AN INTRODUCTION TO THE MEAN CURVATURE FLOW FRANCISCO MART´IN AND JESUS´ PEREZ´ Abstract. The purpose of these notes is to provide an introduc- tion to those who want to learn more about geometric evolution problems for hypersurfaces and especially those related to curva- ture flow. These diffusion problems lead to interesting systems of nonlinear partial differential equations and provide the appropriate mathematical modeling of physical processes. Contents 1. Introduction 2 2. Existence y uniqueness 12 3. Evolution of the Geometry by the Mean Curvature Flow 18 4. A comparison principle for parabolic PDE's 31 5. Graphical submanifolds. Comparison Principle and Consequences 34 6. Area Estimates and Monotonicity Formulas 52 7. Some Remarks About Singularities 72 References 74 Date: July 18, 2014. 1991 Mathematics Subject Classification. Primary 53C44,53C21,53C42. Key words and phrases. Mean curvature flow, singularities, monotonicity for- mula, area estimates, comparison principle. Authors are partially supported by MICINN-FEDER grant no. MTM2011- 22547. 1 2 FRANCISCO MARTIN AND JESUS PEREZ 1. Introduction Mean Curvature Flow is an exciting and already classical mathemati- cal research field. It is situated at the crossroads of several scientific dis- ciplines: Geometric Analysis, Geometric Measure Theory, PDE's The- ory, Differential Topology, Mathematical Physics, Image Processing, Computer-aided Design, among other. The purpose of these notes is to provide an introduction to those who want to learn more about these geometric evolution problems for curves and surfaces and especially curvature flow problems. They lead to interesting systems of nonlinear partial differential equations and provide the appropriate mathematical modeling of physical processes such as material interface propagation, fluid free boundary motion, crystal growth,... In Physics, diffusion is known as a process which equilibrates spatial variations in concentration. If we consider a initial concentration u0 on a domain Ω ⊆ R2 and seek solutions of the linear heat equation @ (1.1) u − ∆u = 0; @t with initial data u0 and natural boundary conditions on @Ω, we obtain 2 a successively smoothed concentrations futgt>0. When Ω = R , the solutions to this parabolic PDE coincides with the convolution of the initial data with the heat kernel (or Gaussian filter) 1 2 2 Φ (x) = e−|xj /σ σ 2πσ with standard deviation sigma, i.e., ut2=2 = Φt ∗ u0 (see Remark 6.18.) In general, derivatives of ut are bounded for t > 0 in terms of bounds on u0. It follows that, even if you start with a heat distribution which is discontinuous, it immediately becomes smooth. Moreover, solutions converge smoothly (in C1) to constants as t ! 1 (eventual simplicity). The heat equation has some surprising properties which carry over to much more general parabolic equations. The Maximum Principle: : At a point where ut attains a max- imum in space (that is, in Ω), the second derivatives in each di- rection are non-positive. By the heat equation, the time deriva- tive is non-positive. It follows that the maximum temperature, umax(t) = supx2Ω u(x; t), does not increase as time passes. Gradient Flow: A further useful property which holds for many but not all heat-type equations is the gradient property: The MEAN CURVATURE FLOW 3 heat equation is the flow of steepest decrease of the Dirichlet Energy: 1 Z E(u) = j(Du)(x)j2dx: 2 Ω Figure 1. A surface moving by mean curvature. If we are interested in the smoothing of perturbed surface geometries, it make sense to think in analogues strategies. So, the source of inspi- ration diffused throughout everything that follows is the classical heat equation (1.1). The geometrical counterpart of the Euclidean Laplace operator ∆ on a smooth surface M 2 ⊂ R3 (or more generally, a hypersurface M n ⊂ n+1 R ) is the Laplace-Beltrami operator, that we will denote as ∆M . 4 FRANCISCO MARTIN AND JESUS PEREZ Thus, we obtain the geometric diffusion equation @ (1.2) x = ∆ x; @t Mt for the coordinates x of the corresponding family of surfaces fMtgt2[0;T ): A classical formula by Weierstraß (see [DHKW92], for instance) says that, given an orientable1 (hyper)surface in Euclidean space, one has: ~ ∆Mt x = H; where H~ means the mean curvature vector. This means that (1.2) can be written as: @ (1.3) x(p; t) = H~ (p; t) @t The mean curvature is known to be the first variation of the area R functional M 7! M dµ (see [DHKW92,CM11,MIP12].) We will obtain for the Area(Ω(t)) of a relatively compact Ω(t) ⊂ Mt that Z d ~ 2 (Area(Ω(t)) = − jHj dµt: dt Ω(t) In other words, we get that the mean curvature flow is the correspond- ing gradient flow for the area functional: The Mean Curvature Flow is the flow of steepest decrease of surface area. Moreover, we also have a nice maximum principle for this particular diffusion equation. Theorem (Maximum/Comparison principle). If two properly immersed hypersurfaces of Rn+1 are initially disjoint, they remain so. Further- more, embedded hypersurfaces remain embedded. In this line of result, we would like to point out that: • If the initial hypersurface M is convex (i.e., all the geodesic cur- vatures are positive, or equivalently M bounds a convex region n+1 of R ), then Mt is convex, for any t. • If M is mean convex (H > 0), then Mt is also mean convex, for any t. 1 Throughout these notes we shall always assume that the hypersurfaces of Rn+1 are orientable. MEAN CURVATURE FLOW 5 Moreover, mean curvature flow has a property which is similar to the eventual simplicity for the solutions of the heat equation. This result was proved by Huisken and asserts: Theorem. Convex, embedded, compact hypersurfaces converge to points p 2 Rn+1: After rescaling to keep the area constant, they converge smoothly to round spheres. There is a rather general procedure for producing heat-like curva- ture flows. In general, we wish to evolve hypersurfaces M n in Rn+1 (or in a complete, Riemannian, (n + 1)-dimensional manifold). Then any (smooth) symmetric function f of n variables, which is monotone increasing in each variable, determines a suitable speed function: F (p; t) := f(k1(p; t); : : : ; kn(p; t)); where ki, i = 1; : : : ; n, represent the principal curvatures of Mt. This yields a general class of curvature flows: @ (1.4) x(p; t) = F (p; t) · ν(p; t); @t where ν(·; t) is the Gauß map of Mt: Some of the most interesting examples are: n X (a) Mean Curvature Flow: f(x1; : : : ; xn) = xi; (F = H): i=1 n !−1 X 1 (b) Harmonic Mean Curvature Flow: f(x ; : : : ; x ) = . 1 n x i=1 i n Y (c) Gauß curvature flow: f(x1; : : : ; xn) = xi; (F = K): i=1 1 (d) Inverse Mean Curvature Flow: f(x1; : : : ; xn) = −Pn ; (F = i=1 xi −H−1): Applications of the mean curvature flow (and its variants: harmonic mean curvature flow, inverse mean curvature flow,...) are numerous and cover various aspects of Mathematics, Physics and Computing. In the following paragraphs we will briefly describe some of these applications, with particular emphasis on two of them. The inverse mean curvature flow was used by Huisken and Ilmanen to prove the Riemann Penrose inequality [HI01]. Similarly, Andrews got an alternative proof of the topological version of the sphere theorem [And94] making use of the harmonic mean curvature flow. 6 FRANCISCO MARTIN AND JESUS PEREZ 1.1. Riemannian Penrose Inequality. The Riemannian Penrose in- equality is a special case of the unsettled Penrose Conjecture. In a seminal paper [Pen73] (see also [Pen82]), in which he proposed the cel- ebrated cosmic censorhip conjecture, R. Penrose also proposed a related inequality, which today is know as \Penrose Inequality". The inequality is derived from cosmic censorship by using a heuristic argu- ment relying on Hawking's Area Theorem [HE73]. Consider a space- time satisfying the so called dominant energy condition (DEC), which contains an asymptotically flat Cauchy surface with ADM mass m (see definition below), and containing an event horizon (roughly, the area of a black hole) of area A = 4πr2, which undergoes gravitational collapse and settles to a Kerr-Newman solution of mass m1 and area radius r1. Physical arguments imply that the ADM mass of the final state m1 is no greater than m (no new mass appear, even though ra- diation may imply some loss of mass), then the area radius r1 is no less than r, and the final state must satisfy 1 m ≥ r : 1 2 1 The evolution of black holes (assuming that it is deterministic, i.e., no naked singularity appears) implies that the area of its event horizon must increase, so it must have been the case that 1 m ≥ r; 2 also at the beginning of the evolution. A counterexample to the Penrose inequality would therefore suggests data which leads under the Einstein evolution to naked singularities, and a proof of the Penrose inequality may be viewed as evidence in support of the cosmic censorship. The event horizon is indiscernible in the original slice without knowing the full evolution, however one may, without disturbing this inequality, replace the event horizon by the (possible smaller) apparent horizon, the boundary of the region admitting trapped surfaces. The inequality is even more simple in the time-symmetric case, in which the apparent horizon coincides with the outermost minimal surface, and the dominant energy condition reduces to the condition of nonnegative scalar curvature. This leads to the Riemannian Penrose inequality: the ADM mass m and the area radius r of the outermost minimal surface in an asymptotically at 3-manifold of nonnegative scalar curvature, satisfy r r A m ≥ = ; 2 16π MEAN CURVATURE FLOW 7 and the equality holds if and only if the manifold is isometric to the canonical slice of the Schwarzschild spacetime.