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Collapsing fibres under Kähler Ricci flow on Hirzebruch manifolds Dixon, Kael 2010

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Collapsing Fibres under Kähler Ricci Flow on Hirzebruch Manifolds by Kael Dixon  B.Math., The University of Waterloo, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 c Kael Dixon 2010  Abstract In this article we study the Kähler Ricci flow on a class of CP1 bundles over CPn−1 known as Hirzebruch manifolds. These are defined by Mn,k := P(H k ⊕ CPn−1 ) where H is the canonical line bundle and n, k ∈ N (we refer to §2 for a detailed description of these). We follow the work in [11], where Song and Weinkove study solutions to the Kähler Ricci flow for a Calabi symmetric Kähler metrics on Hirzebruch manifolds (see §2 for definitions). They were able to show that, depending on the initial Kähler class, the Ricci flow would reach a finite time singularity corresponding to the manifold either shrinking to a point, contracting the zero section to a point, or collapsing the fibres. In this paper, we investigate how the fibres collapse in the latter case with the further assumptions that the singularity is formed at a type I rate, and that the length of a generic vector does not decay too quickly in some sense. In this case we show that the fibres converge to round spheres after blowing up around a singular point on a fibre.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Kähler Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . .  1 1  2 Hirzebruch Manifolds . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . 2.2 Local Coordinates on Mn,k . . . . . 2.3 Radially Symmetric Metrics . . . . 2.4 A Closer Look at the Fibres . . . .  . . . . .  3 3 4 4 6  3 Singularity Analysis for Ricci flow . . . . . . . . . . . . . . . 3.1 Parabolic Dilations . . . . . . . . . . . . . . . . . . . . . . .  8 11  4 The Singularity Model . . . . . . . . . . . . . . . . . . . . . . 4.1 A Candidate for the Limit Manifold . . . . . . . . . . . . . . 4.2 An Isometry Between M∞ and Cn−1 × CP1 . . . . . . . . . .  13 15 17  5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 5.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21 21 21  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  iii  Acknowledgements I gratefully aknowledge the support of the Natural Sciences and Engineering Research Council and the University of British Columbia. I wish to thank my supervisor Dr. Albert Chau for all that he has done to help introduce me to this interesting field of Ricci flow, including many of the ideas in this paper. Thanks to my family for their constant support and encouragement.  iv  Chapter 1  Introduction The goal of the thisis is to prove the following theorem: Theorem 1. Assume (Mn,k , g(t)) is radially symmetric solution to the Kähler Ricci flow (1.1) on the Hirzebruch manifold Mn,k with initial metric as in case 3 described in §2.1. Suppose the solution develops a Type I singularity at time T and that for some p ∈ Mn,k \{D0 , D∞ } and vector X ∈ Tp Mn,k tangent to the fibre, there exists c > 0 such that kXk2g(t) ≥ c(T − t). Then for any p ∈ Mn,k , the singularity model at p is a Riemannian product (Cn−1 × CP1 , δCn−1 + h(t)) where δCn−1 is the standard flat metric on Cn−1 and (CP1 , h(t)) is the standard shrinking sphere soliton. We refer to §1.1, §2 and §3 for the definitions, terms and notations in Theorem 1.  1.1  Kähler Ricci Flow  The Ricci flow is the following system of PDE’s on the space of metrics of a manifold: ∂g (t) = −Ricg(t) ∂t (1.1) g(0) = g0 , This equation is only weakly parabolic, yet local existence and uniqueness of smooth solutions given smooth initial metrics was proved by Hamilton in [4]. We say that T > 0 is the singular time of a solution to Ricci flow if [0, T ) is a maximal interval on which a smooth solution to the flow exists. If T < ∞, we say that the solution forms a finite time singularity. Kähler Ricci flow is simply the study of Ricci flow in the case when the initial metric g0 is Kähler. In this case, solution g(t) will remain Kähler as long as the solution exists (See [7]) and thus induces the corresponding flow of Kähler forms: d dt ω(t)  = −Ric∧ ω(t)  ω(0) = ω0  (1.2) 1  1.1. Kähler Ricci Flow where ω is the Kähler form of g and Ric∧ is the Ricci form. Since both of these are closed real-valued (1,1)-forms, this system descends to the following flow on H 2 (M, R) ∩ H 1,1 (M ): d dt [ω(t)]  = −[Ric∧ ω(t) ]  [ω(0)] = [ω0 ]  (1.3)  We call the set of all cohomology classes corresponding to Kähler metrics the Kähler cone: {α ∈ H 2 (M, R) ∩ H 1,1 (M ) : α > 0}, where we write α > 0 if there exists a form τ > 0 such that [τ ] = α. Since ω is Kähler, [Ric∧ ω ] = 2πc1 (M ), where c1 (M ) is the first Chern class of M which is an invariant of the complex structure, and thus is independent of ω and in particular of t. In particular the family of classes α(t) := [ω0 ] − 2πtc1 (M ) clearly solves the above cohomological