,80Higher Education Press and International Press, Beijing and Boston, 2010,pp. 517542 \u2192 pages 5, 28, 29, 30, 47[34] B. Yang, On a problem of Yau regarding a higher dimensional generalizationof the Cohn-Vossen inequality, Math. Ann. 355(2) (2013), 765-781. \u2192 pages29[35] B. Yang, F. Zheng, U(n)-invariant Ka\u00a8hler-Ricci flow with non-negative cur-vature, Comm. Anal. Geom . 21, (2013) no. 2, 251\u2013294. \u2192 pages 5, 3081AppendixA.1 Interior estimatesLet us first fix some notations and terminology. (Mn, g\u02c6) is said to have boundedgeometry of infinite order if the curvature tensor and all its covariant derivativesare uniformly bounded on M. In particular, the solution g(t) in Theorem 2.1.2 hasbounded geometry of infinite order for t > 0.Also, we will denote the geodesic ball with respect to the metric g with centerat p and radius r by Bg(p,r). The following theorem can be found in [32].Theorem A.1.1. Let (Mn, g\u02c6) be a complete non-compact Ka\u00a8hler manifold withbounded geometry of infinite order. Let h(t) be a solution of Ka\u00a8hler-Ricci (1.0.1)on M\u00d7 [0,T ) with initial condition h0 which is a complete Ka\u00a8hler metric. For anyx \u2208M, suppose there is a constant N > 0, such thatN\u22121g\u02c6\u2264 h(t)\u2264 Ng\u02c6 (A.1.1)on Bg\u02c6(x,1)\u00d7 [0,T ). Let \u2207\u02c6 be the covariant derivative with respect to g\u02c6. Then(i)|\u2207\u02c6kh|2g\u02c6 \u2264Cktkon Bg\u02c6(x,1\/2)\u00d7 (0,T ), for some constant Ck depending only on k, g\u02c6, n, Tand N.(ii) If we assume |\u2207\u02c6kh0|2g\u02c6 is bounded in Bg\u02c6(x,1) by ck, for k \u2265 1, then|\u2207\u02c6kh|2g\u02c6 \u2264Ck,82on Bg\u02c6(x,1\/2)\u00d7 [0,T ) for some constant Ck depending only on k, ck, n, T andN.Proof. Since g\u02c6 has bounded geometry of infinite order, by [30], for any x\u2208M thereexists a local biholomorphism \u03c6x : D\u2192M, where D=D(1) is the open unit ball inCn, satisfying the following in D(a) \u03c6x(0) = x, \u03c6x(D)\u2282 B\u02c6(x,1), \u03c6x(D)\u2283 B\u02c6(x,2\u03b4 ) for some \u03b4 > 0 which is inde-pendent of x.(b) C\u22121\u03b4i j\u00af \u2264 (\u03c6 \u2217x (g\u02c6))i j\u00af \u2264C\u03b4i j\u00af for some C independent of x.(c)\u2223\u2223\u2223\u2223\u2223\u2202 l(\u03c6 \u2217x (g\u02c6))i j\u00af\u2202 zL\u2223\u2223\u2223\u2223\u2223\u2264Cl for any l, i, j and multi index L of length l for some con-stant Cl which is independent of x.Consider \u03c6 \u2217x (h(t)), which clearly will solve (1.0.1) on D(1)\u00d7 [0,T ). By theEvans-Krylov theory for fully non-linear elliptic and parabolic equations (see also[32] for a maximum principle proof in the case of Ka\u00a8hler Ricci flow), the resultfollows.A.2 Plurisubharmonic functionsLet M be a compact complex manifold with smooth Ka\u00a8hler metric \u03b7 .Definition A.2.1. \u03d5 is called plurisubharmonic on the compact Ka\u00a8hler manifold(M,\u03b7), written \u03d5 \u2208 Psh(M,\u03b7), when \u03d5 : M \u2192 R is upper semi-continuous andbounded above, and for any local holomorphic coordinate domain U\u03b1 , \u03b7\u03b1 +\u03d5 is aclassical plurisubharmonic function in U\u03b1 where \u03b7\u03b1 is a local Ka\u00a8hler potential. Inthis context \u03d5 \u2208C\u221e(M)\u22c2Psh(M,\u03b7) is said to have zero Lelong number if for anyc > 0 we havelimd(x,D)\u21920\u03d5(x)c log\u2016S\u2016h \u2192 0where d(\u00b7,D) is the distance to D\u2282M relative to \u03b7 .Let \u03d5 \u2208 Psh(M,\u03b7)) be given. By [15], or [1] for a simpler proof in our setting,there exists a decreasing sequence \u03d5 j \u2208C\u221e(M)\u22c2Psh(M,\u03b7) converging pointwise83to \u03d5 . By a slight modification of the proof in [1], we may assume this convergenceactually holds in C\u221eloc(M) when \u03d5 \u2208 C\u221e(M). We include the statement and proofof this below for completeness.Theorem A.2.1. Suppose