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 {ut}t>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−|x| /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 C∞) to constants as t → ∞ (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) = supx∈Ω 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) = |(Du)(x)|2dx. 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 {Mt}t∈[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)) = − |H| 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 ∈ 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 m∞ and area radius r∞. Physical arguments imply that the ADM mass of the final state m∞ is no greater than m (no new mass appear, even though ra- diation may imply some loss of mass), then the area radius r∞ is no less than r, and the final state must satisfy 1 m ≥ r . ∞ 2 ∞ 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. Note that this charac- terizes the canonical slice of Schwarzschild as the unique minimizer of m among all such 3-manifolds admitting an outermost horizon of area A. For a more precise explanation about the physical interpretation of the inequality we recommend [MS13]. In these notes, we will focus on the mathematical aspects of the Riemannian Penrose inequality and how the ICMF has been a key tool in their demonstration.
3 Consider (N , g = (gij)) a Riemannian 3-manifold. Assume N is asymptotically flat which means that:
(1) N is realized by an open set which is diffeomorphic to R3 \ K; K compact, C (2) |g − δ | ≤ , as |x| → ∞; ij ij |x|
∂gij C (3) ≤ 2 , as |x| → ∞. ∂xk |x| (4) We also assume C Ric ≥ − · g. |x|2
In this setting we define Definition 1.1 (Arnowitt-Deser-Misner (ADM) mass). The total en- ergy, or ADM mass, of the end is defined by a flux integral through the sphere at infinity: Z 1 X ∂gij ∂gij m := lim − · nj dµ, r→+∞ 16π 2 ∂x ∂x i,j S (0,r) j i where n represents the “outward” pointing Gauß map of the Euclidean sphere S2(0, r).
Although this flux is defined using local coordinates, it is global invariant of the end. Theorem 1.2 (Riemannian Penrose Inequality [HI01]). Let N be a complete, connected 3-manifold (with boundary.) Suppose that
(a) N is asymptotically flat, with ADM mass m, (b) N has nonnegative scalar curvature, 8 FRANCISCO MARTIN AND JESUS PEREZ
(c) N has compact boundary which consists of minimal surfaces, and N contains no other compact minimal surfaces.
Then r Area(M) m ≥ , 16π where M is any connected component of ∂N. Moreover, equality holds iff N is isometric to one-half of the spatial Schwarzschild manifold.
The spatial Schwarzschild manifold is (R3 − {(0, 0, 0)}, g) where m 4 g := 1 + · g , 2|x| 0 3 and g0 represents the Euclidean metric of R .
• It possesses an inversive isometry fixing S2(0, m/2), which is an area minimizing sphere of area 16πm2. • The manifold of the Riemannian Penrose inequality is R3 − B(0, m/2).
Huisken-Ilmanen’s proof of the Riemannian Penrose inequal- ity. Huisken and Ilmanen proved the inequality, including the rigidity part, in [HI01] by using the inverse mean curvature flow, an ap- proach proposed by Jang and Wald [JW77]. In this introduction, we would like to give a rough idea of the structure of their proof. A classi- cal solution of the Inverse Mean Curvature Flow (IMCF from now on) is a smooth family of (hyper)surfaces F : M × [0,T ] → N, satisfying the equation ∂ ν (1.5) F = − , ∂t H where ν represents the Gauß map of Mt := F (M, t) and 0 < H(·, t) is its mean curvature.
Without extra geometric hypotheses, the mean curvature could de- velop a zero and then the equation would present a singularity. That was the reason because Huisken and Ilmanen introduced a level-set for- mulation of (1.5), where the evolving surfaces are given as level sets of a scalar function φ:
Mt = ∂ ({x ∈ N / φ(x) < t}) , so (1.6) is replaced by the degenerate elliptic equation MEAN CURVATURE FLOW 9
Dφ (1.6) div = |Dφ| N |Dφ|
They were able to overcome the problem that the evolving surface can become singular before reaching infinity by formulating and analysing a suitable weak notion of the solution of (1.6). These weak solutions are locally Lipschitz continuous functions and the treatment was inspired in a work of Evans and Spruck on the MCF [ES91]. It appears that neither (1.5) is a gradient flow nor (1.6) is an Euler- Langrange equation. The idea of these two authors consists of freezing |Dφ| in the right-hand side of (1.6) and consider (1.6) as the Euler- Lagrange equation of the functional: Z K Jφ(ψ) = Jφ (ψ) := (|Dψ| + ψ|Dφ|) d µ, K where K is a compact subset of N. Definition 1.3 (Weak solution). Let φ a locally Lipschitz function on the open set Ω ⊆ N. Then we say that φ is a weakk solution of (1.6) on Ω provided K K Jφ (φ) ≤ Jφ (ψ), for all ψ locally Lipschitz and such that {ψ 6= φ} ⊂⊂ Ω, where we are integrating over any compact {ψ 6= φ} ⊆ K ⊂ Ω.
In order to understand the value of the above definition in the solution of our problem, we need to introduce a new concept; the Hawking mass. Given a compact surface M in N, the Hawking mass is defined by: s Z Area(M) 2 mH (M) := 3 16π − H dµ . (16π) M
Hawking already noticed that mH approaches the ADM mass for large coordinate spheres. r Area(M) If M is minimal then m (M) is precisely . Moreover, the H 16π Hawking mass has an especially nice behavior respect to the inverse MCF:
I. Geroch Monotonicity Formula. Geroch [Ger73] introduced the IMCF and realized that the mass mH of a family of surfaces 10 FRANCISCO MARTIN AND JESUS PEREZ
evolving by the inverse MCF is monotone nondecreasing, pro- vided that the surface is connected and the scalar curvature of N is nonnegative. II. One of the main achievements of [HI01] consists of proving that Geroch Monotonicity Formula also works for the Huisken-Ilmanen weak solutions of the inverse MCF, even in the presence of jumps. III. The derivative vanishes precisely on standard expanding spheres in flat 3-space and Schwarzschild example.
Taking these properties into account the proof of the case of a connected horizon (M = ∂N connected) works as follows. We move M by the IMCF, obtaining a family of compact surfaces Mt which collapses at the point of infinity. By the monotonicity formula, we know that r Area(M) m (M ) ≥ m (M) = , H t H 16π
(recall that M is minimal.) For t big enough, we have that mh(Mt) approximates the ADM mass m, so r Area(M) m ≈ m (M ) ≥ . H t 16π
Moreover, the equality holds iff mH (Mt) is constant along the flow, i.e. its derivative vanishes. According to Property III, this only happens for standard expanding spheres in Schwarzschild’s example. Finally, we would like to mention that the inequality was proven in full generality (non-connected horizon) by Bray [Bra01] using a conformal flow of the initial Riemannian metric, and the positive mass theorem [SY79].
1.2. The Sphere Theorem. It is known that if a manifold is simply connected and has constant positive sectional curvatures, then it is a sphere with the standard Riemannian metric.
• In the 1940’s, Heinrich Hopf asked whether we can also wig- gle the geometry a little, instead only requiring that the sec- tional curvatures be close to some constant. Let’s say 1 − ε < K ≤ 1. • In 1951 Rauch [Rau51] proved a simply connected manifold with curvature in [3/4, 1] is homeomorphic to a sphere. • At the beginning of the 1960’s, Berger [Ber60] and Klingen- berg [Kli61] confirmed the conjecture, with the optimal value MEAN CURVATURE FLOW 11
of ε : A simply connected Riemannian manifold with sectional curvatures in the interval (1/4, 1] is homeomorphic to a sphere. If the value 1/4 is allowed, there are counterexamples. Ac- tually, any compact symmetric space of rank 1 admits a metric whose sectional curvatures lie in the interval [1, 4]. The list of these spaces includes the following examples: k – The complex projective space CP , for 2k ≥ 4. k – The quaternionic projective space HP , for 4k ≥ 8. – The projective plane over the octonions (dimension 16)
It remained an open conjecture for over 50 years that the conclusion of homeomorphism should be improvable to diffemorphism. The problem was solved in the affirmative by S. Brendle and R. Schoen in [BS09] using the Ricci Flow. However, as we mentioned before, the sphere theorem was also proved by B. Andrews using curvature flows. Idea of the proof: Using the pinching assumption, it is not difficult to construct a large disk D(p, r) in M whose boundary is smooth and convex in the “outwards” direction. We would like to flow this boundary in the outwards direction to a point via a suitable curvature flow. This would demonstrate that the manifold is formed from gluing two disks together along their bound- aries, and hence is a sphere. We know that the mean curvature flow doesn’t work, but there’s at least one flow speed that makes the job; namely, the harmonic mean curvature: n !−1 ∂ X 1 x = f · ν, f(k , . . . , k ) = . ∂t 1 n k i=1 i
Note that the conclusion in this case is stronger than homeomor- phism: The manifold is diffeomorphic to a twisted sphere (two disks glued by a diffeomorphism along their boundary). But this is still slightly weaker than diffeomorphism.
1.3. Image processing. These ‘smoothing” properties that we men- tioned of mean curvature flow make it an ideal tool in image processing, computer aided geometric design and computer graphics. Here, is- sues are fairing, modeling, deformation, and motion. Constructive and 12 FRANCISCO MARTIN AND JESUS PEREZ more explicit approaches based for instance on splines are nowadays already classical tools. More recently geometric evolution problems and variational approaches have entered this research field as well and have turned out to be powerful tools. For those readers interested in the applications of curvature flows in image processing we recommend [CDR03].
Since our background is closer to Geometric Analysis, we have mainly used the monographs [Eck04], [Man11] and [RS10] in the elaboration of these notes. For readers who are more familiar with the language and techniques of Geometric Measure Theory, we recommend [Bra78] and [Ilm95]. These notes correspond to the contents of a mini-course given by the first author at the Program on Geometry and Physics, Granada 2014. The authors are extremely grateful to all participants in this program who have sent us corrections and suggestions that have helped to improve these notes. In that sense, we feel especially indebted to Miguel S´anchez for his valuable comments.