equation. Conversely, as described in [11], if α(t) above lies in the Kähler cone for all t ∈ [0, T ), then (1.2) will in fact have a solution ω(t) defined for t ∈ [0, T ). It follows that if T is finite, then α(T ) lies on the boundry of the Kähler cone.  2  Chapter 2  Hirzebruch Manifolds 2.1  Introduction  Let Mn,k := P(H ⊗k ⊕ CPn−1 ) be an n−dimensional Hirzebruch manifold, where H is the canonical line bundle over CPn−1 (see §2.2 for detailed description). It is known that in this case the first Chern class c1 (Mn,k ) is positive and thus the Kähler Ricci flow starting from any Kähler metric on Mn,k must develop a singularity in finite time (see [11]). As discussed above, this corresponds to reaching the boundry of the Kähler cone. Conveniently, the Kähler cone is easy to describe in this case. In fact, it is generated by only two classes: [D∞ ], the fundamental class of the divisor D∞ corresponding to the ∞ section of the bundle, and [π −1 ωF S ], the pull back of the Fubini-Study metric on the base. We can write our initial Kähler class as α0 = k1 (b0 − a0 )[D∞ ] + a0 [π −1 ωF S ], for some 0 < a0 < b0 . Suppose now that the we have a solution to the Kähler Ricci flow on Mn,k which becomes singular at time T > 0, in other words that α(t) := [ω0 ] − 2πtc1 (M ) reaches the boundry of the Kähler cone at time T . Then under the assumption that the initial metric is Calabi symmetric (see §2.3 for definition) it was shown in [11] that the behaviour of the Ricci flow near time T can be classified in terms of which part of the boundry of the Kähler cone α(T ) lies on: 1. Case: a0 (n + k) = b0 (n − k). Then α(T ) = 0, so the manifold shrinks to a point. This is the situation where it is possible for the solution to be a shrinking Kähler Ricci soliton, and in fact, there exists one on Hirzebruch manifolds called the Koiso soliton. This soliton is constructed using the same symmetry assumptions that we are using. See [6] for details. A theorem of Tian and Zhu then tells us that if we normalized the Kähler Ricci flow to keep the volume constant, then the solution must converge to the soliton. See [10] for details. 3  2.2. Local Coordinates on Mn,k 2. Case: a0 (n + k) < b0 (n − k). Then α(T ) lies on the part of the boundry generated by [D∞ ] and under the symmetry assumption, the Ricci flow solution exhibits GromovHausdorff convergence to the orbifold Mn,k /D0 , with the divisor D0 contracted to a point. 3. Case: a0 (n + k) > b0 (n − k). This is the case that we will concern ourselves with, namely the case when α(T ) lies on the part of the boundry generated by [π −1 ωF S ]. In this case the volume is proportional to (T − t) and under the symmetry assumption the Ricci flow solution exhibits Gromov-Hausdorff convergence to the base manifold with metric aT ωF S where ωF S is the Fubini-Study metric on the base and α(T ) = aT [π −1 ωF S ].  2.2  Local Coordinates on Mn,k  We now describe the manifolds Mn,k through local coordinates. Let {xi }ni=1 be homogeneous coordinates for CPn−1 . Consider the charts Ui given by xi 6= 0. Mn,k can then be realised as a CP1 bundle over CPn−1 via projective fibre coordinates yi on π −1 (Ui ) where the transition functions are given by  k x yj = xjl yi on π −1 (Ui ∩ Uj ). Notice that there are two natural sections (divisors) of this bundle: D0 and D∞ , corresponding to the fibre coordinate yj taking the values 0 and ∞ respectively. Consider the action of Zk on Cn \0 generated by 1 : Cn \0 → Cn \0 : x 7→ e  2πi k x.  We identify (Cn \0)/Zk with Mn,k \(D0 ∪ D∞ ) via the following k-to-one maps on each Vi := {xi 6= 0} ⊂ Cn \0 ψi : Vi → π −1 (Ui )\(U0 ∪ U∞ ) ∼ = Ui × CP1 \{0, ∞} : x 7→ ([x], xki ). Note that these maps commute with the transition maps, allowing us to extend this local identification to the whole space Cn \0. Also note that the divisors are obtained by taking the limit as xi goes to 0 or ∞ respectively.  2.3  Radially Symmetric Metrics  In this section we describe a class of rotationally metrics on the Hirzebruch manifolds Mn,k . These were first investigated by Calabi in [1] from where 4  2.3. Radially Symmetric Metrics we derive most of the calculations in this section. We will begin by defining a class of radially symmetric metrics on Cn \0. Then by assuming certain asymptotic conditions for these metrics (at 0 and ∞) we will then show that these push forward to a smooth metric on Mn,k via the coordinate descriptions of the last section. Consider the function on Cn \0 given by ρ := log k~xk2 . We will consider Kähler metrics on Cn \0 with potentials u(ρ), which are functions of ρ only, and hence are radially symmetric. In order for u to be a potential for a Kähler metric, we need u00 > 0, where the derivatives are taken with respect to ρ. Now consider the Kähler metric ¯ = i∂(u0 ∂ρ) ¯ = iu00 ∂ρ ∧ ∂ρ ¯ + iu0 ∂ ∂ρ. ¯ ω = i∂ ∂u Claim 1. The radially symmetric metric ω descends to a metric on (Cn \0)/Zk . Proof. It suffices to show that the function ρ is invariant under the action. In other words 