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Topics in mean curvature flow of hypersurfaces Hikspoors, Samuel 2004

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TOPICS IN M E A N C U R V A T U R E F L O W OF H Y P E R S U R F A C E S by SAMUEL HIKSPOORS B.Sc. (Mathematics and Physics) Universite de Montreal, 2002 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 2004 © SAMUEL HIKSPOORS, 2004 Library Authorization In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. V V v W * P O O R S Date (dd/mm/yyyy) Name of Author (please print) Title of Thesis: o J U O <a D e 9 r e e : v w w 4 ^ Department of ^ W ^ r A Year: a e m ^ The University of British Columbia Vancouver, BC ' C a n a d a Abstract In this thesis we study the possible solutions of the mean curvature flow problem restricted to hypersurface geometries: We give a complete exposition of the theory contained in some of the articles included in the bibliography. The text is divided in three parts. The first part consist of an informal discussion on some useful knowledge in partial differential equations of parabolic type. The second part of the text is the core of the thesis: It contains a detailed exposition (with full proof of all main results) of the now classical work of Huisken and Hamilton on the MCF of compact-convex initial surfaces. The main result state the existence of a long time solution of the mean curvature flow and that this smooth solution converge to a round point. The third part is meant to be an introduction to some further developments under less restrictive initial data. In some situations the flow can generate singular points and then a smooth solution do no longer exist after a given finite time interval: We characterize in a simple way some of these-singularities. An informal discussion on some important results of MCF of entire graphs then conclude our work. ii Table of Contents Abstract ii Table of Contents iii Part I. S O M E P D E R E S U L T S 1 0.1 Introduction . . . 2 0.2 Bounded Domain 2 0.3 Unbounded Domain 5 Part II. M C F OF C O M P A C T C O N V E X S U R F A C E S 7 Chapter 1. Introduction 8 Chapter 2. Evolution Equations and Convexity 10 2.1 Evolution ....10 2.2 Convexity 13 Chapter 3. The Relative Eigenvalue Difference of A 19 Chapter 4. Some other Bounds on Curvature 29 4.1 Bounding \VH\ ....29 4.2 Derivatives of A 32 Chapter 5. The Finite Time Interval 37 Chapter 6. Mean Curvature Ratio 40 Chapter 7. Rescaling The M C F Solution 43 Chapter 8. Limiting Shape of the Rescaled Hypersufaces 47 Part III. N O N - C O N V E X INITIAL D A T A 54 Chapter 9. Introduction 55 Chapter 10. Formal Statement of the Problem 56 Chapter 11. Normalized Solutions 58 Chapter 12. Monotonicity Formula and Consequences 60 iii Chapter 13. Non-Compact Initial Data and Entire Graph Solutions 63 13.1 Equivalent Formulation • 63 13.2 A Priori Height Estimate • • . . . 6 4 13.3 A Priori Gradient Estimate ; •••••• • 6 5 13.4 Two Existence Results • • 6 5 Appendix 67 Bibliography 7 0 iv I wish to thank my supervisor, Jingyi Chen, for orienting me toward this master degree as well as for financial support. My debt to Jim Bryan and David Brydges is also quite high: Thanks for everything. I could hardly end these acknowledgements without mentioning the french connection, my group of friends here at UBC: Wish you all well guys. v Part I SOME PDE RESULTS 1 0.1 Introduction In this first part of the exposition we discuss some of those results of non-linear partial differ-ential equations that are the most closely related to our study of mean curvature flow (MCF) of hypersurfaces, that is, short time existence of quasi-linear parabolic differential equations of second order with given initial data (initial hypersurface in our case) and various types of constraints on the equation itself. It is not our goal to provide a complete section on PDE theory but rather to present in an in-formal way some useful results for our subsequent investigation of MCF. We will state existence and uniqueness results of various types and classify them according to whether the equations are studied on a bounded domain or not. For more details on this subject and complete proofs of all results, one can consult [E], [F] and [T]. 0.2 Bounded Domain We first consider the following Cauchy problem for fully non-linear parabolic equations of the form: dtu = F(t,x,D2xu), u(0):=f (1) where x € M a boundary free compact n-dimensional Riemannian manifold, u take its values in RK and F is smooth in its arguments. We say that (1) is strongly parabolic if, when we write F — F(t,xr,D$DmUj) , p + q < 2 and 1 < j < K, we have: for any 0 € Rn, C > 0 and I the K x K identity matrix. We also write Wp'q(M) for the usual Sobolev spaces on M. With these conventions we have the following uniqueness result for fully non-linear equations of parabolic type: Proposition 1. If (1) is strongly parabolic andu(0) = / e HS(M) := WS'2(M), with s > |+2, E dFi ^2\d(D?DqmUj) >C\0\2-IKxK 2 then there is a unique sulution: u€C([0,T),Hs{M))nC°°((0,T)x M) which persists as long as \\u(t)\\(p+r. is bounded, for some r > 0. Notice that this proposition do not give us any information about the existence of a short time existence of a solution but rather tells the condition under which (bound on u) such a solution exist. We will see in Proposition 4 that a proper short time existence is possible when we impose more restrictive constraints on (1): Such a result will then be useful to us in our investigation of the MCF problem. We now specialize to a more restrictive class of evolution equations than (1) which will be of greater importance for us later on. We consider the case where (1) is a quasi-linear equation, that is, of the form: dtu = Aij(t, x, Dlu)didjU + B(t, x, Dlxu), u(0) := / (2) where we make use of Einstein summation convention. For such a class of equations, we have at our disposal the following proposition: . Proposi t ion 2. / / the quasi-linear system (2) is strongly parabolic and u(0) = / € HS(M) , s > f + 1, then there is a unique solution satisfying: u e C([0,T),Hs(M)) n.Coo((0,T) x M) which persists as long as \\u(t)\\c2+ris bounded, forr > 0. We see that in restricting the class of equations under consideration we obtain a result similar to the one in Proposition 1 with a slightly weeker assumption on the initial data. We can go a bit further on these lines by restricting even more the equations of type (1) to: dtu = Aij(t,x,u)didjU + B(t,x,Dlu), , «(0) := / (3) 3 Proposition 3. Assume the system (3) is strongly parabolic. If f € HS(M), s > | or s > 1 if n := 1, then there is a unique solution satisfying: u € C([0,T),HS(M)) nC°°((0,T) x M) which persists as long as \\u(t)\\cr is bounded, r > 0. Remark that in this last particular case of (1) we also have a condition under which a long time solution exist that is weaker than the previous ones. We now return to the case of quasi-linear equations (2): In our investigation of MCF of hy-persurfaces we will need a short time existence for such a class of parabolic PDE. In order to formulate our next result (the most useful for us) in its full generality, we need to define a more general type of parabolicity criterion than strong parabolicity is: We say that the quasi-linear system (2) is Petrowski-parabolic if we have: Spec [-Aij(t,x,v)0i9j] C {z € C : Re(z) < -Co|0|2} for some Co > 0 and where Spec denote the spectrum of the operator. Proposition 4. Given the Petrowski-parabolicity hypothesis, if f € HS(M) and s > f + 3, then the quasi-linear system (2) has a unique solution: u e C([O,T),HS(M)) nc°° ( (o , r ) x m ) for some T > 0, which persists as long as \\u(t)\\ci+r is bounded, for some r > 0. This last result will be very convenient since it provides us not only the short time existence of a smooth solution of a type of equations that effectively arise in Parti but also gives us a condition under which we get a long time existence. We will not make use of that second part of Proposition 4 but our development could have been slightly rearranged in order to do so (in the compact and convex case: see next chapter). 4 0.3 Unbounded Domain Here we state a short time existence result for quasi-linear equations defined on unbounded domain (we take M := Rn). We are not assuming that the coefficients of the equations are smooth anymore but shall define a more general setting for these. We start with some defini-tions: By n [ l 0 i t l ) we mean a layer in R n + 1 : IJ. [ t 0 i t l ) := {(t,xu ...,xn) € Rn+1 :t0<t< ti}. C(m>a>0)(TT[tOitl]) is the space of functions u(t, x) defined in II[t0)tl] with continuous and bounded x-derivatives up to order m, which also satisfy (uniformly in n[to t lj) the Holder condition with respect to x and t with exponents a and ft respectively. C(m,aAl)(Q) i s the space of functions u(t,x,y) defined in a region Q and having continu-ous and bound x and y derivatives up to order m, satisfying the Holder condition with respect to x, t and y (uniformly in Q) with exponents a, (5 and 1 respectively. These last definitions are also valid for vector valued functions: We simply require that each component satisfy the previous conditions. We now consider the following quasi-linear Cauchy-problem (of order 26, with b an integer): dtu= J2 Am(t,x,Dku)Dmu + B(t,x,Dku) \m\=2b = P0(t,x,Dku;D)u + B(t,x,Dku) u{t = to) := where \k\ < (2b — 1) and x e W1. In what follows, we will denote by Q the domain Q := {t € [to, T], x 6 M", [yi] < M, i = 1 ,u} where the y^ 's will represents the coordinates of the image space of the vector valued function 5 u. With these notations we can now formulate an important result: Proposi t ion 5. Let our previous quasi-linear system be Petrowski-parabolic with the coeffi-cients of the operator Po(t,x,y;D) be in C(°'Q'a/'26'1)(C}), and the vector function B(t,x,y) be in CQ'^XQ), 0 < a < 1. We also require that <j)(x) e C(26'a)(Rn). Then the Cauchy problem has a unique solution: u 6 C ( 2 6 , a , a / 2 6 ) ( n [ t o t o + £ ) ) j for some e > 0. The magnitude e of the time interval depends on the upper bounds of the absolute value of the coefficients and their first derivatives, on their Holder and Lipschitz constants as well as on the upper bounds of the abolute value of Dk<j)(x), \k\ < 2b, and their Holder constants; it also depends on the constants provided by the condition of parabolicity. With these last results and some of the background material on differential geometry included in the appendix, we are now ready to start our investigation of the mean curvature flow (MCF) of hypersurfaces. 