2. Existence y uniqueness
Along this section M will represent a n-dimensional submanifold of Rn+1. Although the most part of the results are also valid in a more general setting, we will restrict our attention to the study of hyper- surfaces in Euclidean space. Definition 2.1. We say that M moves by the mean curvature if there exists a smooth family of immersions F : M × [0,T ) → Rn+1 such that
• F (·, 0) is the original immersion of M in Rn+1; • For each p in M and for each t in [0,T ) one has ∂ F (p, t) = H~ (p, t), ∂t where H~ (p, t) is the mean curvature vector of F (M, t) at F (p, t).
Remark 2.2. We shall introduce the following notation. Given a map F : M × [0,T ) → Rn+1 as in the previous definition, we will write Ft := F (·, t). Thus, we will use the same notation for all the elements MEAN CURVATURE FLOW 13 associated to the smooth family of immersions {Ft}t∈[0,T ) like, for in- ~ ~ stance Mt := F (M, t), the mean curvature vector Ht(p) := H(p, t), etc. For simplicity in the computations, we will often suppress the subscript t when no confusion is possible.
We will start with some local computations. Let (U, x1, x2, . . . , xn) be local co-ordenates around p ∈ U ⊂ M n. We have F :(M n, g) → (Rn+c, g¯) whereg ¯ = h·, ·i is the usual scalar product in Rn+c and g = dF ∗(¯g). So ∂ ∂ ∂ ∂ ∂F ∂F gij = g( , ) =g ¯ dF , dF = , , ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj where we are writing dF ∂ = ∂F .2 ∂xi ∂xi Recall that we can locally identify M n with F (M n). In this way, we can say that ∂F is a local vector field on n+k which extends the ∂xi R smooth field on M n given by ∂ . ∂xi On the other hand, the Levi-Civita connection ∇¯ in the Euclidean space Rn+c is the standard flat connection in Rn+c, i.e., given X,¯ Y¯ ∈ n+c ¯ ¯ ¯ ¯ X(R ) one has ∇X¯ Y = DX¯ Y , where DX¯ Y means the directional ¯ ∂F derivative of Y alongX. In what follows it will appear ∇ ∂F ∂x , that ∂xi j we will denote as ∂2F . It is an abuse of notation whose justification ∂xi∂xj is as follows: ∂F ∂F ∂F ∂2F ∇¯ ∂F = D ∂F = D ∂ = , ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj ∂xi∂xj where the second equality has taken into account the identification ∂ ∂F between the fields ∂x and ∂x given by F . We have also used that ¯ ¯ ¯ i i ∇X¯ Y = DX¯ Y where D means the standard directional derivative in Rn+c. Recall that the mean curvature vector can be computed in terms of the connection as follows3: ⊥ ij ∂F H~ = g ∇¯ ∂F . ∂xi ∂xj
2We are considering dF as a map from X(U) into X(F (U)), in such a way that dF ∂ can be seen as a local field on n+c. ∂xi R 3 Given a vector v ∈ Rn+c and p ∈ M we can decompose v = v> + v⊥ where > ⊥ ⊥ v ∈ TpM and v ∈ (TpM) 14 FRANCISCO MARTIN AND JESUS PEREZ
And we can go further getting that: ⊥ > ij ∂F ij ∂F ij ∂F g ∇¯ ∂F = g ∇¯ ∂F − g ∇¯ ∂F = ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj ij ∂F ij ∂F ∂F kl ∂F = g ∇¯ ∂F − g ∇¯ ∂F , g . ∂xi ∂xj ∂xi ∂xj ∂xk ∂xl
Proof. Indeed, we only have to check that > ij ∂F ij ∂F ∂F kl ∂F g ∇¯ ∂F = g ∇¯ ∂F , g . ∂xi ∂xj ∂xi ∂xj ∂xk ∂xl To do this, fix m ∈ {1, 2, . . . , n}, then one has ij ∂F ∂F kl ∂F ∂F ij ∂F ∂F kl ∂F ∂F g ∇¯ ∂F , g , = g ∇¯ ∂F , g , = ∂xi ∂xj ∂xk ∂xl ∂xm ∂xi ∂xj ∂xk ∂xl ∂xm ij ¯ ∂F ∂F kl ij ¯ ∂F ∂F k = g ∇ ∂F , g glm = g ∇ ∂F , δm = ∂xi ∂xj ∂xk ∂xi ∂xj ∂xk > ij ∂F ∂F ij ∂F ∂F = g ∇¯ ∂F , = g ∇¯ ∂F , . ∂xi ∂xj ∂xm ∂xi ∂xj ∂xm Using the non-degenerancy of the metric h·, ·i we complete the proof.
∂ ~ Hence, the Mean Curvature Flow equation ∂t F (p, t) = H(p, t) can be written in this new form:
∂ ij ∂F ij ∂F ∂F kl ∂F F (p) = g ∇¯ ∂F − g ∇¯ ∂F , g , ∂t ∂xi ∂xj ∂xi ∂xj ∂xk ∂xl ¯ ∂F ∂2F taking into account that ∇ ∂F ∂x = ∂x ∂x we obtain ∂xi j i j ∂ ∂2F ∂2F ∂F ∂F F (p) = gij − gij , gkl . ∂t ∂xi∂xj ∂xi∂xj ∂xk ∂xl If we express the previous equation in coordinates we get:
n+k ∂ ∂2F α X ∂2F β ∂F β ∂F α F α(p) = gij − gijgkl , ∂t ∂xi∂xj ∂xi∂xj ∂xk ∂xl β=1 where we clearly observe that we are dealing with a non-linear PDE.4
4The second order coefficients are gij, which depend on the map F . MEAN CURVATURE FLOW 15
Another way of writing the MCF equation is the following: ⊥ > ij ∂F ij ∂F ij ∂F H~ = g ∇¯ ∂F = g ∇¯ ∂F − g ∇¯ ∂F = ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj 2 > 2 ij ∂ F ∂F ij ∂ F ∂F (2.1) = g − ∇¯ ∂F = g − ∇ ∂F . ∂xi∂xj ∂xi ∂xj ∂xi∂xj ∂xi ∂xj At this point, we would like to remind some notions related to the hes- sian and laplacian operators.
Definition 2.3. Let (M n, g) be a Riemannian manifold and let ∇ be its Levi-Civita connection. The hessian of f ∈ C∞(M) is defined as the operator ∇2f : X(M) × X(M) → C∞(M) given by ∇2f(X,Y ) := X(Y (f)) − (∇X Y )(f) for any X,Y ∈ X(M).
It is well known that ∇2f is symmetric (to prove this we use that ∇ is torsion free) and C∞(M)-bilinear.
Definition 2.4. Given (M n, g) a Riemannian manifold and f ∈ C∞(M) we define the Laplacian of f as ∆f := Trace(∇2f).
Notice that the map F : M n → Rn+c does not have a well defined Laplacian. However, it makes sense to define the laplacian of each com- ponent F α. So, we define the laplacian of F as ∆F := (∆F 1, ∆F 2,..., ∆F n+c). Analogously, we can define also the hessian of F . Bearing this in mind, we have that 2 ∂F ∂F ∂F ∂F ∂F ∇ F , = (F ) − ∇ ∂F (F ) = ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj ∂ ∂ ∂F = (F ) − ∇ ∂F (F ) = ∂xi ∂xj ∂xi ∂xj ∂2F ∂F = − ∇ ∂F (F ) ∂xi∂xj ∂xi ∂xj and so equation (2.1) becomes ∂F ∂F H~ = gij · (∇2F ) , = Trace(∇2F ) = ∆F, ∂xi ∂xj Remark 2.5. Recall that H~ depends on the point p ∈ M and the instant t. In particular, we want to emphasize the temporal dependence of the mean curvature. Moreover, the intrinsic metric g also depends 16 FRANCISCO MARTIN AND JESUS PEREZ on t. For this reason, it is more precise to denote the intrinsic laplacian as ∆g(t)F .
Taking into account the above remark, one has: ~ H = ∆g(t)F, which provides a new way of writing the MCF equation: ∂F (2.2) = ∆ F. ∂t g(t)
Using this expression we will prove the existence of the mean curva- ture flow for a small time period in the case of compact manifolds. In the demonstration, we will use a technique known as “de Turck trick”. As we shall see, this trick consists of reducing the original problem to a strictly parabolic quasilinear problem.
Theorem 2.6 (Short Existence and Uniqueness). Let M be a n+1 compact manifold and F0 : M → R a given immersion. There exists a positive constant T > 0 and a unique smooth family of immersions F (·, t): M → Rn+1, t ∈ [0,T ), such that ∂ F (p, t) = H~ (p, t) for all (p, t) ∈ M × [0,T ), ∂t F (·, 0) = F0.
Proof. First of all, notice that F0 = F (·, 0) is an immersion, so F (·, t) also would be an immersion for t small enough (see, for instance, [GP74, p. 35].) That means that we only take care about the existence and solution of the above PDE. Assume that for some vector field V = vk ∂ in M (the field V will ∂xk be fixed later) we have that the equation: ˜ ˜ ∂F ˜ k ∂F (2.3) = ∆g(t)F + v ∂t ∂xk ˜ n+1 has solution for initial data F0, F : M × [0,T ) → R . We are going to see that the same happens for the MCF equation with initial data F0. Indeed, consider a family ϕt : M ×[0,T ) → M of diffeomorphisms of M. ˜ ˜ ˜ Let Ft(p) := Ft(ϕt(p)) = F (ϕt(p), t), where F is the aforementioned MEAN CURVATURE FLOW 17 solution of (2.3) (for now, we are assuming that such a solution exists.) ∂Ft Using the chain rule and (2.3) we compute ∂t (p):
˜ ˜ k ˜ ∂Ft ∂F (ϕt, t) ∂F ∂ϕt ∂F (p) = (p) = (ϕt(p), t) (p) + (ϕt(p), t) = ∂t ∂t ∂xk ∂t ∂t ˜ k ˜ ∂F ∂ϕt ˜ k ∂F = (ϕt(p), t) (p) + ∆g(t)F (ϕt(p), t) + v (ϕt(p), t) = ∂xk ∂t ∂xk ˜ k ˜ ∂F k ∂ϕt = ∆g(t)F (ϕt(p), t) + (ϕt(p), t) v + (p) . ∂xk ∂t
Hence, to get a solution to the MCF equation it suffices to find a family ϕt such that:
∂ϕ t = −V, ∂t ϕ0 = id .