1∗ ρ = ρ. 1∗ ρ(~x) = ρ ◦ 1(~x) = ρ(e  2πi k ~ x)  = log ke  2πi k ~ x k2  = log k~xk2 = ρ(~x)  In particular, we can pushforward ω to Mn,k \(D0 , D∞ ) via the maps {ψi }ni=1 from §2.2. Now we further impose the condition that there exist 0 < a < b < ∞ such that a := lim u0 , ρ→−∞  b := lim u0 . ρ→∞  These asymptotic conditions basicallly guarantee that the Kähler form ω on Mn,k \(D0 ∪ D∞ ) will extend smoothly to all of Mn,k ([1]). Now we will explicitely compute ω in the local trivialization π −1 ({Uj }), for some fixed j ∈ {0...n}. First, choose holomorphic coordinates {zi := xxji }i6=j for Uj and y for the fibre. Claim 2. The function ρ pushes forward to a function ρ on Mn,k \(D0 ∪D∞ ) which can be written in coordinates as ρ = k1 log y ȳ + 2πvF S , where vF S is the potential for the Fubini-Study metric on the base CPn−1  5  2.4. A Closer Look at the Fibres Proof. Our identification is through the maps ψi : Vi → π −1 (Ui )\(U0 ∪ U∞ ), so we compute: ! n n X X x x̄ i i (ψj )∗ ρ = (ψj )∗ log k~xk2 = (ψj )∗ log xi x̄i = (ψj )∗ log xj x̄j x x̄ i=0 i=0 j j ! n X xi x̄i = (ψj )∗ k1 log(xj x̄j )k + log x x̄ i=0 j j   X = k1 log y ȳ + log 1 + zi z̄i  = k1 log y ȳ + 2πvF S . i6=j  For ease of notation, we will consider ρ as a function on each trivializatoin Uj via the identification above, writing ρ instead of (φj )∗ ρ. Note that ∂ ∂¯ log y ȳ = ∂ yyȳ dȳ = ∂ ȳ1 dȳ = 0, ¯ = i∂ ∂v ¯ V S = π −1 ωF S is the pull-back of the Fubini-Study so that i∂ ∂ρ metric on the base. Since the Kähler form of the radially symmetric metric 0 ∂ ∂ρ, ¯ ¯ this shows that the second term is degenerate along the is iu00 ∂ρ∧∂ρ+iu fibre. Song and Weinkove showed that case 3 of Ricci flow on Mn,k basically corresponds to having u0 converging to a constant under the flow. From the above observation, this intuitively corresponds to: as u0 moves towards being a constant, making u00 go to zero, the metric becomes proportional to the Fubini-Study metric on the base and degenerate on the fibre (see Theorem 2).  2.4  A Closer Look at the Fibres  Claim 3. Restricting ω to the fibres produces pairwise isometric Riemannian manifolds. Proof. It suffices to prove the claim on every local trivialization, since we can then compose isometries to map any fibre onto any other one. Choose a local trivialization π −1 (Uj ) for some j ∈ {1, . . . , n}. Using the holomorphic coordinates {zi }i6=j on Uj , it suffices to show that the fibre at z is isometric to the fibre at 0 with respect to the the induced metric for each z ∈ Uj .  6  2.4. A Closer Look at the Fibres Using the previous claim and the fact that the vF S term does not contribute to the fibre direction, we get that the induced metric on the fibre at z is ¯ π−1 (z) + iu0 |π−1 (z) ∂ ∂ρ| ¯ π−1 (z) ω|π−1 (z) = iu00 |π−1 (z) ∂ρ|π−1 (z) ∧ ∂ρ| ¯ 1 log y ȳ) + iu0 |π−1 (z) ∂ ∂( ¯ 1 log y ȳ) = iu00 |π−1 (z) ∂( 1 log y ȳ) ∧ ∂( k  k  k  iu00 |π−1 (z) dy ∧ dȳ. = k 2 y ȳ Note that the only dependence on z is from u00 being a function of ρ, and hence of both y and z. We will define our isometry by fz : π −1 (0) → π −1 (z) : y 7→ 1 + kzk2  k  2  y.  Then k 1 (fz )∗ ρ|π−1 (0) = (fz )∗ log y ȳ = k1 log y ȳ 1 + kzk2 k  = k1 log y ȳ + log 1 + kzk2 = ρ|π−1 (z) . Hence (fz )∗ u00 |π−1 (0) = u00 |π−1 (z) . This gives (fz )∗ ω|π−1 (0)  iu00 |π−1 (0) = (fz )∗ dy ∧ dȳ k 2 y ȳ k 1 + kzk2 iu00 |π−1 (z) = dy ∧ dȳ (1 + kzk2 )k k 2 y ȳ iu00 |π−1 (z) dy ∧ dȳ k 2 y ȳ = ω|π−1 (z) , =  so that fz is an isometry. The above claim allows us to concentrate on the fibre π −1 (0) without losing any generality. By definition, π −1 (0) ∼ = CP1 ∼ = C∪∞. Our coordinate −1 y will act as a coordinate for C here. On π (0), we have ρ = k1 log y ȳ.  7  Chapter 3  Singularity Analysis for Ricci flow In this section we set up the necessary background to study singularities for the Ricci flow in our setting: case 3 from §2.1. In [11], Song and Weinkove show that under the condition a0 (n + k) > b0 (n − k) (case 3), the Ricci flow with an initial metric satisfying the Calabi symmetry condition will form −a0 . They showed that this corresponds to a singularity at time T := b02k studying a corresponding family u(ρ, t) of evolving potentials as above such ¯ that ∂ ∂u(ρ, t) is the corresponding family of evolving metrics. In particular, the the assymptotics “a(t) and b(t)” of u(ρ, t) from the previous section will also vary in time such that Theorem 2. [Song-Weinkove [11]] In case 3 for the Ricci flow of a Calabi symmetric metric on Mn,k , if T is the singular time then (b(t) − a(t)) = 2k(T − t). In particular, the Ricci flow solution exhibits Gromov-Hausdorff convergence to the base manifold with metric aT π −1 ωF S where ωF S is the Fubini-Study metric on the base. We now discuss singularities for the Ricci flow in more detail. We will define and discuss type I singularities of the flow. In particular, we will show that the assumption on the existence of a singular point p on Mn,k in Theorem 1.1 is natural under the assumption of type I singularity. As for the additional assumption we make on the rate of metric decay at p, we observe that this is natural in the sense as shown in §4.2 Lemma 4, the volume of the fiber with respect to g(t) is precisely a(T − t) for some constant a. To reduce clutter in the following sections, I will often write f . g for functions f and g to mean that there exists a positive constant C such that f ≤ Cg on the common domain of f and g. This constant will often only be dependent on the initial metric, but occasionally it may depend on the point of evaluation (but not the time), in which case it will be mentioned that it is a pointwise estimate.  