that \u03d5 \u2208 Psh(M,\u03b7)\u2229C\u221e(M). Then there exists a se-quence \u03d5 j \u2208C\u221e(M) with \u03d5 j \u2193 \u03d5 pointwise on M and smoothly uniformly on com-pact subsets of M.Proof. Let S j be an increasing sequence of open sets exhausting M where eachS j is compact. Let m j = min{\u03d5(x) : x \u2208 S j} and define \u03d5 j = max{\u03d5,m j}+ 1j .Then \u03d5 j \u2208C0(M)\u22c2C\u221e(S j)\u22c2Psh(M,\u03b7) with \u03d5 j \u2193 \u03d5 pointwise on M and smoothlyuniformly on compact subsets. Now for each j, suppose there exists a sequence\u03d5 j,k \u2208 Psh(M,\u03b7)\u2229C\u221e(M) with \u03d5 j,k \u2193 \u03d5 j pointwise uniformly on M and smoothlyuniformly on S j\u22121. Then for any diverging sequence a j, we have \u03d5 j,a j \u2193 \u03d5 point-wise on M and smoothly uniformly on compact subsets. Moreover, by the fact\u03d5 j\u2212\u03d5 j+1\u2265 1j\u2212 1j+1 , it is clear that we may choose some sequence a j with \u03d5 j,a j \u2193\u03d5 .From the above, it suffices now to prove the followingCLAIM: if \u03d5 \u2208 C0(M)\u22c2C\u221e(U)\u22c2Psh(M,\u03b7) for some open set U and V is aprecompact open subset of U , there exists a sequence \u03d5 j \u2208 C\u221e(M)\u22c2Psh(M,\u03b7)with \u03d5 j \u2193 \u03d5 pointwise on M and smoothly uniformly on V .The claim follows from a slight modification of the proof of the main Theo-rem in [1]. In [1], an arbitrary open cover U\u03b1 of M is first chosen, then in eachU\u03b1 a smooth local approximation \u03d5\u03b1,\u03b4 of \u03d5 is constructed through the use of localKa\u00a8hler potentials and mollification. Then for fixed \u03b4 , a global smooth approxima-tion of \u03d5 on M is defined as the pointwise regularized maximum (from [15]) of the\u03d5\u03b1,\u03b4 \u2019s where the maximum is taken over all \u03b1 . Then, by letting \u03b4 \u2192 0, it is shownthere exists a sequence \u03d5 j \u2208C\u221e(M) with \u03d5 j \u2193 \u03d5 pointwise on M. To have smoothconvergence uniformly on V we modify this construction slightly as follows.First we choose some finite open cover {V\u03b1}k\u03b1=1 of the compact set M\\U .Moreover, by compactness we may assume for each \u03b1 > 0 we have a proper in-clusion of open sets V\u03b1 \u2282U\u03b1 \u2282W\u03b1 where: W\u03b1\u22c2V = \/0 and W\u03b1 is a holomorphiccoordinate neighborhood with smooth local Ka\u00a8hler potential \u03b7 = i\u2202 \u2202\u00af f\u03b1 . Thenletting U0 :=U we take {U\u03b1}k\u03b1=0 as our open cover of M.84Next we define local approximations \u03d5\u03b1,\u03b4 of \u03d5 on each U\u03b1 . For \u03b1 = 0, let g\u03b1be a smooth function which is equal to 0 in U\\\u222a\u03b1 6=0 V\u03b1 and equals \u22121 outsidesome compact subset of U . For all \u03b1 6= 0, let g\u03b1 be a smooth function in U\u03b1such that g\u03b1 = 0 in V\u03b1 and g\u03b1 =\u22121 outside some compact subset of U\u03b1 . Assumethat i\u2202 \u2202\u00afg\u03b1 \u2265 \u2212C\u03c9 for some C independent of \u03b1 . Now we define \u03d5\u03b1,\u03b4 on U\u03b1 asfollows. If \u03b1 6= 0, then as in [1] we let\u03d5\u03b1,\u03b4 = u\u03b1,\u03b4 \u2212 f\u03b1 +\u03b5Cg\u03b1where u\u03b1,\u03b4 is a mollification of u\u03b1 := \u03d5+ f\u03b1 \u2208 Psh(U\u03b1) in W\u03b1 so that u\u03b1,\u03b4 \u2193 u\u03b1 as\u03b4 \u2192 0. If \u03b1 = 0, then we let\u03d50,\u03b4 = \u03d5+\u03b5Cg0In both cases, we have \u03d5\u03b1,\u03b4 \u2208 Psh(U\u03b1 ,(1+ \u03b5)\u03b7) and it is non-increasing as \u03b4decreases.Now given our open cover {U\u03b1}k\u03b1=0, and local approximations \u03d5\u03b1,\u03b4 in these,the proof of the claim follows exactly as in [1] involving the regularized maximum(from [15]) of the \u03d5\u03b1,\u03b4 \u2019s where the maximum is taken over all \u03b1 . Then, by letting\u03b4 \u2192 0. 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