This is a initial value problem for a system of ODE’s and so we can find a solution. Moreover, taking T > 0 small enough we can assume that ϕt is a diffeomorphism, for any t ∈ [0,T ]. This is due to the fact that the initial data is a diffeomorphism (the identity) and the fact that the diffeomorphisms from a compact manifold into itself form a stable class (see again [GP74, p. 35].) ˜ Hence, Ft(p) = F (ϕt(p), t) verifies:
∂F t (p) = ∆ F˜(ϕ (p), t) = ∆ F (p), ∂t g(t) t g(t) t
˜ ˜ ˜ F (p, 0) = F (ϕ0(p), 0) = F (id(p), 0) = F (p, 0) = F0(p), in other words, it represents a solution of the MCF equation with initial data F0. Summarizing, we only have to see that (2.3) has a solution. To do this we take the vector field V whose coordinates are given by vk := ij k k k k g (Γij − (Γ0)ij), being Γij the Christoffel symbols of M ((Γ0)ij means 18 FRANCISCO MARTIN AND JESUS PEREZ the Christoffel symbol at t = 0.) Then (2.3) becomes ˜ ˜ ∂F ˜ k ∂F = ∆g(t)F + v ∂t ∂xk 2 ˜ ˜ ˜ ij ∂ F ∂F ij k k ∂F = g − ∇ ∂F˜ + g (Γij − (Γ0)ij) ∂xi∂xj ∂xi ∂xj ∂xk 2 ˜ ˜ ˜ ij ∂ F k ∂F ij k k ∂F = g − Γij + g (Γij − (Γ0)ij) ∂xi∂xj ∂xk ∂xk 2 ˜ ˜ ij ∂ F k ∂F = g − (Γ0)ij . ∂xi∂xj ∂xk Thus, this particular choice of V implies that the equation: ˜ ˜ ∂F ˜ k ∂F = ∆g(t)F + v ∂t ∂xk can be written (in coordinates) as follows: ˜ 2 ˜ ˜ ∂F ij ∂ F ij k ∂F = g − g (Γ0)ij , ∂t ∂xi∂xj ∂xk which a system of quasilinear parabolic PDE’s because (gij) is a positive definite matrix which only depends on the first derivatives of F . The local theory of parabolic PDE’s [Tay96] and the fact that M is compact, gives us the existence and uniqueness of the solution in a short interval of time [0,T ). This concludes the proof.
3. Evolution of the Geometry by the Mean Curvature Flow
In this section we study how the usual geometric quantities evolve under the mean curvature flow.
Remark 3.1 (Notation). From now on, we will denote ∇iX = ∇ ∂ X ∂xi for any X ∈ X(M). For the sake of simplicity, we will often write:
∂f ∞ ∇if = for any function f ∈ C (M), ∂xi ∂ ∇iX = X for any field X ∈ X(M). ∂xi MEAN CURVATURE FLOW 19
Theorem 3.2 (Evolution of the intrinsic geometry). The intrin- sic metric and the volumen form evolve as follows: ∂ (3.1) g = −2hH,A~ i ∂t ij ij
∂ (3.2) pdet g = −|H~ |2pdet g, ∂t ∂ ∂ where Aij := II , and II(·, ·) denotes the second fundamental ∂xi ∂xj form of the corresponding immersion.
Proof. As all the computations are local, then we can assume that F is an embedding. Given a vector field X¯ in Rn+c we shall write: ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∇iX := ∇ ∂F X, ∇tX := ∇ ∂F X. ∂xi ∂t Using Schwarz’s formula, the MCF equation and the symmetry of the Levi-Civita connection, one has ∂ ∂ ∂F ∂F ¯ ∂F ∂F ∂F ¯ ∂F gij = , = ∇t , + , ∇t = ∂t ∂t ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj ¯ ∂F ∂F ∂F ¯ ∂F = ∇i , + , ∇j = ∂t ∂xj ∂xi ∂t ¯ ~ ∂F ∂F ¯ ~ = ∇i(H), + , ∇j(H) . ∂xj ∂xi
As ∂F is tangent to M and H~ is normal to M, then H,~ ∂F = 0. In ∂xj ∂xj particular, ∂ ~ ∂F ¯ ~ ∂F ~ ¯ ∂F 0 = H, = ∇i(H), + H, ∇i = ∂xi ∂xj ∂xj ∂xj ¯ ~ ∂F ~ ¯ ∂F = ∇i(H), + H, ∇ ∂F . ∂xj ∂xi ∂xj Therefore ¯ ~ ∂F ~ ¯ ∂F ∇i(H), = − H, ∇ ∂F . ∂xj ∂xi ∂xj 20 FRANCISCO MARTIN AND JESUS PEREZ
And so, substituting in the previous equation, we obtain ∂ ~ ¯ ∂F ¯ ∂F ~ gij = − H, ∇ ∂F − ∇ ∂F , H = ∂t ∂xi ∂xj ∂xj ∂xi ∂F ⊥ ∂F ⊥ = − H,~ ∇¯ ∂F − ∇¯ ∂F , H~ = ∂xi ∂xj ∂xj ∂xi ∂ ∂ ∂ ∂ = − H,II~ , − II , , H~ = ∂xi ∂xj ∂xj ∂xi ~ ~ ~ = −hH,Aiji − hAji, Hi = −2hH,Aiji, where we have used that, by the symmetry of II: Aij = Aji. √ Our next step is to study the evolution of det g 5; the volumen form: ∂ 1 ∂ pdet g = √ det g. ∂t 2 det g ∂t
∂ In order to compute ∂t det g, we are going to use equation (3.1), Jacobi’s formula for the derivative of a determinant: ∂ ∂ det g = Trace adj(g) g = Trace (det g)gij · (−2)hH,A~ i = ∂t ∂t kl n ij ~ X ij ~ = −2 det g Trace g · hH,Akli = −2 det g g hH,Aiji = i,j=1 n n X X ∂ ∂ = −2 det ghH,~ gijA i = −2 det g H,~ gijII( , ) = ij ∂x ∂x i,j=1 i,j=1 i j n ⊥ X ij ∂F = −2 det g H,~ g ∇¯ ∂F = −2(det g)hH,~ H~ i = ∂xi ∂x i,j=1 j = −2(det g)|H~ |2, where we have used the expression of H~ in local coordinates. Therefore, ∂ 1 ∂ 1 pdet g = √ det g = √ (−2)(det g)|H~ |2 = −|H~ |2pdet g. ∂t 2 det g ∂t 2 det g
5 p This is a simplified notation, we should write det gij(p, t). MEAN CURVATURE FLOW 21
3.1. Evolution of the Extrinsic Geometry. Below we will study the evolution of the extrinsic geometry. For simplicity assume that the codimension is 1, since in higher codimension many of the quantities in- volved are tensors. The interested reader can find a detailed exposition of the situation in higher codimension in [Smo11]. Let ν ∈ X⊥(M) be a unit normal field. We define the ν-second ∞ fundamental form hν : X(M) × X(M) → C (M) as the map given by hν(X,Y ) := hν, II(X,Y )i.
Notice that hν is a bilinear, symmetric form. The self adjoint oper- ator associated to hν is called the Weiengarten operator and it will be denoted by Sν. Recall that hν(X,Y ) = hSν(X),Y i.
For a compact hypersurface in Rn+1, there is a unique (up to the sign) Gauß map. So, for the sake of simplicity, we will write h and S instead of hν and Sν. Fix p ∈ M and consider a set of normal coordinates around p. For ∂ k simplicity we are going to label Ei = . Then we know that Γ (p) = 0 ∂xi ij k for all i, j, k (in particular , ∇Ei Ej(p) = Γij(p)Ek(p) = 0). It is well known that if we work with the Levi-Civita connection 6 we can assume that gij(p) = δij. Finally, let us define
hij := h(Ei,Ej).
At this point we are going to deduce the so called Simons’ identities, that play an important role here and in other key aspects of the theory. These are “Bochner-type” formulae relating the Laplacian of the second fundamental form and the Hessian of mean curvature. But first, we are going to get a local expression for the Laplace operator in terms of the notation introduced in Remark 3.1. Given ∞ f ∈ C (M) and {Ei} a local basis of smooth tangent fields, then ∆f = Trace(∇2f) = gij∇2 f = gij(∇ ∇ f − ∇ f). ij Ei Ej ∇Ei Ej
Given T is a symmetric 2-tensor on M, then T 2 is the symmetric 2-tensor defined (for codimension 1 sub manifolds) by:
2 X T (X,Y ) := T (X,Ei) · T (Ei,Y ), i
6To get the existence of a set of normal coordinates we only need that the con- nection is symmetric, not necessarily torsion free. 22 FRANCISCO MARTIN AND JESUS PEREZ where {Ei} is an arbitrary orthonormal frame. With this notation, we have that: Lemma 3.3 (Simons’ Identities). The following identities hold: (3.3) ∆h = ∇2H + H · h2 − |h|2 · h,
1 2 2 2 3 4 (3.4) ∆|h| = hh, ∇ Hi + ∇h + H Trace(h ) − |h| , 2 2 P 2 where hh, ∇ Hi = i,j hij∇ijH
Proof. We are going to work locally around a point p ∈ M, with the orthonormal frame {Ei} with ∇Ei Ej(p) = 0 and define
hij := h(Ei,Ej).
Remark 3.4. To avoid confusion, “the squared of hij” will be denoted 2 2 2 by (hij) , while the coefficients of the tensor h will be denoted as hij. We will proceed similarly with h3.