8  Chapter 3. Singularity Analysis for Ricci flow For Ricci flow to form a singularity at time T , one must necessarily have sup kRmg(t) kg(t) & M  1 . T −t  This motivates the following definitions: Definition 1. A sequence (pi , ti )i∈N ⊂ M × [0, T ) is an essential blow-up sequence if ti % T and kRmg(ti ) kg(ti ) (pi ) &  1 T − ti  Definition 2. A point p ∈ M is a singular point if there exists an essential blow up sequence (pi , ti )i∈N with p = limi→∞ pi .1 Finite time singularities are classified according to the speed at which sup kRmg(t) kg(t) M  diverges to ∞. We say that a solution for the Ricci flow equation is a Type I solution if 1 . sup kRmg(t) kg(t) . T −t M We say that a finite time singular solution is a Type IIa solution if it is not of type I. It is conjectured that Type IIa singularities will only form under highly symmetric initial conditions. From now on we assume that (Mn,k , g(t)) is a solution to Kähler Ricci flow as in Theorem 1.1. Lemma 1. Let p ∈ M . If there exists c > 0 and t0 ∈ [0, T ) such that kRmg(t) kg(t) (p) ≤  c T −t , ∀t  ∈ (t0 , T ),  then kXp kg(t) & (T − t)c for all X ∈ Tp M . Proof. Let α > 0 be a constant to be chosen later. Then for t ∈ (t0 , T ), we have Ricg (αX, αX) d c log kαXk2g = − ≥ −kRicg kg ≥ −kRmg kg ≥ − . 2 dt kαXkg T −t 1  This is not the most general definition of singular point, but for type I solutions they are equivalent. See [3] for details.  9  Chapter 3. Singularity Analysis for Ricci flow Integrating from t0 to t gives s=t log kαXk2g(s) |s=t s=t0 ≥ c log(T − s)|s=t0 .  Choosing α so that log kαXk2g(t0 ) = c log(T − t0 ) gives log kαXk2g(t) ≥ c log(T − t). Finally, exponentiating gives (T − t)c ≤ kαXk2g(t) . kXk2g(t) .  Proposition 1. For any p ∈ Mn,k \{D0 ∪ D∞ }, there exists a sequence (ti )i∈N with ti % T such that (p, ti )i∈N is an essential blow-up sequence, and thus p is a singular point. Proof. Assume that there does not exist such an essential blow-up sequence. Then for any c > 0, the set   c t ∈ [0, T ) : kRmg(t) kg(t) (p) ≥ T −t is bounded. Thus there exists a t0 ∈ [0, T ) such that kRmg(t) kg(t) (p) <  c , ∀t ∈ (t0 , T ). T −t  Without loss of generality, by the symmetry of the metric, we can consider p to be in π −1 (Un ). Then, as before, we have local holomorphic coordinates {y, zi }n−1 i=1 . Moreover, the symmetry allows us to assume that the z coordinate of p is 0. It is clear by looking at the metric in these coordinates that u00 (t) ∂ 2 kg(t) (p) = 2 (p). k ∂y k y ȳ ∂ 2 From Lemma 4.3 in [11], we u00 (t) . (T − t), so that k ∂y kg(t) . (T − t). Combining this with the preceeding lemma gives ∂ 2 (T − t)c < k ∂y kg(t) . T − t.  In other words, (T − t)c−1 is bounded, which is impossible for c < 1. This contradicts c being chosen arbitrarily. 10  3.1. Parabolic Dilations Note that it is not surprising that the singularity is always found in our case by using an essential blow-up sequence with each term using the same point. In fact, in [3], Enders, Müller and Topping show that every singularity for a type I Ricci flow solution on a complete manifold occurs in this way. For the rest of the discussion, we will fix a fibre and the singular point p such that without loss of generality, in local coordinates we have y(p) = 0 and z(p) = 0. In particular, ρ(p) = 0.  3.1  Parabolic Dilations  Let (pi , ti )i∈N be an essential blow-up sequence along the Ricci flow. We will now describe the parabolic scalings of the flow along such a sequence. We will use this in the next section to define the singularity model for the flow along a blow up sequence. For each i, we define Ki = kRmg kg (pi , ti ) and a time dependent metric    t 2 ∗ gi : − Ki ti , Ki (T − ti ) → Sym T M : t 7→ Ki g ti + . Ki   Note that each gi is a solution to the Ricci flow equation, since d d gi (t) = Ki g(ti + tKi−1 ) = −Ricg(ti +tK −1 ) = −RicK −1 gi (t) = −Ricgi (t) . i i dt dt Definition 3. The sequence of Ricci flow solutions (gi )i∈N are the parabolic dilations associated to the essential blow-up sequence (pi , ti )i∈N . We will analyze the parabolic dilations through their behaviour at t = 0. Note that gi (0) = Ki g(ti ). Thus kRmgi (0) kgi (0) = kRmKi g(ti ) kKi g(ti ) = Ki−1 kRmg(ti ) kg(ti ) = Ki−1 Ki = 1. Let p ∈ Mn,k \{D0 ∪ D∞ }. From proposition 1, there exists an essential blow-up sequence (p, ti )i∈N . Proposition 2. The distance between fibres with respect to gi (0) diverges to infinity as i → ∞. 