6 Part II M C F OF COMPACT CONVEX SURFACES 7. Chapter 1 Introduction In this second part of the exposition we begin our study of the mean curvature flow of hyper-surfaces. The idea is to consider a smooth n-manifold M with a smooth immersion F : M —> F(M) 6 Rn+1 (F(M) is a hypersurface in E n + 1 ) and then look for a family of such hypersur-faces parametrized by a continuous time variable which are solutions of the following evolution equation: dtF(p,t) = -H{x,t)-v, peM, xeFt(M) (1.1) F(-,0) = F0(M) C l " + 1 . where H(x, t) is the mean curvature, v is the outward unit normal vector of the hypersurfaces and some further conditions are required on F(M) in order to get a well defined Cauchy prob-lem. Remark that, with our choice of sign, the previous flow makes the hypersurfaces move in the direction of the inner unit normal vector at a rate proportional to the curvature at each points, which is the geometric idea behind the formal statement of the problem. For the basic definitions and results related to hypersurfaces theory (metrics, connections, mean curvature,...) we refer the reader to the appendix. The initial data FQ(M) to be considered in this part of the exposition will be restricted to the case of (boundary-free) compact and uniformly convex initial hypersurface. By FQ(M) to be strictly convex we mean that all of the eigenvalues of the second fundamental form A(x, t = 0) are strictly positive at every point of FQ(M). This last assumption can be written more formally 8 if we notice that a consequence of it is that there must exist a e > 0 such that: hij > eHgij holds on all of the initial hypersurface (see the appendix for the definitions of the metric gij and the second fundamental form hij): This is what we will recall as our main hypothesis throughout the text. Our final goal will be to prove the main theorem of the section (see below) which roughly say that our intuition is right, that is, convex and compact hypersurface without boundary shrinks down to a round point (to be defined) in a finite time interval under MCF: Theorem 1. [Main Theorem] Let n := dim(M) > 2 and assume that Fo(M) is compact and uniformly convex. Then the evolution equation (1.1) has a smooth solution Ft(M) on a finite time interval 0 <t <T and the hypersurfaces Ft(M) converge to a round point ast—*T with respect to an homothetic expansion that preserves the area of the hypersurfaces. This result first appeared in [HI]. The next seven chapters are entirely written in order to prove the main theorem: Most of the results in our subsequent devlopments (in this part of the text) are from [HI] and [H]. Similar results has been proved in the lower dimensional case of plane curves (n := 1). It turns out that a long time existence of a smooth solution always exist for embedded closed plane curves, regardless of their original shape: It has been proved in [GH] and [G] that the curves becomes convex in finite time and then results from [GH] implies that they also shrinks to a round point. A reason for such a difference between the dimension one case and higher dimensional MCF evolution can be attributed to the fact that Sobolev spaces results differ when the dimension of the domain of the functions is one or strictly higher than one. 9 Chapter 2 Evolution Equations and Convexity 2.1 Evolution Before we derive the evolution equations of some important geometrical quantities, it is quite natural to determine if the system (1.1) has any solution; at least for some short time and for given initial conditions. Our basic existence result is given by PDE theory of strictly parabolic type whenever our initial hypersurface FQ(M) is compact. We will not reproduce the proof of this result here but one can verify its validity in [E], section 3.4 (Proposition 4 would also be appropriate in our case): Theorem 2. The MCF evolution equation (1.1) has a solution F(-,t) for a short time with any smooth compact initial hupersurface FQ(M). Now that we have this last result in hand, we can derive some evolution equations: The most important objects here are the metric and the second fundamental form, which gives us much of the information there is to know on a given hypersurface. It will thus be natural to use there evolution equations to get some further knowledge about how our immersions evoluate with time. Lemma 1. The metric of Ft(M) satisfies the following evolution equation: dtgtj = -IHhij (2.1) Proof. Since the djF's are vectors in the tangent space of Ft(M) at a given point of the hyper-10 surface, we find: 9t9i3= 9t(9iF,djF) = (di(-Hu),djF) + (diF,dj(—Hu)) • We need another result in order to derive the evolution equation of A and other important related quantities such as |A| 2 and H, the mean curvature. Lemma 2. The evolution of the unit normal vector v to Ft(M) has the form: Proof. dtv = (dtu,diF)djFgij = -(v,dtdiF)djFgii • =(v,di(Hv))djFgi> = {diH)djFgi\ = VH where the first equality holds because the vector dtv is in the tangent space, as it may easily be verified (0 = dt < v, v >= 2 < dtv,y >). • Theorem 3. The second fundamental form has the following evolution equation: . dthj = Ahtj-2Hhiiglmhmj + \A\2hij Proof. We recall the Gauss-Weingarten equations (see [KN]): didjF = T^dkF-hijU and djV = hjiglmdmF 11 Using these and the previous result, we do some computations in a similar fashion as in the proof of our previous lemmas: dthij = -dt{didjF,v) = {didj(Hv),v)-{didjFAH-dmglm) = didjH + H(ai(hjmgmldlF)t u) - (tfjdkF - htju, dtH • dmFglm) = didjH - Y^dkH + Hhjmg^iTldpF - huv, v) = ViVjH - Hhil9lmhmj The conclusion now follows from the first part of Lemma 29 in the Appendix. • With this last theorem we can now deduce a corollary that will.be very useful in the next section where we need to introduce a mesure of the distance between the eigenvalues of the second fundamental form that will be written in terms of some of these quantities. Corollary 1. The following evolution equations holds: 1. dtH = AH + \A\2H 2. dt\A\2 = A\A\2 - 2\VA\2 + 2|A|4 3- dt{\A\\~ ? ) = A(|A| 2 - - 2(|VA|2 - + 2\A\\\A\2 - f ) Proof. The first identity is an easy calculation: Use Lemma 1 and Theorem 1 with H — g^hij. The validity of the second identity can be seen with the help of the following two derivations: dt\A\2 = dttfX^hijhkt) = Wgimgknhmn!?lhi:jhkl + 2gikgJlhkl{Ahij - 2Hhimgmnhnj.+ |^|2^) = 2gik<?lhklAhij + 2\A\4 A\A\2 = g^VkVi^g™hpmhgn) = 2fgmnhpmAhqn + 2gklg™gmnVkhprnhqn = 2gPqgmnhpmAhqn + 2|VA|2 12 ; The third part of our corollary follows from the second one combined with: i dtH2 = 2H{AH + \A\2H) = AH2 - 2\VH\2 + 2\A\2H2 • We conclude the first part of this chapter with a corollary which tells us that MCF makes the area of the hypersurfaces FT(M) decrease with time and keeps the mean curvature H strictly positive: Corol lary 2. With g the determinant of g^, we have that: 1. If Ht is the measure on Ft(M), then fit — (9) * and dtfit = —H2fit, which also implies that the total area \Ft(M)\ of our hypersurfaces decrease with time. 2. If the mean cuvature of FQ(M) is strictly positive everywhere, then it will stay so on Ft(M), for all time, as long as the solution exist. Proof. Since d(gzsgSj) = 0, we find that dtgli = —gipgqjdtgpq = 2Hhij, by the evolution equation of gij. We also have from linear algebra that the metric can be written in term of the cofactor matrice Gij-. <fJ = With these two last facts, we easily deduce the first part of the corollary. The second part follows by the evolution of H in Corollary 1 and the maximum principle. • 2.2 Convexity We now turn our attention to the following problem: Do the hypersurfaces preserve their (initial) convexity along their MCF evolution? As we will now demonstrate, the answer to that question is that they do so as long as equation (1.1) has a solution. We start with some notation and what will later on (next chapter) appear to be a useful lemma. 13 We say that a symmetric tensor. Sy is nonnegative (and write Sy > 0) if all its eigenvalues are nonnegative. Since we have as initial data on FQ(M) that all eigenvalues of the second fundamental form are strictly positive, we can find a small number e > 0 (as we did in the introduction) such that the following expression holds everywhere on FQ(M): hij > eHgij. (2.2) Lemma 3. If H is bigger than zero and (2.2) holds, we get: 1. Z>ne2H2(\A\2 - . . . 