2 lm 3 ki lm Recall that hij = hilg hmj, huv = hukg hilg hmv, and |h| = ij kl 1/2 (g g hikhjl) is the norm of the tensor h with respect to the metric g. 2 2 2 First we prove ∆hij = ∇ijH + Hhij − |h| hij. As we are working in normal coordinates around p, kl ∆hij = g (∇E ∇E hij − ∇∇ E hij) = δkl∇E ∇E hij = ∇E ∇E hij. k l Ek l k l k k
In the abbreviated form we write ∆hij = ∇k∇khij. On the other hand, from the Codazzi equation, we have
(3.5) ∇khij = ∇ihkj.
Thus, we get ∆hij = ∇k∇ihkj. It is not hard to check that ∇k∇ihkj = ∇i∇khkj + Rikjαhαk + Rikkβhβj, ¯ where R(Ei,Ek,Ej,Eα) = Rikjα and similarly for R. Taking into ac- count that the ambience space is Euclidean, the Gauß equation gives:
0 − Rikjα = hAij,Akαi − hAiα,Akji, that is, Rikjα = hAiα,Akji − hAij,Akαi.
By definition, we have Aij = hijν and taking into account that hν, νi = 1, then we obtain
Rikjα = hAiα,Akji − hAij,Akαi =
= hiαhkjhν, νi − hijhkαhν, νi = hiαhkj − hijhkα. MEAN CURVATURE FLOW 23
Analogously, Rikkβ = hiβhkk − hikhkβ. Therefore
∆hij = ∇i∇khkj + Rikjαhαk + Rikkβhβj =
= ∇i∇khkj + (hiαhkj − hijhkα)hαk + (hiβhkk − hikhkβ)hβj =
= ∇i∇jhkk + hiαhkjhαk − hijhkαhαk + hiβhkkhβj − hikhkβhβj. In the last equality we have used the symmetry of h and (3.5):
∇khkj = ∇khjk = ∇jhkk. Recalling that we are using normal coordinates around p, we deduce ij H = g hij = δijhij = hii, 2 ij kl |h| = g g hikhjl = δijδklhikhjl = hikhik, 2 lm hij = hilg hmj = hilδlmhmj = hilhlj, including all this facts in our previous computations, we get 2 2 2 2 ∆hij = ∇i∇jhkk + hiαhαj − hij|h| + Hhij − hikhkj = 2 2 = ∇i∇jH + Hhij − |h| hij = 2 2 2 = ∇ijH + Hhij − |h| hij, this concludes the proof of the first identity. Now we want to prove that
1 2 2 2 3 4 ∆|h| = hij∇ H + ∇h + H Trace(h ) − |h| . 2 ij
2 2 As we are working on normal coordinates we have that |h| = (hij) . Then 1 1 ∆|h|2 = ∆(h )2. 2 2 ij 2 2 Moreover, we already showed that ∆(hij) = ∇k∇k(hij) , therefore: 1 1 ∆(h )2 = ∇ ∇ (h )2. 2 ij 2 k k ij Thus, 1 1 1 ∆(h )2 = ∇ ∇ (h )2 = ∇ (2h ∇ h ) = (∇ h )2 + h ∇ ∇ h = 2 ij 2 k k ij 2 k ij k ij k ij ij k k ij 2 = (∇khij) + hij(∆hij) 2 2 2 2 = (∇khij) + hij(∇ijH + Hhij − |h| hij) 2 2 2 4 = (∇khij) + hij∇ijH + Hhijhij − |h| . 24 FRANCISCO MARTIN AND JESUS PEREZ where in the last two equalities we have used that: 2 2 2 ∆hij = ∇k∇khij = ∇ijH + Hhij − |h| hij, as we already checked in the proof of the previous identity. Now, using the definition of trace of h3 and the fact that we are working with normal coordinates around p, we get 3 uv 3 uv 3 3 ki lm Trace(h ) = g huv = δ huv = huu = hukg hilg hmu = ki lm = hukδ hilδ hmu = hukhklhlu. On the other hand, 2 lm lm hijhij = hij(hilg hmj) = hij(hilδ hmj) = hij(hilhlj) = hijhjlhli =
= hukhklhlu, where we have used that h is symmetric. 3 2 Therefore, Trace(h ) = hijhij, which implies 1 ∆|h|2 = (∇ h )2 + h ∇2 H + H Trace(h3) − |h|4. 2 k ij ij ij 2 2 Finally, let us check that (∇khij) = ∇h . Using our local orthonor- 2 P 2 mal frame{Ei} parallel at p, we have that ∇h = i,j |∇hij| . Then 2 X 2 ∇h = |∇hij| i,j X X = h∇hij,EriEr, h∇hkl,EsiEs i,j k,l X X = h∇hij,Eri h∇hkl,Esi hEr,Esi i,j,r k,l,s X X = h∇hij,Eri h∇hij,Esi δrs i,j,r i,j,s X 2 = h∇hij,Eri i,j,r X 2 = Er(hij) i,j,r X 2 = ∇rhij , i,j,r
Therefore,
1 2 2 2 3 4 ∆|h| = hij∇ H + ∇h + H Trace(h ) − |h| , 2 ij MEAN CURVATURE FLOW 25 as we wanted to prove.
Theorem 3.5 (Evolution of the extrinsic geometry). In our set- ting we have: ∂ ∂H ∂F (3.6) ν = − gij = −∇H ∂t ∂xi ∂xj
∂ (3.7) h = ∆h − 2Hh2 + |h|2h ∂t ij ij ij ij ∂ (3.8) H = ∆H + |h|2H ∂t
∂ 2 2 2 4 (3.9) |h| = ∆|h| − 2 ∇h + 2|h| . ∂t
∂F Proof. Along this proof, we will denote Ei = . ∂xi
i) Let us prove that ∂ ν = − ∂H ∂F gij. ∂t ∂xi ∂xj ∂ We start by decomposing the vector ∂t ν in the base {Ei}, ∂ ∂ ν = gij ν, E E = gij ∇¯ ν, E E = ∂t ∂t i j t i j ∂ = gij hν, E i − ν, ∇¯ E E = ∂t i t i j ∂ ¯ ν ⊥ Ei, implies ∂t hν, Eii = 0. Moreover, we know that ∇tEi = ¯ ∂F ¯ ∂F ¯ ~ ¯ ~ ∇t = ∇i = ∇iH = ∇iH, which implies ∂xi ∂t ij ¯ ~ = −g hν, ∇iHiEj = ij ∂ ~ ¯ ~ = −g hν, Hi − h∇iν, Hi Ej = ∂xi At this point, we use that hν, H~ i = hν, Hνi = Hhν, νi = H ·1 = H, ¯ ~ ¯ ¯ and h∇iν, Hi = h∇iν, Hνi = Hh∇iν, νi = H · 0 = 0 and so: ∂H ∂F = −gij . ∂xi ∂xj ∂ In order to prove that ∂t ν = −∇H we only have to remember that, given a smooth function f then ∇f = gij ∂f ∂ . Applying ∂xi ∂xj 26 FRANCISCO MARTIN AND JESUS PEREZ
the above expression to H we get: ∂ ∂H ∂F ν = − gij = −∇H. ∂t ∂xi ∂xj
∂ 2 2 ii) Let us prove now ∂t hij = ∆hij − 2Hhij + |h| hij. ∂ ∂ ∂ ∂ ∂ ¯ ⊥ ∂ ¯ hij = II , , ν = h(∇iEj) , νi = h∇iEj, νi. ∂t ∂t ∂xi ∂xj ∂t ∂t By deriving, we get ∂ h∇¯ E , νi = h∇¯ ∇¯ E , νi + h∇¯ E , ∇¯ νi. ∂t i j t i j i j t On one hand, we have
¯ ¯ ¯ ¯ ¯ ¯ ∂F ¯ ¯ ∂F ¯ ¯ ~ ∇t∇iEj = ∇i∇tEj = ∇i∇t = ∇i∇j = ∇i∇jH = ∂xj ∂t ¯ ¯ = ∇i∇j(Hν). On the other hand, using (3.6), we deduce
¯ ∂ rs ∂H ∇tν = ν = −g Es, ∂t ∂xr which gives ∂ h∇¯ E , νi = h∇¯ ∇¯ E , νi + h∇¯ E , ∇¯ νi = ∂t i j t i j i j t ¯ ¯ ¯ rs ∂H = h∇i∇j(Hν), νi + h∇iEj, −g Esi. ∂xr Summarizing, we have
∂ ∂ ¯ ¯ ¯ ¯ rs ∂H hij = h∇iEj, νi = h∇i∇j(Hν), νi + h∇iEj, −g Esi. ∂t ∂t ∂xr ¯ ¯ ¯ Now, we deal with ∇j(Hν) = (∇jH)ν +H(∇jν). We can compute ¯ ∇jν as we did in (3.6): ¯ ∂ lk ∂ lk ∂ ¯ ∇jν = ν = g ν, El Ek = g hν, Eli − hν, ∇jEli Ek = ∂xj ∂xj ∂xj lk ∂ ∂ ∂ lk = g 0 − hν, II , i Ek = g (−hjl)Ek. ∂xj ∂xj ∂xl Hence, we get ¯ ¯ ¯ ¯ lk ∇j(Hν) = (∇jH)ν + H(∇jν) = (∇jH)ν − Hg hjlEk. MEAN CURVATURE FLOW 27
Moreover,
¯ rs ∂H rs ∂H ¯ rs ∂H ¯ > h∇iEj, −g Esi = −g h∇iEj,Esi = −g h(∇iEj) ,Esi = ∂xr ∂xr ∂xr rs ∂H rs ∂H k = −g h∇iEj,Esi = −g hΓijEk,Esi = ∂xr ∂xr rs ∂H k k rs ∂H = −g Γijgks = −Γijg gsk = ∂xr ∂xr k k ∂H k ∂H = −Γijδr = −Γij . ∂xr ∂xk Therefore,
¯ ¯ ¯ rs ∂H h∇i∇j(Hν), νi + h∇iEj, −g Esi = ∂xr ¯ ¯ lk k ∂H = h∇i (∇jH)ν − Hg hjlEk , νi − Γij . ∂xk At this point, we have that:
∂ ¯ ¯ lk k ∂H hij = h∇i (∇jH)ν − Hg hjlEk , νi − Γij . ∂t ∂xk Notice that one has ¯ ¯ lk ¯ ¯ ¯ lk ∇i (∇jH)ν − (Hg hjl)Ek = ∇i (∇jH)ν − ∇i (Hg hjl)Ek = ¯ ¯ ¯ ¯ ¯ lk lk ¯ = ∇i(∇jH) ν + (∇jH)∇iν − ∇i(Hg hjl) Ek − (Hg hjl)∇iEk. We make the scalar product with ν, taking into account that ¯ ¯ ⊥ ∂ ∂ h∇iEk, νi = h(∇iEk) , νi = II , , ν = hik, ∂xi ∂xk and we get ¯ ¯ lk ¯ ¯ lk h∇i (∇jH)ν − Hg hjlEk , νi = ∇i(∇jH) − Hg hjlhik. In short, we proved that
∂ ¯ ¯ lk k ∂H hij = ∇i(∇jH) − Hg hjlhik − Γij . ∂t ∂xk Now observe that by definition of Hessian of a differentiable func- 2 tion we have:∇ f(X,Y ) = X Y (f) − (∇X Y )(f). In particu- lar, for f = H, X = ∂ , Y = ∂ , and taking into account ∂xi ∂xj ∂ k ∂f ∇ ∂ ∂x = Γij and using the notation ∂x = ∇if, one obtains ∂xi j i that ¯ ¯ k 2 ∂ ∂ 2 ∇i∇jH − Γij∇kH = ∇ H , = ∇ijH. ∂xi ∂xj 28 FRANCISCO MARTIN AND JESUS PEREZ
lk kl 2 As g y h are symmetric, g hjlhik = hikg hlj = hij. In short, ∂ h = ∇2 H − Hh2 . ∂t ij ij ij Finally, using Simons’ inequality (3.3) one gets 2 2 2 2 ∇ijH − Hhij = ∆hij − 2Hhij + |h| hij, which concludes the proof of (3.7).