11  3.1. Parabolic Dilations Proof. Recall that Mn,k , g(t) converges in the Gromov-Hausdorff sense to the base CPn−1 with the metric aT gF S under the Ricci flow as t approaches T . Thus, the distance between points in two different fibres converges to the distance between the corresponding points on the base with respect to aT gF S . With respect to the parabolic dilations gi (0) = Ki g(ti ), this allows us to approximate the distance between fibres by the distance between the corresponding base points with respect to Ki aT gF S , which diverges since Ki does.  12  Chapter 4  The Singularity Model Let (Mn,k , g(t), p) be the solution to Ricci flow as in Theorem 1 where p is the singular point. We now consider the corresponding parabolic dilations (Mn,k , gi , p)i∈N of our flow as a sequence of pointed solutions to the Ricci flow. We will see that this sequence converges in some sense to some (M∞ , g∞ , p∞ ), which is the singularity model corresponding to (Mn,k , gi , p)i∈N . In this section, we will review some basic definitions and results about singularity models, and describe the one corresponding to (Mn,k , gi , p)i∈N as described above. Definition 4. A sequence (M, gi , pi )i∈N of pointed Riemannian manifolds converges in the Cheeger-Gromov sense to (M∞ , g∞ , p∞ ) if for each i ∈ N there exists neighbourhoods Ui and Vi of p∞ and pi respectively and a diffeomorphism φi : Ui ∼ = Vi satisfying the following three conditions: • φi (p∞ ) = pi . • (φ∗i gi )i∈N converges uniformly to g∞ in every C m norm on any compact subset of M∞ × (α, ω). • Every compact subset of M∞ must be contained in each of {Ui }i≥N for some N ∈ N. Definition 5. Let κ > 0. A Riemannian manifold (M, g) is κ-noncollapsed at all scales if for any x ∈ M and r > 0 satsifying supBr (x) kRmk ≤ r12 , we have Vol(Br (x)) ≥ κrn . Theorem 3. [Hamilton-Perelman] Let (M, g(t)) be a smooth solution to the Ricci flow on a closed manifold M encountering a singularty at a finite time T . Let (pi , ti )i∈N be an essential blow-up sequence such that sup  kRmg(t) kg(t) (x) . Ki .  (x,t)∈M ×[0,ti )  Then there exists a subsequence of the sequence of parabolic dilations (M, gi , pi ) converging in the sense of Cheeger-Gromov to a complete ancient solution to the Ricci flow (M∞ , g∞ , p∞ ). Futhermore, there exists a κ > 0 such that g∞ is κ-noncollapsed at all scales. 13  Chapter 4. The Singularity Model This result follows from Hamilton’s completeness theorem [5] and Perelman’s no local collapsing [8]. Corollary 1. For our singular point p ∈ Mn,k , the corresponding sequence of parabolic dilations (Mn,k , gi , p) converges in the sense of Cheeger-Gromov to a complete ancient solution to the Ricci flow (M∞ , g∞ , p∞ ). Futhermore, there exists a κ > 0 such that g∞ is κ-noncollapsed at all scales. 1 Proof. Since p is a singular point, Ki & T −t . Since we are assuming that i our solution is a type I solution, we have  sup kRmg(t) kg(t) . Mn,k  1 T −ti .  Putting this together, we have kRmg(t) kg(t) (x) .  sup (x,t)∈Mn,k ×[0,ti )  1 . Ki , T − ti  so that we can apply the above theorem to get the desired result. Lemma 2. e−|ρ| .  u00 (ρ, t) . e|ρ| . u00 (0, t)  Proof. We will again use Lemma 4.3 from [11], which this time provides us with the estimate |u000 | . u00 . This gives Z ρ Z ρ 000 u00 (ρ) u (s) log 00 = (log u00 (s))0 ds = ds . |ρ| . 00 u (0) 0 0 u (s) Exponentiating then gives the desired result. Proposition 3. For each ρ ∈ R, we have the pointwise estimate (T − t) . u00 (ρ, t) . (T − t). Proof. By our assumption, there exists an X ∈ T Mn,k with ρ0 = ρ ◦ π(X) ∈ R and kXk2g(t) & (T − t). By symmetry, we can assume that π(X) is in the same fibre as p, where we have local coordinates {y, z} with z = 0 on the fibre. Then we can write the metric as dy dȳ g(t) = u00 (ρ, t) ky kȳ  for ρ = log(y ȳ) ∈ R. Thus kXk2g(t) ∝ u00 (ρ0 , t). It follows from our assumption that u00 (ρ0 , t) & (T − t). Combining this with the preceeding lemma gives the pointwise estimate u00 (ρ, t) & (T − t) for ρ ∈ R. This combines with the uniform estimate u00 (ρ, t) . (T − t) from Lemma 4.3 of [11] to give the desired result. 14  4.1. A Candidate for the Limit Manifold  4.1  A Candidate for the Limit Manifold  In this section, we will consider Cn−1 × CP1 as a candidate for the limit manifold M∞ . We will begin by defining a sequence of holomorphic maps ψi : Cn−1 × CP1 → Mn,k with respect to which we will study the pullback metrics ((ψi )∗ gi (0)). The main goal of this section will be to show that the limit of the pullback metrics factors as product metric. Later we will show this sequence converges to a limit in some sense away from (·, 0), (·, ∞). We will later use this fact to show that in fact Cn−1 × CP1 = M∞ . n−1 × CP1 , where we Let {wj , y}n−1 j=1 be holomorhic coordinates for C abuse notation by allowing y to take the value ∞, thus parametrizing CP1 = C t {∞}. Recall that Un = {[x1 , . . . , xn ] : xn 6= 0} 3 π −1 [0, . . . , 0, 1] Recall that π −1 (Un ) has holomorphic