2. \Vthkl H-ViH- hkl\2 > e 2 t f 2 l v " l 2 Proof. We prove this lemma in normal coordinates (as a pointwise estimate) so that we have gij — Sij and hij is the matrix with principal curvatures (rvj) on the diagonal entries and zeros everywhere else. With this, we get: z = Hc-\A? = Y J - i - Y . ^ - [ Y . ^ 2 n 2 i i=0 j=0 \i=0 n n 2 — ^(ftjKj + K\KJ) — ^ 2 2KJKJ i<j i<j n = ^ ^ KjAtj'(rtj Kj) i<j >e2H2J2(Ki-Kj)2 and the first part of the lemma follows since: n ^ n n n n J2i<j(Ki ~ Kj) n We recall here that C = g%igklgmnhikh\mhnj and that Z = HC — \ A\4 (from the appendix). For 14 the second part, we find that: Wihki-H-ViH-hu? VihkrH-Vihki • H -ViH • hki + V f c H • hit ViH • hkl + VkH • hu ViH • hki + V f c # • ha + \ViH-'hkl + VkH-,hu\2 > Vihki • H -IViH-hu + VkH-hitf ViH • hkl + VkH • hu + \\/iH-hkl + VkH-hu\2 where we got rid of the scalar product in using Codazzi equations (see appendix) and consider-ations with respect to symmetry-antisymmetry of both sides of the product. To complete the proof, we need only to consider points on the hypersurface where the norm of the covariant derivative of the mean curvature do not vanish, otherwise the result is trivial. Around such a point we introduce an orthonormal frame ex, ...,en such that ex = T^J JT : \VH\ i = 1 0 i > 2 In these new coordinates, we evaluate the last term of our previous calculation and prove the result: Ei , f c , * (V i# • hki + Vfcg • hu)2 > {VxH-hn + VzH-hxi)2 + ( V 2 # • + V i i f • hnf > ( f r 2 2 ) 2 | V # | 2 2 e2H2\VH\2 • 15 We will now introduce a generalized max imum principle for tensors on manifolds which is a crucial ingredient in what comes next: This result has been initially proved in [H]. We first introduce some notation. Let uk be a vector field and g^, M^, Nij be symmetric tensors (may depend on time) on a compact manifold M such that = p(Mij,gij) is a polynomial formed by contracting products of M^'s with the metric. Note that all these tensors might depend on time. We further require that satisfy the so called null-eigenvector condition: NijXlX^ > 0 for any null-eigenvector X of M^. T h e o r e m 4. Suppose that for t G [0,T) the following evolution equation holds: dtMij = AMij + ukVkMij + Ni:i Then, if Mij > 0 at t = 0, it remains so on all of [0, T). Recall that My > 0 means that MijV%vi > 0 for all vectors v1. Proof. We will show that My > 0 for t G [0,6) and repeated applications of the argument will cover all of [0, T) in a finite number of steps. We first define: M*j = Mij + e{6 + t)gij and claim M*j > 0 for t G [0,6} and all e > 0. Letting e go to zero will complete the proof if the claim is verified (since S will be bounded in terms of |My| only). Suppose our assumption is false, then there exist a first time n G [0,6] where M*j has a null-eigenvector vl (w.l.o.g.) of unit length, at some point x G M. If we write N*j = p{M^,gij), the null-eigenvector condition gives us that N*jVlvi > 0. Since p is a polynomial, we get: )N*j-Nij\<C\M*j-Mijl where C is a constant depending only on Max(\M*j\ + |My|). Keeping e,S < 1 gives that max\M?j\ depends only on max\Mij\. From this we have: Nijtfvi > -CeS where C only depends on max\Mij\. We now extend the vector vl to a vector field in a neighborhood of x such that = 0 at x and vx independent of time. We define the following 16 function: f = M*ijviiP and have / >0on0<t<?7 and M, which tells us that dtf(x, n) = 0 and A/(:r, rj) > 0. With these and: dtf = (dtMijWvi + e, dpf = dpMijvV, A / = AAfyuV we get from the evolution equation of M^: which gives us that NijVlvi < —e. This gives a contradiction if we choose 8 such that C5 < 1. • • We now go back to our hypersurface evolution problem and use the previous Hamilton's result in the sequel: Corol lary 3. Suppose h^ > 0 on FQ(M), it then remains so for t 6 [0,T). Proof. Apply Theorem 4 to the result of Theorem 3 with Mij = h^, uk = 0 and Nij = -IHhu^hmj + ^hij. • T h e o r e m 5. IfeHgij < h^ < f3Hgij and H > 0 on FQ(M) for some constants 0 < e < ^ < ft < 1, it then remains so on the entire time interval [0,T). Proof. We first prove the left inequality: we want to apply Theorem 4 with Mij = ^ - egij, uk = I ^ . V / i f , Nij = 2eHhij - 2himgm%j With the help of Theorem 3 and Corollary 1 we verify that this choice of tensors satisfies the appropriate equality: 17 A \-H) ~ W H9 V F C I ? V ' \~H so it is now easy to see that the assumption of Theorem 4 are respected if satisfy the null-eigenvector condition. Suppose that Y is a vector for which hijYi — eHYi, we then get: NijYW = 0 as required. The proof of the left inequality is thus complete. For the proof of the right inequal-ity, we simply repete exactly the same arguments with minus signs to reverse the inequality. • ' • • Now that we have this last result, we know that our hypersurfaces will stay convex as long as there is a solution to the MCF system (1.1). The next steps in the development of the theory are then related to the boundedness of some geometrical quantities: This will in turn give us information about the limitting shape in time of the hypersurfaces as well as on the time interval on which a solution of the MCF equation is well defined. We will also see in a later chapter that the maximal time interval of MCF solution must be finite, but in order to prove that and other geometrical properties, we first need to workout the (quite technical) next two chapters. 18 Chapter 3 The Relative Eigenvalue Difference of A In this chapter we bound the distance between the eigenvalues of the second fundamental form with respect to the mean curvature. The main result is Theorem 6 and essentially all the rest of the chapter is dedicated to the development of some relations used in its proof. This part of the exposition is particularly long and technical; the reader might prefer to skip some proofs in a first reading. We will use the following quantity: 2 n ^ i - ^ f n n which measure the distance between the eigenvalues of'A. T h e o r e m 6. There exist some constants S > 0 and C < oo depending only on Fo(M) such that n for all times t e [0 ,T) . |2 H 2 M|2 We will work with the function fa = fji-J1 and will bound it by a constant. To achieve that we need several partial results related to the evolution of different power of fa and its integration. L e m m a 4. For any o and a = 2 — o, we have: dtU = Af* + 2 { a ~ l)gpqVPHVqfa - 1^-pi\Hyihkl-ViH-hkl\2 (2 a ) ( a 1) H 2 „ . 2 ^ 2 ' W - V ) | V H | " + ( 2 a ) l ^ | 2 / ' 19 Proof. With the evolution of H and \A\2 in Corollary 1 we calculate: dt\A\* . o^AfdtH (2 - g) . _ a „ ft/ff = " l T " " H S i ~ ^ a t H A | A | 2 - 2 | V ^ l | 2 + 2|yl|4 a|A|2(Aff + |A|2#j'. H a Ha+1 {2~a)Hl-a(AH+\A\2H) n We also have the following expressions: Vija — fja-fl • n n v ill &f* = 9PiVpVifa rnfffVPViM2 - a l i4 | 2 V p VtH. _ (a + l ) V p H ( i ? V i l A | 2 - a l A p V . g ) V p gV» |A | 2 - aVp\A\2ViH (l-a)(2-a)WpHViH _ ( 2 - o ) F l . n • •• . = HA\A\2-a\A\2AH _ (2 - a ) g l _ a _ (1 - a)(2 - a)|Vff|2 ~ff5+r < V i , i 4 ' , v < i f > + — — F S + 2 The desired result then follows from the expressions found for dtfG and Afa with a simple use of the next relation: , \Vihkl • H - ViH-hkl\2 = H2\VA\2 + |.4|2|Vtf|2 - 2 < V ^ , V;tf • hkl > H = H2\VA\2 + \A\2\VH\2-<Vi\A\2,ViH>H • One might be tempted to use the maximum principle on our last result but we can see that this approach would not work, simply because the term (2 — a)\A\2fa is positive. We take another approach and will need to derive many integral estimates in order to achieve our goal (Theorem 6). 20 Corol lary 4. For any o, we have the estimate: dtfa < A / . + 2 { a ~ 1 ] < Viff, V i / f f > - ^ l V f f | 2 + a\A\2fa (3.1) Proof. By Lemma 3 we have 2 • |Vih/u 1 H - VjF • / i f e / | 2 > e2H2\S7H\2. Also, by Theorem 5 the main convexity hypothesis remains true on all of 0 < t < T. Using these with Lemma 4 gives the result. • L e m m a 5. Let p>2. Then for any n > 0 and 0 < o < ^, we have the estimate: ne 2J PGH2dii < (2Vp + 5) J ±.fP-^H\2dfi + r)-l(p-l) J / r 2 | V / C T | 2 ^ Proof. We write = hy — ^ L for the trace-free part of the second fundamental form. Using Lemma 29 from the appendix and making a substitution in the expression of A/ C T found in the 21 proof of.Lemma 4 gives us: )<hij,ViVjH> , 2|VA|2 j r2^ <Vi\A\\ViH> A/a = 2-£ z ^ ) # i - " A f f - 2 a -n + a(a + l)\A\ Ha+2 ( 2 - a ) ( l - a ) | V F | 2 n 2<h°ij,ViVjH> H + 2Z 2 AH W + n ~H° _ a | , 4 | 2 ^ - V^H^AH -2a + 2 l ^ 4 f V <V<|i4|2,V<H> Ha+1 n 2 |V i? | 2 (2-a)( l -a) |VJf | 2 n 2<h°ij,ViVjH > H 4 a < v d w > + 2 i ^ y + 2 a + "2Z" + 2 < V i | A | 2 , V i j r Y > Ha+2 + (a(a + 1) - 2 ) ' ^ a V + f 2 - (2 - a)(l - a) |Vtf| 2 2<h%,ViVjH >' 2Z' [a/aAFl -+ \Vihkl-H -ViH -hki Ha+2 <Vi\A\\ViH> \VHf l V „ , 9 J 4 2 - ( a - D ( a - 2 ) ^ ] 2<h0ij,ViVjH> 2Z__ afgAH 2\Vjhkl • H - VjH-hkl H Ha H H<*+2 H so that we can write the following estimate: <h0iPViVjH> 2Z Ma > 2-We now multiply this last estimate by f% 1 and integrate over the hypersurface. With the use 22 of Codazzi equations and integration by parts, we get (since Ft(M) is compact) : - 2(a - l ) J ^ - < VtU, ViH>dli-aJ ^AHdfi + 2 J iL^g^tf'hifViVjHdLi = (p- 1) j / r 2 | V i M 2 d M + 2 j ^j^du -2(0,-1) j iL-<VifA,S7iH>dfi + 2a JJ^<h°ij,VIHS7jH> - 2 J | ^ 5 V V 4 • VjHdn " « / S-j^\VH\2d» - 2 ( p - l ) | ^r<h%,ViH-Vjfa>dn ' , • + a p J ^ - < ViH,Vifa >du By Codazzi equations, we get: 0 > (p - 1 ) | #- 2 |ViM 2d/x + 2 J ^ r d » - 2(a -1) Jl°~< S7ifa, VtH > dfi + toJ^<$,ViHVjH>dp-2!^ j f ^ V i z f ^ -2{p-\)jf-^<h%,ViH-Vjfa>d^-aJ^\VH\2dfji + ap J < ViH, V i / a > dfi We will now use Young's inequality and some other identity: ab< ?a 2 + -*-62, a < 2 — 2 277' 7/2 U<H2-a, \h°ij\2 = \A\2- — = faH<* 23 Rearranging terms in our last estimate and using the identities above allows us to get: ^Wr*" * J ^WH?d, + j A | V i f i > + J ^r\<Vifa,ViH>\dn + lJ§^\<h%,Vi-VjH>\dii + (p-l)^\<h°j,ViH-VjU>\dv + P f^f-\< V i ^ , V i / , > \dn - ^-=-^ f / r 2 | V / C T | 2 d M + 2 / ^ r r l ^ l l v ^ M + (P-V j f^\h%\\VH?\Vfc\d» + P j.