∂ 2 iii) Now we prove ∂t H = ∆H + |h| H. This time, we will use that the mean curvature can be also com- ij puted like H = g hij. ∂ ∂ ∂ ∂ H = gijh = gij h + gij h . ∂t ∂t ij ∂t ij ∂t ij As we saw in the proof of the previous item, ∂ h = ∆h − 2Hh2 + |h|2h . ∂t ij ij ij ij ∂ ij To compute ∂t g recall that we knew, from (3.1), that ∂ g = −2hH,A~ i = −2hHν, A i = −2Hhν, A i = −2Hh . ∂t ij ij ij ij ij ij il kj 7 As g = g glkg , then ∂ ∂ gij = gilg gkj = ∂t ∂t lk ∂ ∂ ∂ = gil g gkj + gil g gkj + gilg gkj = ∂t lk ∂t lk lk ∂t ∂ ∂ ∂ = gil δj + gil g gkj + δi gkj = ∂t l ∂t lk k ∂t ∂ ∂ ∂ = gij + gil g gkj + gij = ∂t ∂t lk ∂t ∂ ∂ = 2 gij + gil g gkj, ∂t ∂t lk so substituting, ∂ ∂ gij = −gil g gkj = −gil(−2Hh )gkj = 2Hgilh gkj = ∂t ∂t lk ij ij i kj ij = 2Hhkg = 2Hh .
7 il kj i kj ij Indeed, g glkg = δkg = g . MEAN CURVATURE FLOW 29
Therefore,
∂ ∂ ∂ H = gij h + gij h = ∂t ∂t ij ∂t ij ij ij 2 2 = 2Hh hij + g (∆hij − 2Hhij + |h| hij).
2 2 2 Simons’ identity (3.3) implies ∆hij = ∇ijH + Hhij − |h| hij, land 2 2 2 2 so ∆hij − 2Hhij + |h| hij = ∇ijH − |h| hij. Then,
∂ H = 2Hhijh + gij(∇2 H − |h|2h ) ∂t ij ij ij ij ij 2 2 ij = 2Hh hij + g ∇ijH − |h| g hij ij 2 2 = 2Hh hij + Trace(∇ H) − |h| H ij 2 = ∆H + H(2h hij − |h| ).
Note that
2 ij kl ij lk ji l lj jl ij |h| = g g hikhjl = g g hkihjl = g hihjl = h hjl = h hjl = h hij,
therefore
ij 2 2 2 2 2h hij − |h| = 2|h| − |h| = |h| .
In short ∂ H = ∆H + |h|2H. ∂t
∂ 2 2 2 4 iv) Finally, we will prove ∂t |h| = ∆|h| − 2 ∇|h| + 2|h| . By definition, we have
2 ij kl |h| = g g hikhjl.
From the proof of the former item, we already know
∂ gij = 2Hhij, ∂t
∂ h = ∆h − 2Hh2 + |h|2h (it is precisely (3.7)). ∂t ij ij ij ij 30 FRANCISCO MARTIN AND JESUS PEREZ
Then ∂ ∂ |h|2 = (gijgklh h ) = ∂t ∂t ik jl ∂ ∂ = gij gklh h + gij gkl h h + ∂t ik jl ∂t ik jl ∂ ∂ gijgkl h h + gijgklh h = ∂t ik jl ik ∂t jl ij kl ij kl = 2Hh g hikhjl + g 2Hh hikhjl+ ij kl 2 2 g g (∆hik − 2Hhik + |h| hik)hjl+ ij kl 2 2 g g hik(∆hjl − 2Hhjl + |h| hjl). If in this last expression we rename the indices of the second term by changing i for k, j for l, k for i and l for j, and in the fourth addend changing i to j, j for i, k for l and l for k, then you get ∂ |h|2 = 4Hhijgklh h + 2gijgkl(∆h − 2Hh2 + |h|2h )h . ∂t ik jl ik ik ik jl In the proof of Lemma 3.3 we got 2 kl hij = g hikhjl, 2 3 hijhij = Trace(h ), therefore ij kl 3 4Hh g hikhjl = 4H Trace(h ). Using (3.3), 2 2 2 2 ∆hik − 2Hhik + |h| hik = ∇ikH − Hhik. So we get ∂ |h|2 = 4H Trace(h3) + 2gijgkl(∇2 H − Hh2 )h . ∂t ik ik jl It is sufficient to check the equality at a point p in which it can be assumed that are considered normal coordinates (in particular, ij kl g (p) = δij and g (p) = δkl), thus the last addend of the above expression can be simplified as follows ij kl 2 2 2 2 2g g (∇ikH − Hhik)hjl = 2hik(∇ikH − Hhik) = 2 2 2 3 = 2hik∇ikH − 2Hhikhik = 2hik∇ikH − 2H Trace h . So far, we have proven ∂ (3.10) |h|2 = 2H Trace(h3) + 2h ∇2 H. ∂t ik ik MEAN CURVATURE FLOW 31
At this point recall that we want to see
∂ 2 2 2 4 |h| = ∆|h| − 2 ∇h + 2|h| . ∂t According to Simons’ identity (3.4),
2 2 2 3 4 ∆|h| = 2hij∇ijH + 2 ∇h + 2H Trace(h ) − 2|h| , thereby substituting the expression we want to show, we get ∂ (3.11) |h|2 = 2h ∇2 H + 2H Trace(h3). ∂t ij ij Comparing (3.10) and (3.11) we see that both coincide. This com- pletes the proof.
4. A comparison principle for parabolic PDE’s
As we have seen in section 2, understanding and working with the mean curvature flow involves a good knowledge about parabolic partial differential equations. As it is well known, these equations generally can not be solved, forcing us to look for results about the qualitative behavior of the solutions. In this section we prove a theorem in that direction which is known as the principle of comparison. Throughout this section we will use Lieberman’s book [Lie96].
Let Ω be a domain (open, connected subset) in Rn+1. In this setting we will write the points of Rn+1 as X = (x, t), where x ∈ Rn. Definition 4.1 (Parabolic Boundary). We define the parabolic bound- ary of Ω, denoted as ∂P Ω, as the set of points X = (x, t) ∈ ∂Ω (that is, the topological boundary of Ω) such that for all > 0 the parabolic cylinder Q(X, ) contains points in the complement of Ω; the definition of parabolic cylinder Q(X0, ) is n+1 Q(X0, ) := {Y ∈ R : |Y − X0| < , t < t0}, p n where X0 = (x0, t0) and |(x, t)| = max{|x|R , |t|}.
In the simplest case Ω = D × (0,T ), D a domain in Rn and T > 0, we have that the parabolic boundary of Ω coincides with ∂P Ω = BΩ ∪ SΩ ∪ CΩ, where BΩ := Ω × {0} ( the “bottom” Ω), SΩ := ∂Ω × (0,T ) (the “side” of Ω) and CΩ := ∂(Ω) × {0} (the “corner” of Ω.) 32 FRANCISCO MARTIN AND JESUS PEREZ
Given u ∈ C2,1(Ω) we define the quasi-linear, second-order operator P as ∂u (4.1) P u := − + aij(X, u, Du)D2 u + a(X, u, Du). ∂t ij We assume that aij(X, z, p) y a(X, z, p) are defined for any (X, z, p) ∈ Ω × R × Rn. We say that P is parabolic in a subset S of Ω × R × Rn if the matrix whose coefficients are aij(X, z, p) is positive definite for any (X, z, p) ∈ S. We distinguish two especial cases for S:
• If S = Ω × R × Rn, we say that P is parabolic; • If S = {(X, z, p) ∈ U × R × Rn : z = u(X), p = Du(X)} for some function u ∈ C1(U) where U ⊂ Ω, then we will say that P is parabolic at u. Lemma 4.2. Let A and B symmetric real matrices of order n, with A positive definite and B negative semidefinite. Then Trace(A · B) ≤ 0.