fibred coordinates {zj , y}n−1 j=1 , again allowing y to take the value ∞ in order to parametrize each fibre CP1 . We will identify Cn−1 × CP1 and π −1 (Un ) for each i via the biholomorhisms: p ψi : Cn−1 × CP1 → π −1 (Un ) : (w, y) 7→ ( Ki w, y), In particular, recall that we chose our singular point p = (0, 1) in the fiber coordinates on π −1 (Un ), and thus we have ψi−1 (p) = (0, 1) in (w, y) coordinates on Cn−1 × CP1 for all i. We thus have a sequence ((ψi )∗ gi (t))i∈N of solutions to the Ricci flow on Cn−1 × CP1 . The goal in this section will be to calculate the components of (ψi )∗ gi (0) with respect to the (w, y) coordinates. Recall that the Kaher form corresponding to gi (0) is simply ¯ + iKi u0 (ti )∂ ∂ρ. ¯ ωi (0) = Ki ω(ti ) = iKi u00 (ti )∂ρ ∧ ∂ρ ¯ term We will expand the two terms above separately. For the u0 (ti )∂ ∂ρ above we have:  ¯ = Ki (ψi )∗ u0 (ti )∂ ∂¯ 1 log y ȳ + log(1 + kzk2 ) (ψi )∗ Ki u0 (ti )∂ ∂ρ  k  dz ∧ dz̄ (z̄ · dz) ∧ (z · dz̄) ∗ 0 = Ki (ψi ) u (ti ) + 1 + kzk2 (1 + kzk2 )2  −1  −2 Ki dw ∧ dw̄ Ki (w̄ · dw) ∧ (w · dw̄) 0 = Ki u (ti ) + , 1 + Ki−1 kwk2 (1 + Ki−1 kwk2 )2 15  4.1. A Candidate for the Limit Manifold where we simplify notation by using dz ∧ dz̄ :=  z · dz :=  n−1 X i=1 n−1 X  dz i ∧ dz̄ i zi dz i .  i=1  Thus since limi→∞ Ki = ∞ and limi→∞ u0 (ti ) = limt→T u0 (t) = aT , we have ¯ = aT dw ∧ dw̄. lim (ψi )∗ Ki u0 (ti )∂ ∂ρ  (4.1)  i→∞  uniformly on compact subsets of (0, 1) ∈ Cn−1 × CP1 in the C 0 sense. ¯ term, first we compute: For the u00 ∂ρ ∧ ∂ρ   dy + ∂ log(1 + kzk2 ) (ψi )∗ ∂ρ = (ψi )∗ ky   dy z̄·dz + 1+kzk = (ψi )∗ ky 2 =  dy ky  +  Ki−1 w̄·dw , 1+Ki−1 kwk2  thus giving ¯ = Ki u00 (ti ) (ψi )∗ Ki u00 (ti )∂ρ ∧ ∂ρ =    dy ky  +  Ki−1 w̄·dw 1+Ki−1 kwk2    ∧    dȳ kȳ  +  Ki−1 w·dw̄ 1+Ki kwk2    Ki u00 (ti ) dy ∧ dȳ + Ki u00 (ti )o(Ki−1 ), k 2 y ȳ  where the asymptotic above is uniform on any neighborhood of (0, 1) uniformly away from (·, 0), (·, ∞). From proposition 3, we have a pointwise estimate (T − t) . u00 (ρ) . (T − t). Since our solution is of type I, we also have 1 T −ti  . Ki .  1 T −ti .  Combining these tells us that u00 Ki is pointwise bounded away from {0, ∞}. This gives the convergence ¯ =h lim (ψi )∗ Ki u00 (ti )∂ρ ∧ ∂ρ  i→∞  (4.2) 16  4.2. An Isometry Between M∞ and Cn−1 × CP1 where h is some Kähler metric on C\0 = CP1 \ {0, ∞} and the convergence is uniformly on compact subsets of C\0 in the C 0 sense. Combining (4.2) and (4.1) this gives lim (ψi )∗ ωi (0) = g̃ := aT δCn−1 + h  i→∞  (4.3)  where δCn−1 is the standard Euclidean metric on Cn−1 and the convergence is uniformly on compact subsets of (0, 1) in the C 0 sense. In particular, we have convergence to a product metric on Cn−1 × (CP1 \ {0, ∞}).  4.2  An Isometry Between M∞ and Cn−1 × CP1  In this section we show that Cn−1 × CP1 = M∞ . Consider the sequence of maps Ψi = φ−1 i ◦ ψi We begin by describing the domain of these maps. We will consider × C as a subset of Cn−1 × CP1 using the coordinates (w, y) defined in the previous section. Now it is not hard to show that from the estimates in the previous section, for any compact subset K ⊆ Cn−1 × C we may have Cn−1  (i) ψi (K) ⊂ Bgi (0) (r, p) ⊂ π −1 (Un ) for some r > 0 some N > 0 and all i > N. (ii) Ψi is defined on K and Ψi (K) ⊂ Bg∞ (2r, p∞ ) ⊂ M∞ for some r > 0 some N > 0 and all i > N . Theorem 4. There exists a smooth subsequence limit map Ψ∞ = limi→∞ Ψi from Cn−1 × (C\0) to M∞ . Moreover, Ψ∞ is holomorphic. Proof. Let K be any compact set K ⊆ Cn−1 × C. Let r be the radius of K with respect to g∞ (0). Then by (ii) above, the sequence Ψi is a uniformly bounded sequence on K with respect to g∞ on M∞ , and is therefore likewise uniformly bounded with respect to (φ−1 )∗ gi (0) on M∞ by the locally uniform convergence (φ−1 )∗ gi (0) → g∞ . Moreover, each Ψi is holomorphic into (M∞ , Ji ) where Ji = φ∗i J is the pullback of the complex structure J on Mn,k . By derivative estimates for holomorphic functions, it follows that any fixed order derivative of the Ψi ’s will be uniformly bounded with respect to (φ−1 )∗ gi (0) on M∞ on any compact subdisc contained strictly within K. From this fact and the smooth local convergences (φ−1 )∗ gi (0) → g∞ and Ji → J∞ , we can conclude that any fixed order derivative of the Ψi ’s 17  4.2. An Isometry Between M∞ and Cn−1 × CP1 will be uniformly bounded with respect to g∞ on M∞ on any compact subdisc contained strictly within K. From this and a diagonal subsequence argument we see that some subsequence of Ψi converges smoothly uniformly on compact subsets of Cn−1 × (C\0) to a smooth limit Ψ∞ . That Ψ∞ is holomorphic then follows from the convergence Ji → J∞ . We now want to show that Ψ∞ in fact extends to give a bijective holomorphic map from Cn−1 × CP1 to M∞ . Note that such a map must then be a biholomorphism. We will establish this through several lemma’s. Recall from (4.3) that we had the locally uniform convergence lim (ψi )∗ ωi (0) = g̃ := aT δCn−1 + h.  i→∞  in the C 0 sense on compact subsets of Cn−1 × (C\0), and that the later is a product metric on Cn−1 × (C\0). Note that we have g∞ = (Ψ∞ )∗ g̃, ∗ since (Ψ∞ )∗ (limi→∞ (ψi )∗ ωi ) = limi→∞ (Ψi )∗ (ψi )∗ ωi = limi→∞ (φ−1 i ) ωi = ω∞ .  