^\VH\\VU\dii-^^.j / r 2 | V / a | 2 We now take 772 = 1 and 771 — 773 = 774 = 77* to get: 24 Finally, if we write rf = 2n, then since p > 2 we find fp-i r fp-i It is now easy to get what we wanted from Lemma 3 and Theorem 5. • Given some conditions on the possible values of a and p, we are now ready to produce a bound on the Lp-norm of fa: . Lemma 6. There is a constant C < oo depending only on FQ(M) such that for allp > 100e-2, a < |e 3p -^, we have : ( f f%du\ <C, for all te[0,T). \JFT(M) J Proof. We first fix the constant C: C~{\FQ{M)\ + \)-Supce%l]{SupFQ{M)fa) With this choice of C, we see that in order to prove our lemma it is sufficient to show that dt J f<jdu < 0 on the entire time interval. In the following derivation, we use Corollary 4 and the identity dtfJ-t = —H2IM'-dt j f„du = JpfrldtUdfi- J H2f^dn -Jf%H2du <-JP(p-i)fr'2\vf<r\2d,i+Jp^jplfr^m-ivfaidv - _/>/r^g'2d^ + jopf^du - j flH2da Rearranging terms in the inequality such that all signs become positive (coefficients) and using (3.2) and (3.3) on the term with coefficient 2(p— l)p gives us: dt J tfdu + P - ^ - J / r 2 | V / C T | 2 ^ + ^ / / r ^ ! i ? | 2 ^ < apj H2fadn (3.4) 25 where we, also used \A\2 < H2. By hypothesis we know that a < §e 3 p - ^, so this and Lemma 5 in (3.4) gives: •p-i <|p*(2»B» + 5J | ^ I V ^ + ^ - ^ b - l ) / / r 2 | V / | 2 — 1/2 which is true for all 77 > 0. The result then follows with 77 = — . Corol lary 5. If we assume p > (^)228e 6 and a < n e 3 ^ 6 1 / 2 , we <7e£: I /P • {$H™fUnYIP <c, Proof. Simple consequence of {fH mfUiJ) llp = ( J 7 £ . ^ ) 1 / p with * 1 m n e 3 p - 1 / 2 , n e 3 p ~ x / 2 ^ n e 3 ? - 1 / 2 a — ° p - 16 " l " 16 - 8 ' . • We are now ready to bound fa: For doing so, we will need the following generalized Sobolev inequality on submanifolds of R n (see [MS]). L e m m a 7. [Sobolev Inequality] For all Lipschitz continuous functions v on F(M), we have: (fF(M) \ v \ n ^ d „ ) { n ~ 1 ) / n < C{n) • ( / F ( M ) |V«|d/i + H | t ; | d / i ) P r o o f of T h e o r e m 6: We define f^k '•= max{fa — k, 0), for all k > ko :— Sup^fc, and A(k) := {x € Ft{M)\fc > k}. If we multiply by pf^1 the inequality found in Corollary 4, we can derive the following estimate exactly as in the proof of Lemma 6 (for p > 100e~2): 9t [ fad? + / I V M 2 / ^ < op f H2fjfadn. JA{k) 1 JA(k) JA(k) On A(k) we have > | V / ^ 2 | 2 = ^ / ^ 2 | V / C T | 2 and thus obtain, with v := f^: dt f v2dfx+ f \Vv\2dn<op f H2fldii. JA(k) JA(k) JA(k) 26 From now on we will denote by Cn any constant depending only on n. Then our generalized Sobolev inequality (Lemma 7) and Holder inequality gives us: (\ i/<? • • / \ 2 / n / \ 1/9 / v2qdu\ <Cn f \Vv\2dfi + Cnl I Hndv) If v2(>du) (3.5) JF(M) J JF(M) \JSupp(v) J \JF(M) J with q = 2 ^ 2 . Since Supp(v) C A(k) and fa > k on A(k), we deduce from Corollary 5: (f Hndu\ < fc-2p/n (f Hnffdu\ < k~2pln- C2p'n \Jsupp(v) ) \JA(k) J as long as p > 28e~6 and a < — . So if we put this last bound in (3.5), we obtain: 1/9 / \Vv\2dfl > Cn • [ f V^d/A JF(M) \JF(M) ) which implies: 1/9 lA(k) \J ) JA(k) and then integration w.r.t. time yields: 1/9 dt f v2dix + Cn • ( f Aik^dfj) ' q <ap f H2fpdu JA(k) \J / JA('k) Sup[0T] f v2dfi + Cn- f I f v2qdu) dt<apf H2f§du (3.6) JA{k) JO \JA(k) J JA(k) for k > k\ = k\(ko,Ci,n,e) since v = 0 at t = 0. We now recall the formula for the interpolation inequality for Lp-spaces: (\ l/d / \ a/q / \ 1—a / v2ddu) <[[ v2qdA If v2du) , ! = £ + ( ! - a ) JA(k) ) \JA(k) J \JA(k) J a q with a = 3 and 1 < d < q. We now get from (3.6): (j , \ l/d t / / v2ddfidt ) < Cnop f f H2fPdudt JO JA(k) J Jo JA(k) ( T \ f f H2rfprdfidt) Jo jA(k) I 27 by Holder inequality with any r > 1 and \A(k)\ := fQT fA^d(j,dt. With v := (fa,k)p/2 and (/QT fmv2ddiMdt)l'd > \A{k)\^ld / 0 T JA{k) H2r(fa,kyd»dt, we then get: / T / y? f cd/id« < Cnop\A(k)\2~z~r ( [T f H^iUrdfidt JO JA{k) ' \J0 JA(k) We take r sufficiently large so that j :— 2 — ^ — £ > 1, which in turn implies we can take p > re~6210 and a < e 6 2- 9 r - 1 / 2 such that by Corollary 5 for h>k>k1: \h-k\P-\A(h)\ <C2(n,Cl,e)-\A(k)\i - ' By a result of Stampacchia ([S], Lemma 4.1) stated in the appendix (Lemma 31), we easily deduce from our last inequality that: fa<ki + d with dP = C22^^-^\A(k1)\^-1. Since Corollary 2 provide us with d^ < |Fo(M)|, the completion of a proof of Theorem 6 only requires us to show that T is finite: L e m m a 8. T < oo Proof. We already found that dtH = AH + H\A\2 > AH + Let G be the solution of dtG = § with G(0) = Hmi/n(0) > 0. We then have: dt{H - G) > A(H - G) + IZ^-l, since AG = 0. By the maximum principle we have H > G on 0 < t < T. The ODE solution for G(t) is easily found to be G{t) = Hmin(0)/(1 — ^Hmin(0)2 • t)1/2 and since G —>• oo as t -> %Hmin(0)-2, it follows that T < oo. . D The proof of Theorem 6 is now complete. 28 Chapter 4 Some other Bounds on Curvature In order to know the shape of the limitting object our hypersurfaces evolution problem converge to, we need to be able to compare the curvature at different points of a given hypersurface Ft(M) at a given time: That is the main reason why we devote the first section of this chapter to show that \S7H\ is bounded. We will then turn our attention to the derivatives of the second fundamental form: The fact that our flow has a well defined limit requires in its proof some additional knowledge on the derivatives of A, in particular of its bounds. 4.1 Bounding \VH\ In this section we prove the following result: T h e o r e m 7. For any n > 0, there is a constant C(n,Fo(M),n) such that |Vtf| 2 <VHA + C{Ti,Fo(M),n). We will need the following ecolution equation: L e m m a 9. dt\VH\2 = A|Vtf | 2 -2|V 2 tf | 2 + 2|A|2|Vtf|2 + 2 < ViH-hmj,V jHVhim > +2H < ViH,Vi\A\2>. Proof. By the evolution of g%i and H, we find: dtlVH^dtigVViHVjH) = 2Hh^ViHVjH + 2gij(Vi(AH + \A\2H) • VjH) = 2H< hi:j, ViHVjH > +2gijVi(AH) • VjH + 2gijVi(H\A\2)VjH 29 Using the interchange of covariant derivatives formulas (see appendix), we get: A(VkH) = Vk(AH) + gi'ViHiHhkj + hkmgmnhnj) A simple check also gives us: A|V#f = 2gklA(VkH) • VtH + 2|V 2#| 2 Reassembling everything proves the lemma. We now have an easy consequence of Lemma 9: Corollary 6. dt\VH\2 < A\VH\2 - 2|V2tf|2 + 4|A| 2|V#| 2 + 2H < VtH, Vt\A\2 > Lemma 10. dt (&f.) < A + 3|A|2 + 2 < ViH, Vi\A\2 >. Proof. , \VH\2\ _ 0t\VH\2 • H - \VH\2dtH H J H2 < [[iJ(A|Viy| 2-2|V 2H| 2+4|A| 2 |Vif| 2 + 2H < ViH, Vi\A\2 >) - \VH\2 {AH + \A\2H)]]/H2 _ HA\VH\2 - |Vffl 2Aff AV2H\2 0\A\2\VH\2 H2 H + H + 2<ViH,Vi\A\2 > A i \VH\2\ _ HA\VH\2 - \VH\2AH \VH\2 A \ T X ) 7-T-O 4" 2" H J H2 ' H* KHVjVjH, VjHVjH^ H3 These last two together with — 21 H 1 2' ^ < —4 H* — — (by Schwarz inequality) give the desired result. 30 Lemma 11. We have the following evolution equations: 1. dtH3 = AH3 - 6H\VH\2 + S\A\2H3 2- dt[{\A\2 - ^) H] < A[(\A\2 - ^) H + m 2 - f ) H] - ^ # | V A | 2 + C 3 |VA | 2 Proof. The first part is a direct consequence of the evolution equation of H. To prove the second one we first use Corollary 1 to find: HA | |A| 2 - —) - 2H (|VA|2 - | V H | 2 n n + (\A\2 - ^) (AH + \A\2H) + 2H\A\2 (\A\2 - ^) = A 14 # 2 2 " ' F n + 3|^| 2F |A| H2^ n ) 2H \WAf | V F | 2 n H2 n From Theorem 6 and Ih^A2— (|A|2 - we get for the last term in (4.1): 2 H2\ ViH,Vi[\A\'-—) = 4| < vy? • Vi/ig, > | < 4\vH\\h°kl\\vA\ <A\VH\cl'2Hl-&l2\VA\ < AnCl'2Hl-*\VA\2 2(n-l) < 3n -H\VA\2 + C(n,C0,6)-\VA\2 (4-1) where we also used Young's inequality and |Viz"|2 < n|VA| 2 as before. This last inequality in (4.1) with the use of Lemma 30 (see appendix) to eliminate the term with coefficient — 2H proves the result. • Proof of Theorem 7: We now bound / := + N (\A\2 - ^ H + NC3\A\2 - nH3, for 31 some laxge N dependind only on n and 0 < 77 < 1. From Lemma 10 and Lemma 11 we find: dtf < A / + 3\A\2 (^1.\+ 2 < Vitf, Vi\A\2 > +6rjH\VH\2 - 3n\A\2H3 V H -N2(n~^H\VA\2-r2NC3\A\4 + 3N\A\2H(\A\2- — on \ n Now since ^ < \A\2 < H2 (pointwise estimate), |VH"|2 < n|V./4|2 and 77 < 1, we can choose N sufficiently large such that: dtf < A / + 2NC3H4 + 3NH3 (\A\2 - —\ - -r>H5. \ n J n By Theorem 6 we see that we have: 2NCZHA + 3NH3 (\A\2 - < 2NC3H4 + 3NC0H5~5 <-r)H5-rC(n,6,n,Co,C3) n since for large H the ^r/H5 term is bigger than the two previous ones. We can now write: dtf < A / + (7(77, FO(M)) =>• Max[f(t)] < Max[f(0)] + C(n, FQ(M)) • t => f(t) < C'(TI,FQ(M)) which in turn implies, since 77 > 0 is arbitrary: | V#| 2 < nH* + C'(n, F0(M)) • H < 2nH4 + C"(n, F0(M)) which complete the proof of Theorem 7. 4.2 Derivatives of A We.now turn our attention to the time derivative of higher order covariant derivatives of the second fundamental form. We start with some notation: Define S*T :— any linear combinations of tensors formed by contraction of S and T with the metric gi:>. For example, we find: 9tX% = lAvMyk,) + v,(%,7) - Vi(dtgjk)} = -gil[Vi{Hhkl) + Vk{Hhjl)-Vl{Hhjk)] 32 The following sequence of arguments and lemmas is developed, following [H], in order to prove a result (Theorem 8) first obtained by R.S. Hamilton in [H]. Lemma 12. Along a MCF, if B and C are tensors satisfying dtC = AC + B, we then have: dt(VC) = A(VC)-)-A*A*VC + A*VA*C + VB. Proof. Because of the expression found for the time derivative of a covariant derivative, we have that: dt(VC) = V(dtC) + A * VA * C. By the interchange of derivatives formula, we also get: V(AC) = A(VC) + A * S/A * C + A * A * VC. Combining these two with the hypothesis of the lemma complete the proof. • From this last lemma and Theorem 3 we find: dt(VA) = A(V-A) + A * A * V A Lemma 13. 6\(Vn,4) = 'A(VM) + J2i+j+k=n ^ A * ^ J A * Proof. For n = 0,1 the result is verified. Suppose it is true for n. By Lemma 12 with C and B = £ i + j + f c = n ^ A * V J A * y f c 4 w e n n d : dt{Vn+1A) = A{Vn+1A) + * W J A * y f e A i+j+k=n+l Theorem 8. For any m, we have: dt\VmA\2 = A\VmA\2 - 2\Vm+1A\2 + £ i + J - + f c = m VM * * V*A * VmA. Proof. dt\VmA\2 = 2 < V"M, ^ ( V M ) > +A * A * V m A * V"M where the last term arise from differentiation of the metric. The result now follows by Lemma 13 with: A |V m v l | 2 = 2 < VmA,A{VmA) > +2\Vm+1A\2. • = VnA • 33 We need another result (Lemma 15, see below) due to Hamilton [H]: We again begin with some useful developments in order to prove the final assertion. Theorem 9. Let Mn be a compact riemannian manifold of dimension n and T be any ten-sor on Mn. Suppose ± + '± = r with r > 1. We then have: (f | V T | 2 r c f y i ) 1 / r < (2r - 2 + n) ( / | V 2 T-| /^/z) • ( / Proof. Following Hamilton's proof, we carry our calculations in an orthonormal basis and con-sider T = {Ti} as a 1-tensor since in the more general case the proof is exactly the same (with many more indices!). Using integration by parts, we find:. J\VT\2rdn = JviTj-ViTj-\VT\2r-2dn = - J T j V i V i T j • | V T | 2 r - 2 d / x - 2(r - 1) J {TjViVkTu ViTr VkTt)\VT\2r~4dn With the following basic inequalities: \Ti • ViViTj\ < n\T\\W2T\ and (T^Vfcl], • VkTt) < |T| |V 2 r| |VT| 2 we can see that: J\VT\2rdn<(2r-2 + n) J \T\\V2T\\VT\2r-2dn. Making use of Holder inequality (with | + ^ + = 1) on the right hand side of this last relation complete the proof. • We will also need the following lemma and two corollaries: Lemma 14. Let' f(k) be a real valued function of the integers k € [0, n\. If f(k) < \{f{k — 1) + f(k + 1)), then: f(k) < (1 - f)/(0) + j/(n). Proof. The assumption still holds on f*{k) := f(k) —.