Proof. As A is positive definite, then A has a squared root, that we denote by A1/2, which is regular and symmetric. By Sylvester’s law of inertia, as B and A1/2B(A1/2)> = A1/2BA1/2 are congruent symmetric matrices8, then they have the same number of positive, negative and 1/2 1/2 null eigenvalues. This means that all the eigenvalues µi of A BA are non-positive. Then, we have n X 1/2 1/2 1/2 1/2 0 ≥ µi = Trace(A BA ) = Trace((A B)A ) = i=1 = Trace(A1/2(A1/2B)) = Trace(AB)
Theorem 4.3 (Lieberman, Comparison Principle). Let P be a quasi-linear operator like in (4.1). Suppose aij(X, z, p) dos not depend on z and that there exists a positive increasing function K(L) such that a(X, z, p) − K(L) · z is decreasing in z on Ω × [−L, L] × Rn for L > 0. 2,1 If u and v are functions in C (Ω \ ∂P Ω) ∩ C(Ω) such P is parabolic at either u or v, P u ≥ P v on Ω \ ∂P Ω and u ≤ v on ∂P Ω, then u ≤ v on Ω.
Proof. First, we fix L in such a way that [−L, L] contains the range of u and v , i.e., L := max{supΩ |u|, supΩ |v|}. 8Notice that B = P >A1/2BA1/2P ; where P = (A1/2)−1. MEAN CURVATURE FLOW 33
Let us define w := (u − v)eλt en Ω, where λ is a real constant to be determine later. Notice that u ≤ v on Ω is equivalent to prove that w ≤ 0 on Ω. From our assumptions, we know that u ≤ v on ∂P Ω, so we have to prove that:
w ≤ 0 en Ω \ ∂P Ω. So would finish if we see that w can not have a positive interior max- imum. We proceed by contradiction: Suppose that X0 = (x0, t0) is an interior maximum such that w(X0) > 0. Classical Analysis says to us that if a function of class C2 reaches the maximum at an interior point, then the gradient at that point is zero and the Hessian matrix is negative semidefinite 9. In particular, we have:
λt0 (4.2) Dw(X0) = 0 ⇔ (Du − Dv)(X0)e = 0 ⇒ Du(X0) = Dv(X0), ∂w ∂ (X ) = 0 ⇔ (u − v)(X )eλt0 + λ(u − v)(X )eλt0 = 0, ∂t 0 ∂t 0 0 which implies: ∂ (4.3) λ(u − v)(X ) = − (u − v)(X ), 0 ∂t 0
10 (4.4) The matrix (Dijw(X0))i,j=1,...,n is negative semidefinite .
Below, for convenience, we denote R := (X0, u(X0), Du(X0)) and S := (X0, v(X0), Dv(X0)). From our hypothesis we know P u ≥ P v on Ω \ ∂P Ω; in particular
0 ≤ P u(X0) − P v(X0) = using the definition of the operator P ∂ = aij(R)D u(X ) − aij(S)D v(X ) + a(R) − a(S) − (u − v)(X ) = ij 0 ij 0 ∂t 0 ij ij Notice that the first two terms a (R) = a (X0, u(X0), Du(X0)) and ij ij a (S) = a (X0, v(X0), Dv(X0)) coincide. This is due to the fact that
9We are assuming that n ≥ 2. If n = 1 would have that the derivative vanishes at that point and the second derivative is positive at that point. We no longer do this distinction, but it is clear that what follows is valid in the case n = 1 with the obvious changes. 10 If X0 is an interior maximum of w, then the Hessian matrix w in X0, Hw(X0) = (Dijw(X0))i,j=1,...,n,t, is negative semidefinite. Therefore, by the criterion of the principal minors, the square submatrix obtained by removing the last row and last column of Hw(X0), which is (Dijw(X0))i,j=1,...,n, remains a negative semidefinite matrix. 34 FRANCISCO MARTIN AND JESUS PEREZ aij does not depend on the second argument and because, by(4.2), Du(X0) = Dv(X0). Using the above fact and (4.3) we have: ij = a (R)Dij(u − v)(X0) + a(R) − a(S) + λ(u − v)(X0) ≤ ij Applying Lemma 4.2 to the matrices A := (a (R))i,j=1,...,n and B := ij (Dji(u − v)(X0))i,j=1,...,n, we get a (R)Dij(u − v)(X0) ≤ 0, and so
≤ a(R) − a(S) + λ(u − v)(X0) ≤
At this point, recall that w(X0) > 0 , i.e., u(X0) > v(X0). As a(x, z, p) − K(L)z is a decreasing function of z in Ω × [−L, L] × Rn, then a(R) − K(L)u(X0) ≤ a(S) − K(L)v(X0), or in other words, a(R) − a(S) ≤ K(L)(u − v)(X0),
≤ K(L)(u − v)(X0) + λ(u − v)(X0) ≤ [K(L) + λ](u − v)(X0).
Now, It suffices to take λ < −K(L) to get that (u − v)(X0) ≤ 0, that is, w(X0) ≤ 0, which is contrary to w(X0) > 0. This contradiction proves the theorem.
5. Graphical submanifolds. Comparison Principle and Consequences
The mean curvature flow has been extensively studied in some families which have specific geometric conditions, as is the case of hypersurfaces, Lagrangian submanifolds, graphs, etc. In this section we will focus on the study of smooth graphs.
Let u: Rn → R be a smooth function. It is well known that the graph of u, Graph(u) = {(x, u(x)): x ∈ Rn} is a hypersurface Rn+1. Then, we can study how it evolves under mean curvature flow. The first goal of this section will be to deduce the evolution equation for ut. Next, we will see that we can apply a comparison principle to obtain a result known as avoidance principle.
n n+1 We start with a graph M0 = {(x, u(x)): x ∈ R } ⊂ R . We are looking for a map n n+1 F : R × [0,T ) → R F (p, t) = (x(p, t), u(x(p, t), t)) satisfying the MCF equation ∂F = Hν, ∂t MEAN CURVATURE FLOW 35 and such that F (·, 0) = M0 = Graph(u). At this point, it would be ex- tremely useful to know how to express the main geometrical quantities associated to M0 in terms of u and its derivatives. This is the purpose of the next lemma
Lemma 5.1 (Graphical submanifolds). Let u: Rn → R be a smooth function and M0 = Graph(u). Then, we have:
(1) gij = δij + DiuDju, D uD u (2) gij = δ − i j , ij 1 + |Du|2 (−Du, 1) (3) ν = , p1 + |Du|2 2 Diju (4) hij = , p1 + |Du|2 Du (5) H = div . p1 + |Du|2
Proof.
(1) The proof of the first item is straightforward:
gij = hDiF,DjF i = h(ei,Diu), (ej,Dju)i = δij + DiuDju,
n+1 where {e1, . . . en+1} denotes the canonical basis of R . (2) If we make the matrix product, we get D uD u g gkj = (δ + D uD u)(δ − k j ) = ik ik i k kj 1 + |Du|2 D uD u D uD u = δ δ − δ k j + D uD uδ − D uD u k j = ik kj ik 1 + |Du|2 i k kj i k 1 + |Du|2 D uD u D uD u = δ − i j + D uD u − |Du|2 i j = ij 1 + |Du|2 i j 1 + |Du|2 D uD u = δ − (1 + |Du|2) i j + D uD u = ij 1 + |Du|2 i j
= δij − DiuDju + DiuDju = δij,
ij which means that (g ) is the inverse matrix of (gij). (3) The vectors {DiF = (ei,Diu), i = 1, . . . n} are a global basis of the tangent bundle of Graph(u).Then (−Du, 1) is a normal, 36 FRANCISCO MARTIN AND JESUS PEREZ
non-vanishing vector field. So, we only have to normalized it (−Du, 1) ν = . p1 + |Du|2 (4) ¯ ⊥ ¯ hij = hII(DiF,DjF ), νi = h(∇DiF DjF ) , νi = h∇DiF DjF, νi = 2 2 (−Du, 1) = hDD F DjF, νi = hD F, νi = (0,D u), = i ij ij p1 + |Du|2 D2 u = ij . p1 + |Du|2 (5) On one hand, we have Du Diu div = Di = p1 + |Du|2 p1 + |Du|2 Pn 2 i=1 Diiu 1 1 = − Diu 2hDi(Du), Dui = p1 + |Du|2 2 (1 + |Du|2)3/2
∆u 1 2 = − Diu hD uDju, Dui = p1 + |Du|2 (1 + |Du|2)3/2 ji 2 2 ∆u DiuD uDju ∆u DiuDjuD u = − ji = − ij . p1 + |Du|2 (1 + |Du|2)3/2 p1 + |Du|2 (1 + |Du|2)3/2 We number this auxiliary result that we will use it a few times in later calculations: 2 Du ∆u DiuDjuD u (5.1) div = − ij . p1 + |Du|2 p1 + |Du|2 (1 + |Du|2)3/2 On the other hand, 2 ij DiuDju Diju H = g hij = δij − = 1 + |Du|2 p1 + |Du|2 2 2 2 Diju DiuDjuDiju ∆u DiuDjuDiju = δij − = − . p1 + |Du|2 (1 + |Du|2)3/2 p1 + |Du|2 (1 + |Du|2)3/2 Therefore, Du H = div . p1 + |Du|2
MEAN CURVATURE FLOW 37
Now we use this information to find new partial differential equations for graphs that evolve by mean curvature. We start by deriving with respect to time the application F . Deriving component by component and applying the chain rule, ∂F ∂x ∂u ∂x ∂u ∂x ∂x ∂u = , i + = , Du, + . ∂t ∂t ∂xi ∂t ∂t ∂t ∂t ∂t Using this, the equation of the mean curvature flow becomes: ∂x ∂x ∂u H , Du, + = (−Du, 1). ∂t ∂t ∂t p1 + |Du|2 or equivalently ∂x Du (5.2) = −H , ∂t p1 + |Du|2 ∂x ∂u H (5.3) Du, + = . ∂t ∂t p1 + |Du|2 Notice that, using 5.2, one has ∂x H H Du, = − hDu, Dui = − |Du|2. ∂t p1 + |Du|2 p1 + |Du|2 Therefore, equation (5.3) can be written as follows: ∂u 1 (5.4) = H(1 + |Du|2) = Hp1 + |Du|2 ∂t p1 + |Du|2 Finally, substituting H = div √ Du in (5.2) and (5.4) they be- 1+|Du|2 come: ∂x Du Du (5.5) = − div · , ∂t p1 + |Du|2 p1 + |Du|2 ∂u Du p 2 (5.6) = div · 1 + |Du| . ∂t p1 + |Du|2
We need some extra computations to get an expression for the above divergence. According to (5.1),
Pn 2 2 Du D u DiuDjuD u div = i=1 ii − ij . p1 + |Du|2 p1 + |Du|2 (1 + |Du|2)3/2 38 FRANCISCO MARTIN AND JESUS PEREZ
Then, Pn 2 2 Du D u DiuDjuD u div = i=1 ii − ij = p1 + |Du|2 p1 + |Du|2 (1 + |Du|2)3/2 2 1 2 DiuDjiuDju = δijD u − = p1 + |Du|2 ij 1 + |Du|2 1 DiuDju 2 = δij − D u. p1 + |Du|2 1 + |Du|2 ij Substituting in (5.6), ∂u D uD u D2u(Du, Du) (5.7) = δ − i j D2 u = ∆u − , ∂t ij 1 + |Du|2 ij 1 + |Du|2 where recall that D2u means the Hessian operator associated to u. With all we have seen so far we can prove the following well-known result for mean curvature flow of a sphere.
