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Lectures on Mean Curvature Flow

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Lectures on Mean Curvature Flow Lectures on Mean Curvature flow Mariel S´aez July 17, 2016 2 Contents Preface 5 1 Basic definitions and short time existence 7 1.1 Introduction . 7 1.2 Mean Curvature flow . 10 2 Comparison principle and consequences 15 2.1 Comparison principle . 16 2.2 Maximum principle and consequences . 17 2.3 Geometric estimates . 17 2.4 Some examples of singularities . 20 3 Singularities 21 3.1 Curve Shortening Flow . 21 3.2 Monotonicity formula and type I singularities . 25 3.3 Type II singularities . 28 3.4 Weak solutions . 29 4 Non-compact evolutions 33 4.1 Complete graphs over Rn ...................... 33 4.2 Rotationally symmetric surfaces . 33 4.3 Graphs over compact domains . 34 3 4 CONTENTS Preface This is a first (incomplete) draft on a series of 4 lectures at the XIX School of Differential Geometry 2016 at IMPA (Brazil). The notes are mostly based on the exposition in the references [8, 28]. These notes are by no means a comprehensive survey on the subject. In fact there are many relevant references and results that are not included here. 5 6 CONTENTS Lecture 1 Basic definitions and short time existence 1.1 Introduction Definition 1.1.1 (Wikipedia). In mathematics, specifically differential geome- try, a geometric flow is the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature. They can be interpreted as flows on a moduli space (for intrinsic flows) or a parameter space (for extrinsic flows). These are of fundamental interest in the calculus of variations, and include several famous problems and theories. Particularly interesting are their critical points. A geometric flow is also called a geometric evolution equation. n n+1 Examples. • Consider an immersion F0 : M ! R and the functional Z Area (F0) = dµ, M where µ is the canonical measure associated to the metric g induced by the immersion. If we consider a smooth variation Ft of F0, the first variation of area is given by Z dArea (Ft) = − HhX; νidµ, dt t=0 M where H is the mean curvature, ν the outward unit normal and X = @Ft @t t=0. Hence, the gradient flow of the area functional is given by solutions to the equation @F t = −Hν: @t 7 8 LECTURE 1. BASIC DEFINITIONS AND SHORT TIME EXISTENCE This is the steepest descent direction in the L2-sense and it is known as Mean Curvature Flow. • Consider a Riemannian manifold (M; g) and the energy functional Z E(g) = Rdµ, M where R is the scalar curvature of M and µ the measure induced by g. @g Consider a variation g(t) of the metric g (i.e. g(0) = g and @t = X), t=0 then Z dE (g(t)) ij 1 ij = (−Ric + Rg )Xij; dµ. dt t=0 M 2 Then, the gradient flow associated to E is given by @g = −2Ric(g) + Rg: @t If a geometric flow depends on intrinsic geometric quantities (such as scalar curvature) the flow is known as an intrinsic geometric flow, while if the geometric quantities are extrinsic (i.e. depend on the ambient geometry, such as the mean curvature) the geometric flow is known as extrinsic. Notice that the first example above, can be written as follows: @F = ∆ F; @t g(t) where ∆g(t) is the Laplace Beltrami operator on the immersion of M induced by Ft. This equation is a quasilinear parabolic system of equations that has some nice analytic properties. On the other hand, the second example above does not have nice analytic properties as a PDE. This contrasts with the evolution equation @g t = −2Ric (g); (1.1.1) @t which also has the structure of a quasilinear parabolic system of PDE's , but it is not clearly a gradient flow. Remark 1.1.2. Equation (1.1.1) is known as the Ricci flow and it was used by Perelman to prove the geometrization conjecture. Definition 1.1.3. We will understand as an extrinsic (resp. intrinsic) geomet- ric flow a family of immersions of a submanifold N n ⊂ M n+1 (resp. a family of metrics) indexed by a parameter t such that the immersion (resp. the metric) is induced as family of solutions to a partial differential equation that involves extrinsic (resp. intrinsic) geometric quantities. Examples. Consider F (t): N n ! M n+1 and g(t) a metric on a manifold M 1.1. INTRODUCTION 9 • Mean Curvature Flow @F t = −Hν; @t where H is the mean curvature of N in M and ν the outward pointing unit normal. • Inverse mean curvature