Lemma 3. Ψ∞ is injective. Proof. Let q1 6= q2 ∈ Cn−1 × (C\0). Let r = dg̃ (q1 , q2 ). Since g̃ = limi→∞ ψi∗ gi (0), we can choose an N large enough such that dψi∗ gi (0) (q1 , q2 ) > r ∗ ∗ ∗ 2 for all i > N . Since (Ψi ) φi gi = (ψi ) gi , dg∞ (0) (Φ∞ (q1 ), Ψ∞ (q2 )) = lim dφ∗i gi (0) (Φ∞ (q1 ), Ψ∞ (q2 )) i→∞  = lim dψi∗ gi (0) (q1 , q2 ) > 2r . i→∞  Since r > 0, Ψ∞ (q1 ) 6= Ψ∞ (q2 ) as required. Showing surjectivity is a little trickier. The proof will come after a series of lemmas: Lemma 4. The fibre (C\0, h) has finite volume. Proof. We will establish this by showing that the fibres of the parabolic dilations (π −1 (p), gi |π−1 (p) ) have volume uniformly bounded independent of i. As before, we have local holomorphic coordinate y for the fibre (assuming by symmetry that z = 0). Through the relation y = exp( kρ 2 + i arg y), we 18  4.2. An Isometry Between M∞ and Cn−1 × CP1 can reparametrize π −1 (p) in terms of ρ and θ := arg y. The metric is given by    00 00 g = ku2 yȳ dydȳ = uk2 k2 dρ + idθ k2 dρ − idθ = u00 14 dρ2 + k12 dθ2 . The volume of the fibre with respect to the dilated metric gi (0) = Ki g(ti ) is then Z Z Z Ki u00 (ti ) Ki 00 dθ u (ti )dρ dρdθ = 2k 2k R S1 CP1 Ki 0 πKi u (ti )|∞ (b − a)(ti ) = 2πKi (T − ti ), = −∞ 2π = 2k k where we used that b−a = 2k(T −t). From our assumption that the singular1 ity is type I, Ki . T −t . It follows that the volume of (π −1 (p), gi (0)|π −1 (p)) i is bounded independent of i. Thus, the limit fibre (C\0, h) has finite volume. Lemma 5. There exists an r > 0 such that the inf y∈C\0 Vol(Brh (y)) > 0. Proof. Recall from the proof of corollary 1 that there exists C > 0 such that sup kRmg(ti ) kg(ti ) ≤ CKi .  Mn,k  It follows that sup kRmgi (0) kgi (0) = sup kRmKi g(ti ) kKi g(ti ) =  Mn,k  Mn,k  1 sup kRmg(ti ) kg(ti ) ≤ C. Ki Mn,k  Let r = √1C . From theorem 3, there exists a κ > 0 such that the singularity model is κ-noncollapsed at all scales. Since sup kRmgi (0) kgi (0) ≤ C =  Mn,k  1 , r2  we have that Vol(Br (x)) ≥ κr2n for all x ∈ Mn,k . Recall that Φ∞ is an isometric embedding of the singularity model into the product of Riemannian manifolds (Cn−1 , aT δ) × (CP1 , h). Define modified balls for each x = Φ−1 ∞ (v, y) by aT δ h B 0 (x) = Φ−1 ∞ (Br (v) × Br (y)). Clearly B 0 (x) ⊃ Br (x) for each x ∈ Mn,k . Let c be the volume of the unit ball in Cn−1 with respect to aT δ. Then we have Vol(B 0 (x)) = Vol(BraT δ (v))Vol(Brh (y)) = cr2n−2 Vol(Brh (y)). 19  4.2. An Isometry Between M∞ and Cn−1 × CP1 But we also have Vol(B 0 (x)) ≥ Vol(Br (x)) ≥ κr2n . Combining these gives Vol(Brh (y)) ≥ κc r2 for all y ∈ CP1 . Lemma 6. The diameter of (C\0, h) is bounded. Proof. Let r be chosen from the preceeding lemma. Assume that (C\0, h) has infinite diameter. F = Ψ∞ ({0} × (C\0)). Thus F must also have infinite diameter with respect to g∞ |F = (Ψ∞ ◦ πC\0 )∗ h, where πC\0 is the projection onto the C\0 part of Cn−1 × (C\0) ⊂ Cn−1 × CP1 . Then there exists a sequence (yi )i∈N such that the balls {Brh (yi )}i∈N ⊂ F are pairwise disjoint. Applying the previous lemma, we have X X Vol(F ) ≥ Vol(Brh (yi )) ≥ inf Vol(Brh (y)) 1 = ∞. i∈N  y∈C\0  i∈N  Since (C\0, h) is isometrically mapped onto F , this implies that the volume of (C\0, h) must be infinite. This contradicts Lemma 4. Thus our assumption must be false as required. Lemma 7. Ψ∞ extends to a surjection from Cn−1 × CP1 to M∞ . Proof. That Ψ∞ extends to a holomorphic map on Cn−1 × CP1 follows from the fact that Ψ∞ is a bounded holomorphic map by the previous lemma, and derivative estimates for holomorphic functions. We now show surjectivity. Fix any q ∈ M∞ and consider distance r of this point to p∞ . Then for sufficiently large i we have q ∈ B̄2r (p∞ , g∞ ) ⊂ B3r (p∞ , (φ−1 )∗ gi (0)). In particular, this would imply q ∈ Ψi (K) for some compact set K and all i sufficiently large. From this it is not hard to see that we must have q ∈ Ψ∞ (K). The lemmas above imply the following Theorem 5. Ψ∞ is a bijective holomorphic map from Cn−1 × CP1 to M∞ . Thus Ψ∞ is a biholomorphism. Moreover, h extends to a metric on CP1 , and Ψ∞ is isometric with respect to the metrics g̃ = aT δ + h on Cn−1 × CP1 and g∞ on M∞ .  20  Chapter 5  Conclusion 5.1  Proof of the Main Theorem  Proof. We have shown that for any p ∈ Mn,k \{D0 , D∞ }, under the assumptions of the theorem, the Ricci flow forms a singularity at p at the singularity time. Moreover, the corresponding singularity model at time 0 is biholomorhically isometric to the product (Cn−1 , aT δ) × (CP1 , h), where δ is the Euclidean metric on Cn−1 , and h is some Kähler metric on CP1 . Since the solution is assumed to be type I, [3] tells us that singularity model must be a shrinking soliton. It follows that (CP 1 , h) must be a shrinking soliton. By a theorem of Tian and Zhu (see [9]), Kähler Ricci solitons on compact manifolds are unique modulo automophisms, so (CP1 , h(t)) must in fact be the standard shrinking sphere soliton. This proves the main theorem.  