Proposition 5.2. The spheres of Euclidean space evolve under the mean curvature flow as spheres that concentrically contract until col- lapse in finite time at one point; the common center of the family of spheres.
Proof. The upper n-dimensional hemisphere of radius ρ can be seen as p the graph of the function u: B(0, ρ) → R given by u(x) := ρ2 − |x|2, where B(0, ρ) ⊂ Rn means the Euclidean ball centered at the origin of radius ρ. The abundance of symmetry of the sphere will allow us to solve the partial differential equation of the mean curvature flow in this particular case. Indeed, xi Diu = − , pρ2 − |x|2 and from here we get x |x|2 ρ2 Du = − , |Du|2 = , 1 + |Du|2 = ; pρ2 − |x|2 ρ2 − |x|2 ρ2 − |x|2 x x g = δ + i j ; ij ij ρ2 − |x|2 ρ2 − |x|2 x x x x gij = δ − i j = δ − i j ; ij ρ2 ρ2 − |x|2 ij ρ2 MEAN CURVATURE FLOW 39
pρ2 − |x|2 −x x pρ2 − |x|2 ν = − , 1 = , . ρ pρ2 − |x|2 ρ ρ Thus 2 ∂ ∂ ∂ xi Diju = u = − p = ∂xj ∂xi ∂xj ρ2 − |x|2 p 2 2 xj δij ρ − |x| − xi −√ ρ2−|x|2 = − = ρ2 − |x|2 δ x x = − ij − i j , pρ2 − |x|2 (ρ2 − |x|2)3/2 therefore
p 2 2 ρ − |x| δij xixj 1 xixj hij = − + = − δij+ . ρ pρ2 − |x|2 (ρ2 − |x|2)3/2 ρ ρ2 − |x|2
ij Using that H = g hij, we get: X xixj 1 xixj H = gijh = − δ − δ + = ij ij ρ2 ρ ij ρ2 − |x|2 i,j 1 x x x x P x2x2 = − δ δ + δ i j − i j δ − i,j i j = ρ ij ij ij ρ2 − |x|2 ρ2 ij ρ2(ρ2 − |x|2) 1 X |x|2 |x|2 |x|4 = − δ + − − = ρ ij ρ2 − |x|2 ρ2 ρ2(ρ2 − |x|2) i,j 1 ρ2 − ρ2 + |x|2 |x|4 = − n + |x|2 − = ρ ρ2(ρ2 − |x|2) ρ2(ρ2 − |x|2) n = − , ρ
P 2 P 2 2 where we have used δijδij = i,j δij, δijxixj = |x| , and i,j xi xj = P 2 P 2 2 2 4 i xi j xj = |x| |x| = |x| . Taking into account the previous computations the PDE (5.4) leads to a partial differential equation that we solve (notice that we do not q 2 2 work now with u(x) but with u(x, t) = ρ(t) − x(t) ): ∂u n ρ(t) n = Hp1 + |Du|2 = − = − , ∂t ρ(t) q 2 2 q 2 2 ρ(t) − x(t) ρ(t) − x(t) 40 FRANCISCO MARTIN AND JESUS PEREZ in other words, ∂u n = − , ∂t u(x, t) which is an ODE that we can integrate: u2 udu = −ndt ⇒ = −nt + C ⇒ u(x, t) = pK(x) − 2nt, 2 where K(x) is a “constant” (which depends on x but not on t). As the initial data is u(x, 0) = pρ2 − |x|2, then we deduce q u(x, t) = ρ2 − 2nt − |x|2, x ∈ B(0, ρ2 − 2nt). Finally, notice that: ρ2 (5.8) ρ2 − 2nt ≥ 0 ⇔ ≥ t, 2n ρ2 then, if the starting sphere is Sρ(0), the collapse occurs at time t = 2n (see Figure 2.)
Let us turn our attention to equation (5.7): ∂u D uD u = δ − i j D2 u. ∂t ij 1 + |Du|2 ij If we compare it with the definition of the operator (4.1) we obtain that (in this particular case): p p aij(X, z, p) = δ − i j (therefore it does not depend on z), ij 1 + |p|2 a(X, z, p) ≡ 0. We claim that (5.7) defines a parabolic operator, that is, that the ma- trix whose coefficients are aij(X, z, p) is positive definite. Indeed,
ij • At p = 0, a = δij, which is the identity matrix; • If p 6= 0, consider x ∈ Rn \{0}. p p x p p x x aij(X, z, p)x = x δ − i j x = x δ x − i i j j = i j i ij 1 + |p|2 j i ij j 1 + |p|2
n n (xipi)(pjxj) hx, piR hx, piR = hx, xi n − = hx, xi n − = R 1 + |p|2 R 1 + |p|2 hx, pi2 = |x|2 − . 1 + |p|2 MEAN CURVATURE FLOW 41
Figure 2. A family of concentric spheres collapsing at one point.
Taking this into account, hx, pi2 x aij(X, z, p)x > 0 ⇔ |x|2 − > 0 ⇔ hx, pi2 < |x|2(1 + |p|2), i j 1 + |p|2 and this last inequality holds by the Cauchy-Schwarz inequality (in Rn): hx, pi2 ≤ hx, xihp, pi2 = |x|2|p|2, and because, obvi- ously, |x|2|p|2 < |x|2(1 + |p|2) (recall that |x|= 6 0).
At this point we are ready to prove the comparison principle.
Theorem 5.3 (Comparison principle). Let M0 and N0 be compact, embedded hypersurfaces, without boundary, in Rn+1 that do not inter- sect. If Mt and Nt are their respective evolutions by the mean curvature flow, then they never intersect. 42 FRANCISCO MARTIN AND JESUS PEREZ
First proof. We proceed by contradiction. Assume Mt and Nt first in- tersect at time t0 at a point p (which is an interior point of both surfaces because , by assumption, none of them have boundary.) Then both hy- persurfaces have the same tangent plane at the point p, otherwise this would not be your first point of contact. Then Mt0 y Nt0 can be ex- pressed locally about p as graphs of functions, ut0 and vt0 respectively. As both hypersurfaces have the same tangent plane at p, also have the same unit normal vector at p, except perhaps by the sign. Considering the same orientation on Mt0 and Nt0 we can compare them. Note that, as t0 is the first point of contact, just a moment before both hypersur- faces do not intersect, , so either vt < ut or vt > ut. Without loss of generality (since this depends on the choice of the Gauß map) we can assume vt > ut. So, just a moment before t0, there exists ε > 0 such that vt − ut > ε. Now we apply the mean curvature flow starting in that instant be- fore t0. As we have seen before, we know that ut and vt verifies the quasilinear parabolic equation: ∂u D u, D u P u = − + δ − i j D2 u = 0. ∂t ij 1 + |Du|2 ij As we also check before, P can be seen as an operator satisfying the hy- potheses of the comparison principle for parabolic operators (Theorem 4.3.) It is obvious that vt − is also a solution of the above equation. Moreover, ut y vt − satisfy the assumptions of Theorem 4.3 in a neigh- borhood of the point p, then we deduce that ut ≤ vt − (< vt) in this particular neighbourhood of p, which contradicts u(p, t0) = v(p, t0). This contradiction completes the proof.
As the reader can see, the idea in the proof of the above theorem is very simple. However, we may get a better result if we work a little more on the techniques to compare solutions of a partial differential equation of parabolic type. For this purpose we are going to follow Mantegazza’s monograph [Man11]. But first recall the concept of lo- cally Lipschitz function, then we will use it in a previous result known as Hamilton’s Trick.
Definition 5.4. Let (A, d) be a metric space. A function f : A → R is defined to be Lipschitz (globally on A) if there exists a constant L > 0 such that: |f(x) − f(y)| ≤ L · d(x, y) ∀x, y ∈ A. MEAN CURVATURE FLOW 43
We will say that f is locally Lipschitz if for each x0 ∈ A there exist U0 a neighborhood of x0 and a constant L0 > 0 such that:
|f(x) − f(y)| ≤ L0 · d(x, y) ∀x, y ∈ U0. Remark 5.5.