(1 — ^)/(0) — so w.l.o.g. assume /(0) = /(n) = 0. Define g(k) := f(k) — f(k — 1), then the assumption give: g(k) < g(k + 1). 34 We now define the integer m such that #(1) < ... < g(m) < 0 < g(m + 1) < ... < g{n). For any A; we can see that : f(k) = Yli=i — ~ J2i=k+i We now observe that when k <m the first summation is negative and when k>m the second one is. So, =$> f(k) < 0 for all k. • Corollary 7. / / f(k) satisfies f(k) <}[f(k - 1) + f(k + 1)] + C, then: f(k) < (1 - |)/(0) + £/(n) + Cfc(n-fc). Proof. Apply previous lemma to g(k) := f(k) + Ck2. • Corollary 8. If f{k) satisfies f(k) <Cf{k-l)ll2-f{k+l)ll2,wefind: f{k) < ^ - ^ / ( O ) 1 - ^ " -f{nf/n. Proof. Apply previous result to g(k) := log[/(/c)]. • We are now ready to prove the following Hamilton's estimate: Lemma 15. IfT is any tensor and ifl<i< m'—l, then with a constant C(n,m) independent of the metric and the connection, we have: j | V i T | 2 m / i d / x < c . MaxMn\T\2{rr-V . J\WmT\2dn. Proof. We first apply Theorem 9 to the tensor V ' - 1 T with 2 < i < m -1 and p = , q = ^ , r = j. We get: with C := 2 f - 2 + n. We will now make use of Corollary 8. We define /(0) := MaxM\T\ , /(0:= ( / i V ' T ^ d / z ) ^ , l < i < m . 35 Since from Theorem 9 we have: 7(0<C/(* + l ) 1 / 2 ; / ( * - l ) < / 2 we then obtain the following (and the result): / ( * ) < C / ( 0 ) H s . / ( m ) £ • Theorem 10. From our last Lemma, we derive: dt J\VmA\2dn + 2 J \Vm+1A\2dn<C{m,n)-MaxFt(M)\A\2J \VmA\2dfx Proof. We first use Holder to find (i + j + k = m): J |VM||V J A||V f c A||V r M|d/i < ( / 1 * 4 * 4 . ) * ( / V'AYTd„)* ( / ( / I V - A P ) 1 Using this and dtdfit — — \H\2dfxt while integrating the equation of Theorem 8 gives us, by Lemma 15, the desired result. • 3 6 Chapter 5 The Finite Time Interval The main result of this short chapter is the following self-explanatory Theorem 11; we will dedicate the rest of the discussion to its demonstration. Theorem. 11. The solution of the MCF problem (1.1) exist on a maximal time interval t € [0,T) for T < oo and MaxpT{M)\A^ becomes unbounded as t —• T. We already know from Lemma 8 that T is finite. We will now proceed by contradiction: If MaxpT(M)\A\2 < C for t —*T, then our hypersurfaces Ft(M) converge to a smooth hypersurface Fr{M). Applying short time existence then contradict the maximality of T. In fact, if we have: MaxFt(M)\A\2 <C, . te [(),T), •.- (5.1) from the MCF evolution equation (1.1), we obtain: \F(x,t2) - F(x,h)\ < I2 H(x,t)dt, 0<h<t2<T. Jtx This implies that F(-,t) converge to a continuous limit F(-,T) since H is bounded. In order to show that this limit represent an hypersurface, we will need another result from [H]: Lemma 16. Let Qij be a time dependent metric on a compact manifold M for 0 < t < T < oo. Suppose J 0 Max\dtgij\dt < C < oo. Then the metrics gij(t) for all different times are equivalent and they converge uniformly as t —> T to a positive definite metric tensor gij (T) which is continuous and also equivalent. [We use: \dtgij\2 := glkg^l{dtgij){dtgki)] Proof. We first take a tangent vector v G TM at some point x € M and write: M? ••= ga&tyvi. 37 We proceed by recognizing that from dt\v\2 = (dtgij)vlvi we get (by the Cauchy-Schwartz inequality) that: dt -\og\v\l\ < \dt9ij With this last relation we deduce for 0 < t\ < t<i < T: log such that if the integral is finite, all of the metrics are equivalent (using the exponential func-\v\t ~~* \ V \ T uniformly for \v\j, a continuous function which cannot be equal to zero if v is not. Since the parallelogram law still holds in the limit, we use the Polarization Identity to conclude that the limiting norm is induced from an inner product gij{T) given by: In view of the fact that our hypersurfaces are all diffeombrphic and using Lemma 1 and (5.1), we see that Lemma 16 applies to the pulled back metrics on M (with F(-,t)) of the metrics denned on FT(M), so that it only remains to show that FT(M) is effectively smooth (and then get our contradiction). Prom the evolution equations of Ft and g^, it will be sufficient to bound all derivatives of the second fundamental form A. Lemma 17. / / (5.1) holds on 0 < t < T < 0, then \VmA\ < Cm for all m, with Cm := tion), which is the case by assumption. We also get from this last bound (uniform in x) that g(v,w) = -{\v + w • Cm(n,Fo(M)). Proof. Integration on the result of Theorem 10 gives us: We then use Lemma 15 and obtain: m, q < oo. 38 Using the following Sobolev Inequality for p> m (see [H], section 14): Mdx\f\*^cJ(\f\* + \Vfr)diMi (5.2) and applying it to the function fm := |V"M| 2 , we finish the proof. • This last result with the use of Arzela-Ascoli Theorem (remember that M is compact) tells us that our smooth immersions do converge to a (unique) smooth hypersurface FT(M), which by . short time existence contradicts our assumption. This complete the proof of Theorem 11. 39 Chapter 6 Mean Curvature Ratio We are now ready to study more carefully the shape of our hypersurfaces as they evoluate in time. In this chapter we will compare the maximum value of the mean curvature to its minimum value by taking a close look at their ratio as time approach T. This will already give us a good idea of what Ft(M) tend to as we let t —> T. First observe that since \A\2 < H2, we obtain from Theorem 11 that Hmax is unbounded. Theorem 12. Hmax/Hmin -> 1 as t -> T. In order to prove this result, we will need the following theorem (see [CE], pp 27) as well as the subsequent lemma: Theorem 13. [Myers] Let Mn be a complete Riemannian manifold of dimension n. Then, if Rij > (n — l)Kgij for some constant K, we have that every geodesic of length > •nK~ll2 has conjugate points and the diameter of Mn satisfies: d(Mn) < irK~ll2. Lemma 18. If hij > eHgij holds on Ft(M) with 0 < e < i , then we have Rij > (n — \)e2H2gij. Proof. Remark that H > eHn: The result is then a simple calculation starting with Rij = Hhij — hiPgpqhqj, which is itself a direct consequence of Gauss Equation. Proof of Theorem 12: From Theorem 7 we have that |V.ff| < \n2H2 + C{n), for any n > 0 on 0 < t < T. We know that Hmax becomes unbound as t —• T, so there exist a 6 < T such 40 that C(n) < ^r)2HlaX(6) \VH\ < n2H2max for t > 9. Let x e F6{M) with ff(a:) = F m a a ; . By our previous inequality we easily see that along any geodesies of length at most rf^H^^. starting at x we have H > (1 — rj)Hmax. Then by Myers Theorem and Lemma 18 these paths (geodesies) reach any points of Fe(M) if n is small enough => Hmin > (1 — n)Hmax on Fg(M). These last arguments are valid for all t € [0,T) and every n > 0 : The result follows. To conclude this chapter, we derive some consequences of Theorem 12: These will appear to be useful in some subsequent developments. Theorem 14. / Q T H%iax(t)dt = oo Proof. The ODE dtg = H^ax • g with initial data 5(0) = Hmax has a solution since Hmax is continuous: log(g(t)/g(0)) = /„* H^axdt'. . Also, as already obtained: = AH + \A\2H < AH + H^axH. From these we can see that (since g is independent of any space variables): dt(H - g) < A(H - g) + • (H - g). By the maximum principle, we get: H < g for 0 < t < T. This implies that g —» 00 as t —> T. • Corollary 9. / / h(t) := —-, then f* h(t)dt = 0 0 . ./„ du Proof. Since H^j n < h < if^ a x, the result follows directly from Theorem 12 and Theorem 14. • Corollary 10. - ±) 0 as t -> T. Proof. Consequence of Theorem 6 and Theorem 12. . . . . " • 41 We can see (geometrically) that our MCF solution Ft(M) is confined into the region of R n + 1 determined by Fto(M) for any to < t. This implies, by Theorem 12, that the diameter of Ft(M) goes to zero when t approach the maximal time interval T so that we have achieved a proof of the first part of our main result. It remains to show that the hypersurfaces converge to a round point, which will be achieved in our two next chapters. In order to do that, we will need to study a normalized MCF equation that keeps the area of the hypersurfaces fixed. This process will provide us solutions from which we can study the limiting shape as t —• T. 42 Chapter 7 Rescaling The MCF Solution From now on we will assume without lost of generality that the point (P, say) to which Ft{M) shrinks is the origin of E n + 1 . We will write our MCF equation (1.1) in a normalized form such that its solution keeps the area of the hypersurfaces fixed. This is done in the following fashion: First multiply the solution of equation (1.1) by a function of the time parameter so that the area of the resulting hyperfurfaces equals that of FQ(M): ,' F(-,t):=m-F(;t) [ d~u= j d /x:= |F 0 (M) | , te[0,T). JF(M,t) JFo(M) Define a new time variable in the following way: Jo With these definitions we easily derive the following normalized expressions for important geo-metrical quantities: 9ij-^ 29ij, hij = iphij ff = V~Jff, , \A\2 = i/;-2\A\2. An easy (three steps) calculation also provide us the equation: , - i o / h J o H 2 d u We can now deduce the normalized MCF equation: diF^dtF-^ = ip-2(dti>-F + xpdtF) at - hF • •• -Hv + — n 43 where we used h := ip~2h. We next prove a result that is usefull in computing the normalized evolution equations of other geometrical quantities when we already know their original form. Lemma 19. Suppose P and Q are objects formed in terms of the metric gij and the second fundamental form A which satisfy the evolution equation dtP = AP + Q and P has degree a (P = ipaP). Then Q has degree (a-2) and we have: d t P = AP + Q + %KP. Proof. This is an almost direct consequence of what has been done before the lemma: d-tP = i>-2[ail)a-l{dtil})P + i>a(dtP)} = i/j~2[-hP + ipaAP + ^aQ] = -hP + AP + Q , n • A very useful consequence of this last lemma comes next: Lemma 20. We have the following normalized relations: 1. hij > eHgij 2. Hmax IHmin * 1 t > T 3. \A\2/H2 ^1 /n as i-*f Proof. Direct from Lemma 19, Theorem 5, Theorem 12 and Corollary 10 and the equations there convert to identical form. . - • Lemma 21. There exist C±, C$ such that for 0 < i < T we have: 0 < C4 < Hmin < Hmax < C5 < CO. 44 Proof. For M a compact Riemannian manifold with boundary, the Divergence Theorem tells us that: fM div(X)dV = JdM < X,u > dS where X is a smooth vector field on T M . Using this formula with dM := Fx(M), M the smooth manifold consisting of dM and its interior points in R n + 1 and X being the coordinate vector field of M (on TM), we obtain for the volume V enclosed by Ff(M): V = —!