flow @F 1 t = ν; @t H where H is the mean curvature of N in M and ν the outward pointing unit normal. • Willmore Flow @F t = −∇W (F (t)); @t R 2 where W = M (H − K)dA and K is the Gaussian curvature. • Gauss flow @F t = −Gν; @t where G denotes the Gaussian curvature of N in M and ν the outward pointing unit normal. • Yamabe flow @g = R; @t where R is the scalar curvature • Ricci flow @g = −2Ric(g); @t where Ric is the Ricci curvature. Remark 1.1.4. The extrinsic flows above can be defined for co-dimension bigger than 1. However, from an analytic point of view the study of such flows becomes far more complicated and in this series of lectures we will assume that the co- dimension is 1. We remark that geometric flows are often degenerate parabolic equations. This is not a surprising feature, due to geometric invariances, such as the in- variance of geometric quantities under reparametrizations. 10 LECTURE 1. BASIC DEFINITIONS AND SHORT TIME EXISTENCE 1.2 Mean Curvature flow We start by recalling definitions that we will use throughout the lectures. We note that everything could be defined more generally for a Riemannian manifold (M n+1; g) and a smooth immersion of an k-dimensional submanifold F : N k ! M n+1 for k ≤ n. However, in our context we will always assume that M = Rn+1 and that N is n-dimensional. Let F : N n ! Rn+1 be a smooth immersion and ν a choice of a unit normal vector field that is smooth. We will denote spatial derivatives as @i (as the derivative respect to the i−th parameter). The metric induced by the ij immersion F will written as gij and g its inverse. We denote the Levi Civita connection associated to g by rM and more precisely, for h : M ! R the tangential gradient is given by M ij r h = g @jh@iF: i ij For a smooth tangent vector field X = X @iF = g Xj@iF the covariant deriva- tive tensor can be computed as m j j j k ri X = @iX + ΓikX ; k where the Christoffel symbols Γij are defined by T k (@i@jF ) = Γij@kF: The Laplace Beltrami operator can be computed by 1 p ∆ h = p @ ( ggij@ h): M g i j We define the second fundamental form as @F hij = − ν · ; @!i@!j A =(hij)ij: The mean curvature can be computed as the trace of the second fundamental form: ij H = g hij: Finally, we denote the norm of the second fundamental form by 2 ij kl jAj = g g hikhjl: We say that a hypersurface F (t) evolves under mean curvature flow if @F t = −Hν; (MCF) @t where ν is a smooth choice of unit normal vector field, that points outward if F (!; t) is convex. 1.2. MEAN CURVATURE FLOW 11 1.2.1 Examples • Consider a sphere of radius R0. Let us assume that the solution is a sphere for every time t, that is F (t) = R(t)! with ! 2 Sn. Then the mean n curvature is given by H = R and the equation is equivalent to n R0 = : R Hence, the solution to the equation is p R(t) = R0 − 2nt: Notice that the solutions are well defined only for finite time. Exercise: Repeat the computation for cylinders. • Minimal surfaces are solutions to the equation. All these solutions are non compact. • Consider the curve y = − log cos x, then the curve t(0; 1) + (x; y(x)) is a translating solution to (MCF). This solutions is known as the Grim Reaper and it is defined for every t 2 (−∞; 1). 1.2.2 Short time existence First notice that when we consider a time dependent parametrization '(!; t) of a solution to (MCF) we have that F~(!; t) = F ('(!; t); t) satisfies @F~ @' @F @' = dF i ; t + ('(!; t); t) = dF − Hν: @t @t @t @t Notice that F~ is not longer a solution to (MCF) (although is geometrically the same object). However, we have that the following converse result holds: Theorem 1.2.1. Assume that a family of immersions satisfies @F = − Hν + X; (1.2.1) @t F (!; 0) =F0; (1.2.2) where X is a smooth (possibly time dependent) tangent vector. Then there is a family of reparametrizations of F that satisfies (MCF). Proof. Consider F~ = F ('(!; t); t). As before we have that @F~ @' = dF − Hν + X: @t @t 12 LECTURE 1. BASIC DEFINITIONS AND SHORT TIME EXISTENCE By standard ODE theory, we can choose ' that solves @' = − (dF )−1(X('; t)); @t '(·; 0) =Id: Note that the choice is possible since X 2 TM. Then we have that F~ is a solution to (MCF). The previous theorem implies that an equivalent (and more geometric) equa- tion to (MCF) is @F t · ν = −H: (MCF') @t Theorem 1.2.2 (Short time existence).
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