5.2  Remarks  In this section, we discuss the validity of the assumptions that we made. The assumption that the lengths of of vectors tangent to the fibre bundle don’t decay too fast is used to use the coordinates that we have for the fibre as coordinates for the fibre in the limit manifold. Without making this assumption, the metric in these coordinates could become degenerate even after parabolic dilations. But the limit manifold exists, so there certainly are some coordinates where the limit metric is non-degenerate. We could try to adjust the coordinates for each i, as we did in the base direction, but it is not as easy in the case. The metric in the fibre direction is invariant under linear dilations, so the adjustments would have to be non-linear, which makes things tricky. It is difficult to imagine adjustments that would be holomorphic, injective, and non-linear, so we assume that they could only be linear, so that the initial coordinates suffice. Moreover, our assumption is consistent with the actual precise decay of volume of the fiber under the flow being precisely a(T − t). The other assumption that we made was that the singularity forms at  21  5.2. Remarks a type I rate. As mentioned before, it is conjectured that the alternative, type IIa singularities, only form with very particular initial metrics. There are other things that suggest a type I rate. The volume of the manifold and even each fibre decays at a rate of (T − t), which is typical of a type I singularity (e.g. shrinking Ricci solitons). Moreover, the metric becomes degenerate only in the fibre direction. Because of the symmetry, the Ricci flow on the whole manifold can induce a modified flow ĝ(t) on the fibre, which is of the form dĝ dt  = −Ricĝ + F (u0 , u00 , u000 ).  Moreover, the Ric term dominates the F term. From [2], we know that the Ricci flow on the S2 will always converge to the shrinking round sphere, and hence will form a type I singularity. Perhaps this modified flow on S2 is close enough to the Ricci flow to also deduce that the singularity must be type I. It is tempting to generalize this work. In particular, one could hope that if the Ricci flow on a compact fibration coverges in the Gromov-Hausdorff sense to a metric on the base, then for a type I singularity at some p in a fibre F , the corresponding singularity model should factor as a Riemannian product of a shrinking soliton on the fibre and flat space corresponding to the dilated base. The Gromov-Hausdorff convergence would guarantee that the metric only becomes degenerate in the fibre direction. This implies that the parabolic dilations will spread disjoint fibres infinitely far appart, so that the singularity in the base direction becomes flat. One can then use a local trivialization to find coordinates for each ti factoring apart the base and fibre direction so that they are orthogonal along F with respect to the parabolic dilation gi (0). This orthogonality should allow us to deduce that near F , the cross terms of the metric decay faster than the terms in the F direction, so that the limit metric on the singularity model have the fibre and the base orthogonal. This allows us to use the same argument with Perelman’s no-local-collapsing to conclude that the fibre of the singularity model must be compact, and hence contains all of the original fibre. Of course, these are just rough ideas, and the details still need more work.  22  Bibliography [1] Eugenio Calabi. Extremal Kähler metrics. In Seminar on Differential Geometry, volume 102 of Ann. of Math. Stud., pages 259–290. Princeton Univ. Press, Princeton, N.J., 1982. [2] Bennett Chow. The Ricci flow on the 2-sphere. J. Differential Geom., 33(2):325–334, 1991. [3] Joerg Enders, Reto Müller, and Peter M. Topping. On type I singularities in Ricci flow. arXiv:1005.1624v1. [4] Richard S. Hamilton. Three-manifolds with positive Ricci curvature. J. Differential Geom., 17(2):255–306, 1982. [5] Richard S. Hamilton. A compactness property for solutions of the Ricci flow. Amer. J. Math., 117(3):545–572, 1995. [6] Norihito Koiso. On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics. In Recent topics in differential and analytic geometry, volume 18 of Adv. Stud. Pure Math., pages 327–337. Academic Press, Boston, MA, 1990. [7] Ngaiming Mok. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differential Geom., 27(2):179–214, 1988. [8] Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [9] Gang Tian and Xiaohua Zhu. Uniqueness of Kähler-Ricci solitons on compact Kähler manifolds. C. R. Acad. Sci. Paris Sér. I Math., 329(11):991–995, 1999. [10] Gang Tian and Xiaohua Zhu. Convergence of Kähler-Ricci flow. J. Amer. Math. Soc., 20(3):675–699 (electronic), 2007. [11] Ben Weinkove and Jian Song. The Kähler-Ricci flow on hirzebruch surfaces. arxiv:0903.1900v1. 23  

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