—- f F Pdfi n + l JF-t{M) where F • v must be positive since by our assumption the origin is inside the hypersurfaces at all time. One can also verify the following Isoperimetric Inequality: n n - 1u; n_i|t/| n~ 1 < |df/|n (where \U\ is the volume of the domain in lRn) which adapted to our situation gives: V{ < Cn\Fi(M)\(n+lVn = cn|Fo(M)|^n + 1^n. ; (7.1) We also obtain from the first variation formula (see [J] or page 61) that: |F0(M)| = \Fi(M)\ = ± JH{F • i>)d~u < HmaxV-t which complete the proof of the first inequality (lower bound) by using (7.1) as well as Lemma 20(2). For the upper bound, we first observe that hij > eHmingij enables us to conclude that the en-closed volume V of the hypersurfaces will be smaller than the one of a ball of radius (eHmin)~l since the principal curvatures are locally given by one over the radius of the circle they approx-imate: Vi< C n ( e F m i n ) - ( n + 1 ) Again using the first variation formula gives: which complete the proof by use of Lemma 20(2). • Corollary 11. f = oo. 45 Proof. Since dt(i) = ip2 and H2 = ip-2H2, Corollary 9 yield: [Th(t')dt'= fT h(t')dt' = JO Jo But Lemma 21 also provide us with the following: h ^ Hmax ^ C5 which implies the result. 00 46 Chapter 8 Limiting Shape of the Rescaled Hypersufaces In this last chapter we complete the work that we began in the last one (in particular with Lemma 20(2)) by making the notion of convergence to a sphere of our MCF solution more precise. The final result will be the completion of the proof of the main theorem. From now on we will use the convention that 8 > 0 and C < oo denote various constants depending on known quantities. L e m m a 22. We can find constants 5 > 0 and C < oo such that: We will make use of some previously derived inequalities in Lemma 5 as well as in the proof of Lemma 6. By setting o = 0 in the second inequality (in the proof of Lemma 6) and using Young's inequality on the last term at right, we obtain: Proof. We first define the following zero degree function: lil!-i = <T\<J=0-H2 n + ^  / / - 2 | V / | V „ + % f ^ | V t f | ^ < 0 47 Using Lemma 5 and \A\2 < H2, we find for large p and n = 2/p: ne2 j'fp\A\2d»<ne2 J fpH2du <(2r1p + 5)J^\VH\2d^ + r1-1(p-l)Jfr2\^fc\2dfi = 9 / ^ H l ^ + ^ l j/r2|VM2d/x so that subtitution in the previous relation gives: Jdtfpdu<-ne2 J fp\A\2du (8.1) with p large enough and e small as before. From dp, = il>ng1/2dx, we can also find the expression: didjx = ^ -2[ml>n-l{dti>)gll2dii - i\>nH2gll2dy\ = (h- E2)d~a so that we now easily derive from relation (8.1): d-t J fpd~u = ^ ~ 2 j dtfvd~u + fp(h - H2)d~u < V" 2(-«5) j fp\A\2d~u + J fp(h - H2)d~u = -sJ fp\A\2d~u + j fp(h - H2)d~u . By what we already proved in Lemma 20(2) and Lemma 21, we see that the last term at right of this expression can be absorbed into the last one for all times larger than a given one. Also, since \A\ is bounded, we can write: ^ J fpdfi < -5' J fpdfi =• J fpdp < Ce'5'1 The result now follows from Holder inequality and Lemma 21. • Lemma 23. We have that f(H - h)2dp, < Ce~5i with h := %(M) Proof. Poincare Inequality insures us it is enough to show that J \ VH\2d(l decreases expo-nentially (the constant in the inequality depends on n and Fx(M)). We now define: 48 . Z : = ^ + N(\A\ 2-^)H, of degree-3, with N a constant to be determined. From Lemma 10 and Lemma 11 we obtain the following estimates: d-tz = i>~2dt < -lip-*hz + 3\A\2Hip-5 (\A\2- — n \ n + V~5 V H + 2< V ^ V i l ^ l 2 > H2\ 2(n - 1) 3n H\V A\2 + CZ\V Af Az + 3N\A\2H (\A\2 - — \ - -hz -rip -5 3\A\ 2| v g | 2 H N n j n 2(n - 1) 3n H - C3) \S7A\ + 2 ^ < ViH,Vi\Af > <Az + 3N\A\2H ( \ A \ 2 - ^ ) - h z for all i > t\ (for some ti) since |ViJ| 2 < n|VA| 2, ^ £ and N is large. We now make use of Lemma 22 and the fact that H has lower and upper bound for all time to deduce from our last inequality that for all sufficiently large time i we get: d-t j zdji = j(dtz)dfl + J(h - H2)zdfi < Ce~6i -6 J zdji-r j(h- H2)zdjl since the hypersurfaces are compact. By Lemma 20(2) we have that (h — H2) —> 0, so for large enough times we obtain: 7 d-t ( esi I zdpL-Ci) < 0 =» / |Vff| ; J H djl < Ce -S'i since C'(l + i)e-si = C'(l + ^ e ^ ' V 5 ' ' < Ce'5'1 for some constant C. Since H is boubded, we get what we wanted. 49 We will from now on turn our attention to some higher derivatives of the second fundamental form. In particular we will be interested in finding some bound on their integrated form. We begin by a useful result from [H]: Lemma 24. IfT is any tensor on F(M), then with a constant C := C(n,m) independent of the metric and the connection we have (with 0 < i < m): i/m / f \ l—i/m J\VlT\2du < c(^f |VmT| 2V) (^J \T\2df^ Proof. The result follows from an application of Theorem 9 to the tensor V 1 lT with p = q = 2 and r = 1 and then making use of Corollary 8. • • Now since both sides of the relation in Theorem 10 have the same degree, it keeps the same form once normalized: d-t f | V m i | 2 d A + 2 f |V m + 1 i | 2 d/x < C • Max(\A\2) f |Vmi | 2d/2 (8.2) JF(M) JF(M) . JF(M) (also with Max\A\2 < C|), Define := hij — ^hgij so that we obviously have WmA = S7mE for all m > 0. We make use of Lemma 24 to estimate the R.H.S. of (8.2): . r / f \ m / ( m + l ) / p \ / \VmA\2dfi <C( | V m + 1 i | 2 d M j f / \E\2dfi\ = (*) from which we obtain by Young inequality (p = (m + l)/m , q = m + 1): {*)<Cn J\Vm+1A\2dfi + Cv-m J \E\2dfl . Choosing n such that Cn < 2 we get from (8.2): di J\VmA\2dfi<C J \E\2djl l 2 2 - r . . 1 7 2 = C / \A\l - -Hh+ -h )dfL 1 1 n n ' = C dp, 50 where both of these last two integration decrease exponentially (by Lemma 22 and Lemma 23. We conclude: Lemma 25. For every m we have: fF(M) \S7mA\2dp, < C on all ofO < i < oo and C := C(m). From Lemma 15 we also get bounds on higher Z -^norms of |V"M|: j\VmA\r <Cm,p (8.3) As we have done previously, we can use another version of the Sobolev Inequality (7) to get: MaxF{M)\V™A\2<C(m). Theorem 15. There are constants 8 > 0 and C < oo such that: (\A\2 — < Ce_<s*. Proof. Define A0 = (h°)ij := h\j — ^ Hgtj, the traceless part of the second fundamental form. We can see that \A°\2 = \A\2 - . Since |V"M°| is bounded we obtain from Lemmas 24 and 22: / r fr-2 A !/(-+!) I |VmA°|2dA..< Cm I / \A\2 - —dii < Cme~6t and now from Lemma 15 also the following one: . ' . J \VmA°\Vd~u<Cm,Pe-8i. Our second version of the Sobolev inequality then yields the result. Lemma 26. There are constants 8 > 0 and C < oo such that: !• Hmax Hmin < Ce 2. \hijH -\h~9ii\<Ce •Si 3. MaxF\VmA\ < Cme~Smi 51 Proof. (1): From the proof of Lemma 23 we have (for large enough times): H2 3r-d-tz < Az + SN\A\ZH I \A\Z - — I - ^hz <Az + Ce~si-5z, the last inequality being a direct consequence of Theorem 15. With this we now obtain that: di{esiz - Ci) < A(esiz - Ci) so that the maximum principle yields (again, for large i): zesi - Ci < C => z < C* (1 + t)e-&i. Since 6 > 0 can be chosen as small as we wish, we get: z < C<r5i and the first part holds. (2): Since \hijH — ^ %y| is of degree zero, it is equivalent to study the non-normalized ex-pression. Now, we can see that since H is a continuous function' of its variables, there exists S Ft(M) such that H{XQ)2 = h for each t. At such a point, the eigenvalues of our matrix are of the form (in normal coordinates): which implies: 1/2 1 1/2 < HCe~si < C'e-Si KjY i<j 1/2 At points where H2 is not equal to h, part (1) tells us that the exponential bound will still do. (3): This one simply follows from the use of the version of Sobolev Inequality in (5.2) and the relation (8.3), just as we did after having recalled this version of the "Sobolev Inequality. 52 • Final Steps We are finally ready to complete the proof of the main theorem. We first notice that all of the hypersurfaces i^(M) stay in a bounded region of the origin 0 € R n + 1 , as a consequence of Lemma 21, since we have a bound on the diameter of the hy-persurfaces in M n + 1 . Also, since Lemma 20 implies that the hypersurfaces are contained in arbitrarily close exterior and interior spheres for large enough times, we can see that we indeed have convergence to a round point in some week sense: We now make stronger and clearer this last (quite vague) statement. We first calculate an evolution equation: diSiij = dt(ip2gij) • ip~2 = 2ijj-l(dtt/j)gij + 2 2 - _ ' ' - -= —hga — 2Hha — —hga — 2Hhij n n which with the help of Lemma 26(2) and Lemma 16 tells us that the metrics gij(i) converge uniformly to a positive definite metric ^ ij(oo) as i —> oo. As we already have seen, the bound obtained in Lemma 26(3) also insures us that the metrics converge in the C°°-topology as well, so gij{oo) is smooth. Now Theorem 15 say that ^ y(oo) is the metric of a sphere, which complete the proof of the main theorem. 53 Part III NON-CONVEX INITIAL DATA 54 Chapter 9 Introduction We have seen in the previous sections that the mean curvature flow of an initially compact-convex smooth immersion is well behaved and makes the hypersurfaces converge nicely (without creating any singularity) to a final round point, in the sense of a normalized flow. We will now turn our attention to the similar problem of MCF of compact but not necessarily convex initial hypersurfaces. Here we still can use a great number of results developed in the previous convex initial data case but the main problem arise from the fact that in this more general setting nothing prevent the hypersurfaces to create singular points before it can shrink smoothly to a final point (if it ever do so). We propose to give a very short introduction to the classical machinery used in these further developments on MCF of hypersurfaces. The main results are the so called monotonicity fomula and a consequence of it related to the asymptotic behavior of the flow near a first singularity point. We will also give an overview of some of the most important results of MCF of entire graphs (non-compact!): As we will see, long time existence in this particular case requires less constraining conditions on the initial hypersurface (graph in that case). A great number of deep results has been found on these lines and we simply give an exposition of the simplest technics. Some further information on these initial developments of the theory can be found in the articles cited in the bibliography. 55 Chapter 10 Formal Statement of the Problem We again consider M to be a n-dimensional compact manifold without boundary and Fo : M —> R n + 1 a smooth immersion of M in the (n + l)-dimensional euclidean space. Note that we do not require FQ{M) to be convex. We define as before the (initial value quasilinear parabolic PDE of second order) Mean Curvature Flow equation as follows: dTF = -H • 0, p € M (10.1) . . F(-,0) :=F0 where H is the mean curvature on the hypersurfaces. Because of the compactness assumption on our initial hypersurface FQ(M), we still have a short time existence result available, just as we did in the earlier part of the text. Now remark that nowhere in the proof of Theorem 11 we had to use the convexity assumption of the initial data. That allow us to conclude the following identical result in the actual context (see part I and II): Theorem 16. The evolution equation (10.1) has a smooth solution on a maximal time interval 0 < t < T < oo, and MaxFt(M)\A\2 becomes unbounded as t —* T. We still have another basic assumption to state in order to go further in our investigation of (10.1): The next lemma indicates where this constraint is from. Lemma 27. The function defined by U(t) :— Maxpt^\A\2 is Lipschitz continuous and satis-fies U(t) > ^ ^ y . 56 Proof. From Corollary 1 we have that: 0t\A\2 = A\A\2 -2\SJA\2 + 2\A\A so that we immediatly get: dtU(t) < 2[U(t)}2 o dtp-^t)} > -2. From these we obviously have that U(t) is Lipschitz as long as \A\2 is bounded and integration gives what we wanted. • From now on we will assume that: < 1 0 - 2 > One can check that this is verified for cylinders and convex hypersurfaces, so that we can hope that it is indeed a condition that is not too restrictive. Singularities arising within such restriction are called "type 1 singularities". 57 Chapter 11 Normalized Solutions As we did in the previous part of the exposition we will rescale our hypersurfaces Ft(M), this time in such a way that the second fundamental form associated to these stay bounded as time goes to T (the maximal time interval). Now we can easily derive that: \F(p,t2) - FfaW < [* \H(p,t)\dt < C[(T'-i ,) 1 / 2 - (T - t 2 ) 1 / 2 ] as a direct consequence of the constraint (10.2). That simply tells us that F(-,t) converge uniformly as time goes to T. We now define a blow-up point x € M.n+1 as a point such that there is a p E M with F(p, t) —* x as t —>• T and such that the norm of the second fundamental form \A(p, t)\ goes to infinity as t —> T. From now on we will assume, without lost of generality, that the origin of R n + 1 is a blow-up point of our evolution problem. In view of (10.2), it is quite natural to define our rescaled (normalized) hypersurfaces as: ^• s ) : = (2 ( r - ' ( ) )v 2 - ••«) = -\**W-t) since we then get a bounded normalized second fundamental form (\A\2 < Co). The new time parameter s(t) is thus now defined on: —5 log (T) < s < 0 0 , and we have the following evolution 58 equation: dsF(p,s) = dtF(p,s)-(dts(t))-1 = 2(T - t)[(2(T - t)y3/2F(p,t) + (2(T - J))-'1/2 • dtF(p,t)} = H(p,s) + F(p,s) where H := (2(T — t))ll2H is the mean curvature vector of FS(M). By Lemma 19 we can derive (just as we did before) the normalized evolution equations of other geometrical quantities, as long as we have their original one. Also, with the help of (10.2), we bound all higher derivatives of the rescaled curvature in a fashion that is similar to the one used in the convex initial data case: Proposi t ion 6. For each integer m > 0 there is a constant C(m) < oo such that \VmA\2 < C(m) holds on the hypersurfaces FS{M), where C{m) depends on,n,m,Co and FQ{M). Proof. First note that the degree of |VA| 2 is —2(m + 1). From Theorem 8 and Lemma 19 we thus get: a s | vm i | 2 < A | v m i | 2 - 2 | v m + 1 i | 2 + c n , m I v M H V M H v ^ i i i v ^ i (n.i) i+j+k=m , For m = 0we already have the results, by assumption. We now proceed by induction on m, assuming the result holds until m — 1. From (11.1) we obtain that there must be a constant C such that: a s |V m i| 2 < A | V m i | 2 + C ( l + | V m i | 2 ) ' We now add C | V m _ 1 J i | 2 to the left side of this, last evolution equation in order to keep control on the right side (so that it becomes possible to use the maximum principle): ds(\VmA\2 + C\Vm-lA\2) < A ( | V m i | 2 + CIV™- 1 ^ 2 ) - C | V m i | 2 + Ci < A ( | V m i | 2 + C\Vm-lA\2) + Ci so that the maximum principle gives us what we wanted. • 59 Chapter 12 Monotonicity Formula and Consequences Monotonicity formula (Theorem 17) is a very useful tool in studying MCF of hypersurfaces: In this chapter we derive a version of the formula and use it to prove that the singularities of our evolution problem are selfsimilar (see Theorem 18). We define the backward heat kernel p(x,t) at (0, to) by: p(x,t) :--1 [4n(t0-t)]^'eXP\4(t0-t)J> This definition is from the analogy with the usual heat kernel and will turn out to be very convenient tool in the proof of the following results. - x t < t0-Theorem 17. [Monotonicity Formula] dt / p(x, t)dut - I p{x, t) JFt(M) JFt(M) ' t( ) t{ ) where F1- represents the normal component of F. H + F^ 2(t0 -1) dfH Proof. We already proved that dtfit = —H2Ht (see Corollary 2), so that we can calculate: dtp(x, t) = p(x, t) and then, we obviously get: n <H,F> x 2(t 0-t) 2(t 0-t) 4((t 0-t) 2 ; / pdnt = -JFt(M) JFt H + 2(t 0-t) ' V-ZT. 7\d,Ut Ft(M) 4 f0 - t) dut + JFt(M) 1^*0 - t) 60 We again make use use of the first variation formula: / V-Zdu = - I <H,Z>du JF(M) JF(M) with Z := 2{to-t) *° 0 D t a m : / p^77 ~7\dut = / P JFt(M) Wo-t) JFt(M) n \F T | 2 + 2(tQ-t) 4 ( t 0 - t ) 2 J From there the results follows by a simple substitution. du, • A similar calculation gives us the rescaled version of the monotonicity formula: Corollary 12. With the previously defined rescaling factor and p(x) := exp (—^-j, we have a normalized monotonicity formula: dsf_ pdji,= f ~P\H + FL\2dfLs JF3(M) JFS(M) We are almost ready to prove an important result characterizing the rescaled hypersurfaces as time s(t) goes to infinity (in other words, as they evoluate up to their first singularity point). In order to do that we state a result that can be found in [HI] and which is proved in [HI] and [L]: Lemma 28. Assuming that the constraint (10.2) hold, we then have that for each sequence Sj —• oo there is a subsequence Sjk such that FSJK(M) converges smoothly to an immersed nonempty limiting surface FQO(M). With this last statement and the use of the monotonicity formula (normalized one), we can now prove the following theorem which partly characterize the limit of our MCF evolution: Theorem 18. Each limiting hypersurface FOQ(M) as obtained in Lemma 28 satisfies the equa-tion: H =< x,i> > where x := F € R n + 1 , v and H has the usual meaning. 61 Proof. Making use of the monotonicity formula and integrating w.r.t. time, we get the following estimate: / / p\H + F1\2dflsds < oo Js(0) JFS(M) Suppose $Foo(M) P\H + F±\2djls > C\ > 0. Then, by the uniform bounds on the derivatives of the curvature found in Proposition 6, we know there must be a time sc < oo and a constant C2 > 0 such that /ps(M) p\H + F^^dfLg > C2 > 0 for all s e [sc, 0 0 ) . But then that contradict our previous bound on the double integral so that the integrand must be equal to zero on Foo(M). • We have reached in this last theorem a certain level of understanding of the asymptotic nature of the MCF of hypersurfaces toward the first (type 1) singular point: It is possible to go further in this direction by the use of geometric measure theory and some more sophisticated PDE technics. In such an approach one can run the flow past the first singularity: We shall not pursue on these lines but we refer the interested reader to the classical work of Brakke [B] for an introduction to MCF in that context. 62 Chapter 13 Non-Compact Initial Data and Entire Graph Solutions In this section we give a quick overview (omiting the proofs) of some results of the MCF of (non-compact) entire graphs: We consider an initial hypersurface FQ(M) for which there exists aw 6 R n + 1 such that (v,uj)^n+i > C > 0 on all of FQ(M), where V is the usual unit outward normal vector and C is a fixed constant. With this constraint we again consider the MCF parabolic-type equation of the form: dTF(p,t) = H(p,t), peM (13.1) F(-,0):=FO with further entire graph constraint as mentioned above. We will from now on assume that we already have short time existence for this problem: This approach will be justified later on by the main result of the chapter which states long time existence using the estimates previously found. For the details of the proof of all results in this chapter, we refer to [EH1] and [EH2]. 13.1 Equivalent Formulation It is in general true that a PDE of the form (13.1) can be transformed locally as an evolution problem for a graph. Here we give the explicit form of this equivalent problem in the actual context, which allows such a new formulation globally (since we consider entire graphs). We know that there exists a unit vector in M n + 1 such that Ft(M) can be written as F(p,t) = 63 (F(p,t),u(F(p,t),t),t) 6 R n + 1 . From now on we will write Dp simply as D to indicate differ-entiation with respect to the n coordinates defined by F(p,t). We see that: dtF = (dtF, Du • dtF + dtu) € R n + 1 Since Du is clearly tangent to the graph of u(F(p, t),t), the upward unit vector is easily seen to be: _ = (-Du,l) {l + \Du\2yi2 with the mean curvature expressible as: - H = - E «* = *«((1 + 1^,2)1/2 ) Now by the usual MCF equation, we have that —if = dt.F • 1/ and we deduce from there (simple substitution) that: which is the final form of our equivalent PDE problem. With this last equation and the constraint imposed on the initial hypersurface, we see that we can turn our MCF problem into one that is defined purely as an initial valued quasilinear PDE of parabolic type. 13.2 A Priori Height Estimate We define the height function associated to Ft{M) with respect to the hyperplane orthogonal to Co € M.n+1 byu :=< X,UJ >, where x stand for the coordinate representation of FT(M) £ R n + 1 . The reason for introducing such a height function is that it can be proved that under the conditions of our MCF of graphs, an initial polynomial growth on u2 (to be define) will be preserved during the evolution process: Proposi t ion 7. If for some Co, p > 0, we have u2 < Co(l + |a;|2 — u2)p on FQ(M), then for all t> 0 the folowing also holds: u2 < C 0(l + \x\2 - u2 + (2n + 4(p - l))t)p - 64 The proof of the result follows from the use of the weak maximum principle on an estimate obtained on the evolution equations of v?. To get a good understanding of Proposition 7 it is important to notice that the function (\x\2 — v?) measures the length of the horizontal component of the vector x € M n + 1 : That is the reason why we speak of polynomial growth. 13.3 A Priori Gradient Estimate Here we want to show that Ft(M) stays an entire graph as long as the flow is defined. The useful geometrical quantity to study is the gradient function v :=< v,Q > - 1 : We know that by assumption this quantity is initially bounded and our goal would be reached if we could bound it uniformly in time. As the next result will show, it turns out that the function v is also bounded by a polynomial growth relation, formaly very close to the one we got for v?. Proposition 8. If for some C\ < oo, p > 0, we have: v < C\(l + \x\2 — u2)p on FQ(M), we then also have: v < Ci( l + \x\2 - u2 + 2nt)p on Ft(M). The proof of this result follows just as before by the weak maximum principle. With this last proposition, we know that there exist a uniform upper bound (Ci in previous result) to the gradient function v as long as this bound hold on FQ(M): The solution of the MCF evolution equation with entire graph initial data (where v is bounded) will stay an entire graph as long as a smooth solution of the flow exist. 13.4 Two Existence Results In [EH1], Ecker and Huisken proved the following result for MCF of entire graph with Lipschitz initial data: Theorem 19. If FQ{M) is Lipschitz continuous and satisfies S,wp[F0(M)]u ^ Ci> then the MCF equation (13.1) has a smooth solution for all time. Moreover, each Ft(M) is an entire graph. An even stronger result has been proved by the same authors in [EH2], that is, the long time 65 existence of the flow (13.1) with Lipschitz initial data does not require any initial gradient bound and each of the Ft(M) are again entire graphs. This last result is more subtle to establish and one needs a good working knowledge of parabolic-type equations theory in order to follow the discussion: The fact that the equation is studied on unbounded domain and has non-smooth initial data requires more work. 66 Appendix In this appendix we give a review of some knowledge of hypersurfaces theory and prove two results that will eventually be used in the text. We assume the reader is familiar with basic differential geometry, such as given in the first three chapters of [J] for example. In what follows (•, •) is the usual scalar product in R n + 1 and < •, • > is the one induced by the metric on the hupersurfaces. So suppose M is a n-dimensional manifold immersed in Rn+1 by F, we then define the metric and the second fundamental form on our hypersurface by: gij(x) = (diF(x),djF(x)), hij(x) =-(v(x),didjF(x)) with x € W1 and v the outer unit normal of the hypersurface. The connection induced on F(M) has the same form in local coordinates as usual: The rules for covariant derivatives of tensors on F(M) are then exactly the same as before (see [J], chapter 3). With the help of Gauss equation we can deduce the following useful local expressions for the Riemann and Ricci tensors as well as for the scalar curvature: Rijkl = hikhji - huhjk, Rik = Mint-hugljhjk, R = H2 — \A\2 where H = h\ and \A\2 = h^h^. The following are two formulas for interchanging covariant derivatives that we will use: \7iVjXh - VjViXh = R}ljkXk . V,V , n - VjViYk = RijkiglmYm We also recall the well known Codazzi equations: We are now ready to prove the following two results that will turn out to be important ingredient in the derivation of some estimates in Part II: Lemma 29. With Z := h{h^hfH - |A| 4, we have that: 1. Ahij = ViVjH + Hhuglmhmj -\A\2hi:j 2. =< hij, ViVjH > +|VA|2 + Z. 67 Proof. Ah — nmn\7 V h — nmnV X7h ~ 9 [^i^mhjn + Rmijp{f"*hqn + Rminp9Pq hjq] — 9 [ V j V j T j / i j n + {hmjhip — hmphij) g*"* hqn + (hmnhip hmphin)g pqhqj] = 9 V j V j / i m n + g g*"*{hmjhiphqn + hmnhiPhqj hmphifihqj) - n mnn pqh h-h y y • l'mp'H]' l>qn = ViVjH-\A\2hij + Hgpqhiphqj + (h]hqhqn-h;hnihp) That complete the proof of the first part of the lemma since the last term of the last entity vanish. For the second part, we find: \\A\ 2 = ^g pqg mn[(Ahqn)hpm + hqn(Ahpm)} + l9pq9mn9 ij(VihqnVjhpm + VjhqnVihpm) = 9pq9 mnhmP(VqVnH +Hhqig lahan - \A\ 2hqP) + |VA| 2 =<hqn,VqVnH>+Z + \VA\2 as an almost direct consequence of the first part of the lemma. Lemma 30. We also prove: 1. \VA\ 2>j^\VH\ 2 2. \VA\ 2 - ™^ >  2 { n ~ l ^ A ? Proof. In order to prove the first inequality we decompose the tensor VA in two orthogonal part as follows. Set Vihjk = Eijk + Fijk with , ' _ VjHgjk + Vj(Hgik) + {S/kH)9ij j h ~ n + 2 One can then verify that < E{jk,Fijk >=< Eijk,S7ihjk — Eijk >= 0 (use Codazzi two times). Also, we compute: 2_3(n|Vff1 2) + 6lVff|2 = 3 | V f f l 2 1 1 (n + 2)2 n + 2 by a simple calculation, which proves what we wanted. The second result follows from the first if we use \WH\ 2 < n|VA| 2. • We also state a PDE result of Stampacchia ([S], Lemma 4.1) that will be useful in chapter 6: 68 Lemma 31. Let 4>{t) be a non negative and increasing function defined on (ko, + 0 0 ) such that if h > k > ko, then: m^ih^w m]"  (2) c,a,ft being positive constant. Then (i) If P > 1 we have 4>{ko + d) = 0 where da = c[<l>(ko)}l3-12al3/P-1, (ii) If P — 1 we have <f>(h) < eexp[-C(/i - ko)](j)(ko) where Q = (ec)" (iii) If (5 < 1 and ko > 0 we have 4>{h) < 2^1-®{c1K1-® + (2ko)"<j)(ko)}h-'1 where ft r§-g • 69 Bibliography [B] Brakke, K.A., The Motion of a Surface by its Mean Curvature , Math..Notes Princeton, N.J., Princeton University Press 1978. [CE] Cheeger, J. &i Ebin, D.G., Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975. [E] Eidel'man, S.D., Parabolic Systems, North-Holland, Amsterdam, 1969; [EH1] Ecker, K. & Huisken, G., Mean curvature evolution of entire graphs, Annals of Math., Vol.130 (1989) 453-471. 1 [EH2] Ecker, K. &; Huisken, G., Interior estimates for hypersurfaces moving by mean curva-ture, Invent.Math. Vol.105 (1991) 547-569. [F] Friedman, A., Partial Differential Equations of Parabolic Type, Prentice Hall, Engle-wood Cliffs, N.J., 1964. [GH] Gage, M. k, Hamilton, R.S., The eat equation shrinking convex plane curves, J. Differ-ential Geom. Vol.23 (1986) 69-96. [G] Grayson, M.A., The eat equation shrinks embedded plane curves to round points, J. Differential Geom. Vol.26 (1987) 285-314. [H] Hamilton, R.S., Three-manifolds with positive ricci curvature, J. Differential Geom. Vol.17 (1982) 255-306. [HI] Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. Vol.20 (1984) 237-266. [H2] Huisken, G., Asymptotic behavior for singularities of the mean curvature flow, J. Dif-ferential Geom. Vol.31 (1990) 285-299. [J] Jost, J., Riemannian Geometry .and Geometric Analysis, Springer-Verlag, Universitext (Third Edition) 2002. [KN] Kobayashi, S. & Nomizu, K., Foundations of Differential Geometry, Vol. 1 and 2, Interscience, New York, 1963 and 1969. [L] Langer, J., A compactness theorem for surfaces with Lp-bounded second fundamental form, Math. Ann. Vol.270 (1985) 223-234. [MS] Michael, J.H. & Simon, L.M., Sobolev and mean-value inequalities on generalized sub-manifolds ofW1 , Comm. Pure Appl. Math., Vol.26 (1973) 316-379. . [S] Stampacchia, G., Equations elliptiques du second ordre a coefficients discontinues Sem. Math. Sup 16, Les Presses de l'Universite de Montreal, Montreal, 1966. [T] Taylor, M., Partial Differential Equations, Vol.3, Springer 1996. 70 \ 

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