(1) If we want to be more precise, then we say that f is (locally) L-Lipschitz, specifying the Lipschitz constant. (2) It can be shown that a Lipschitz function f : R → R is differen- tiable almost everywhere (eg, proving that a Lipschitz function is absolutely continuous). Later we will use this fact.
Lemma 5.6 (Hamilton’s Trick). Let u : M × (0,T ) → R a C1- function such that for each t0 there exist δ > 0 and a compact subset K ⊂ M \ ∂M such that for any t ∈ (t0 − δ, t0 + δ) the maximum umax(t) := maxp∈M u(p, t) is reached at least at one point of K. Then, the function umax is locally Lipschitz in (0,T ) and for each t0 ∈ (0,T ) where it is differentiable we have: du (t ) ∂u(p, t ) max 0 = 0 dt ∂t where p ∈ M \ ∂M is any interior point where u(·, t0) reaches its max- imum.
Proof. Consider t0 ∈ (0,T ). Let δ and K be as in the hypotheses of the lemma. We start by showing that u| is Lipschitz with respect to t. K×(t0−δ,t0+δ) Fix p ∈ K, then we have to deduce the existence of a constant C > 0 such that if t1 < t2 in (t0 − δ, t0 + δ) then
|u(p, t2) − u(p, t1)| ≤ C|t2 − t1|. This is essentially a consequence of the Mean Value Theorem for func- tions of class C1. Indeed, without loss of generality we can assume 1 that [t0 − δ, t0 + δ] ⊂ (0,T ). As u is C , we can apply the Mean Value Theorem to the function u(p, ·):[t0 − δ, t0 + δ] → R. Furthermore ∂u ∂t : K × [t0 − δ, t0 + δ] → R is bounded (K is compact). This im- ∂u(p,t) plies the existence of a constant C > 0 such that ∂t ≤ C for all (p, t) ∈ K × [t0 − δ, t0 + δ]. Therefore,
u(p, t2) − u(p, t1) ∂u(p, t3) |u(p, t2) − u(p, t1)| = |t2 − t1| = |t2 − t1| t2 − t1 ∂t
≤ C|t2 − t1|, 44 FRANCISCO MARTIN AND JESUS PEREZ as we wanted to prove.
Let us see that umax is locally Lipschitz in (0,T ). Take t0 in (0,T ) joint with δ and K provided by the hypotheses of the lemma. Consider 0 < < δ. Taking into account that u| is Lipschitz with K×(t0−δ,t0+δ) respect to t, we have
umax(t0 + ) = u(q, t0 + ) ≤ u(q, t0) + C ≤ umax(t0) + C, for some q ∈ K (this point q exists by hypothesis). So u (t + ) − u (t ) max 0 max 0 ≤ C. Analogously,
umax(t0) = u(p, t0) ≤ u(p, t0 + ) + C ≤ umax(t0 + ) + C, for a certain p ∈ K. Therefore u (t ) − u (t + ) max 0 max 0 ≤ C. Summarizing, we have showed that 0 < < δ,
|umax(t0) − umax(t0 + )| ≤ C|(t0 + ) − t0|. If we consider −δ < < 0, then we can prove in a similar way that:
|umax(t0) − umax(t0 + )| ≤ C|(t0 + ) − t0|.
Thus, we have got that umax is locally Lipschitz in (0,T ), which implies that it is differentiable a.e. in t ∈ (0,T ).
Finally, take a point t0 ∈ (0,T ) where umax is differentiable. From our assumptions, there exists p ∈ M \ ∂M so that umax(t0) = u(p, t0). By the Mean Value Theorem, for each 0 < < δ there exists ξ ∈ ∂u(p,ξ) (t0, t0 + ) such that u(p, t0 + ) = u(p, t0) + ∂t . Therefore ∂u(p, ξ) ∂u(p, ξ) u (t + ) ≥ u(p, t + ) = u(p, t ) + = u (t ) + , max 0 0 0 ∂t max 0 ∂t from which, as > 0, we can deduce u (t + ) − u (t ) ∂u(p, ξ) max 0 max 0 ≥ . ∂t Taking limit, as → 0 we get ∂u(p, ξ) u0 (t ) ≥ . max 0 ∂t Applying just the same argument for −δ < < 0 we conclude u (t + ) − u (t ) ∂u(p, ξ) max 0 max 0 ≤ , ∂t MEAN CURVATURE FLOW 45 and taking limit again, as → 0 what we get now is that: ∂u(p, ξ) u0 (t ) ≤ . max 0 ∂t 0 ∂u(p,ξ) Summarizing, umax(t0) = ∂t .
Corollary 5.7. Hamilton’s trick also holds if we consider umin(t) := minp∈M u(p, t) instead of umax.
Proof. We just consider v := −u. n+1 Theorem 5.8 (Comparison principle). Let ϕ: M1 × [0,T ) → R n+1 and ψ : M2 × [0,T ) → R be two hypersurfaces moving by mean curvature. Suppose that M1 is compact, M2 is complete and that ψt is proper11 for any t ∈ [0,T ).Then the distance between the hypersurfaces is non-decreasing in time.
Proof. First notice that, as ϕt(M1) is compact and ψt(M2) is properly immersed, then the distance between both hypersurfaces at time t is given by: d(t) = min |ϕ(p, t) − ψ(q, t)|. p∈M1,q∈M2 We want to apply Lemma 5.6, but the problem is that the Euclidean norm has differentiability problems at the origin. So we are going to consider M := M1 × M2 and the function u: M × (0,T ) → R given by u(p, q, t) := |ϕ(p, t) − ψ(q, t)|2 = hϕ(p, t) − ψ(q, t), ϕ(p, t) − ψ(q, t)i Notice that 2 2 umin(t) := min |ϕ(p, t) − ψ(q, t)| = d(t)) . (p,q)∈M
So, it suffices to prove that umin(t) is a non-decreasing function. Step 1. The function u satisfies the hypotheses of Corollary 5.7 (which are the same as Lemma 5.6.) Clearly u is C1 (even more, it is smooth.)
Fix t0, we want to obtain the existence of δ > 0 and a compact subset K ⊂ M \ ∂M such that for any t ∈ (t0 − δ, t0 + δ) the minimum umin(t) = min(p,q)∈M u(p, q, t) is attained at one point of K.
11 n+1 n+1 ψt : M2 → R is defined proper if for any compact subset K ⊂ R then −1 ψt (K) is also compact. 46 FRANCISCO MARTIN AND JESUS PEREZ
Take δ > 0 satisfying [t0 − δ, t0 + δ] ⊂ (0,T ). Notice that umin([t0 − δ, t0 + δ]) is bounded, because M1 is compact. So, there is a positive constant β > 0 such that
umin([t0 − δ, t0 + δ]) ⊂ [0, β].
Let us define: −1 K := u [0, β]) ∩ (M1 × M2 × [t0 − δ, t0 + δ]) . We have to prove that K is compact. Clearly, K is closed. If we prove that K is bounded, then we have done. Assume that K is not bounded. As M1 and [t0 −δ, t0 +δ] are compact, this means that π2(K) is unbounded, where π2 is the second canonical projection. Thus, we take {qn} a sequence in π2(K) such that neither the sequence itself nor any subsequence is bounded. Consider pn ∈ M1 and tn ∈ [t0 − δ, t0 + δ] such that (pn, qn, tn) ∈ K. Up to taking a subsequence, we can assume 0 that {tn} → t ∈ [t0 − δ, t0 + δ]. From the definiton of K, we have that
ψtn (qn) belongs to the set
n n+1 p o n+1 K := x ∈ R : distR (x, ϕ (M1 × [t0 − δ, t0 + δ])) ≤ β , 0 which is compact. As we are assuming that ψt0 is proper, then K = −1 0 ψt0 (K) is also compact. So, any limit point of {qn} must lie on K , which is absurd because we are assuming that this sequence is unbounded. This contradiction proves that K is compact.
Hence, Corollary 5.7 gives us the function umin is locally Lipschitz in (0,T ) and for each t0 ∈ (0,T ) where it is differentiable we have: d ∂ u (t ) = u(p , q , t ), dt min 0 ∂t 0 0 0 where (p0, q0) ∈ M1×M2 is any point where u(·, t) reaches its minimum. ∂ Step 2. Let (p , q ) be a minimum of u(·, t), then u(p , q , t ) ≥ 0. 0 0 ∂t 0 0 0 We distinguish two cases:
∂ Case 1. u(p0, q0, t0) = 0, then ∂t u(p0, q0, t0) cannot be negative, otherwise u(p0, q0, t) would be negative for t in a small interval [t0, t0 + s), which is absurd.
Case 2. u(p0, q0, t0) 6= 0. As (p0, q0) is a minimum of u, we have that ϕ(p0, t0) − ψ(q0, t0) is normal to both hypersurfaces; ϕt0 (M1) and
ψt0 (M2). In other words, the respective tangent hyperplanes are paral- lel. MEAN CURVATURE FLOW 47
This allows us to write the respective hypersurfaces (locally around p0 and q0) as graphs of functions, f and h, defined on (a part of) one of their tangent spaces, Π, in a small time interval (t0 −, t0 +). Without n+1 loss of generality, we can fix a reference in R so that {e1, . . . , en}, the canonical basis of Rn, is a base of the hyperplane Π. Assume that ϕ(p0, t0) = (0, f(0, t0)) and ψ(q0, t0)) = (0, h(0, t0)) with f(0, t0) > h(0, t0). Note that in this reference we have
ϕ(p0, t0) − ψ(q0, t0) en+1 = . |ϕ(p0, t0) − ψ(q0, t0))|
From (5.7) we have: ∂f ∇2f(Df, Df) ∂h ∇2h(Dh, Dh) (5.9) = ∆f − , = ∆h − . ∂t 1 + |Df|2 ∂t 1 + |Dh|2
On the other hand, as the function f(x, t0) − h(x, t0) has a minimum at x = 0, then its gradient vanishes at this point and the Hessian is positive semidefinite at x = 0. In particular, the Laplacian of f(x, t0)− h(x, t0) is non-negative. 0 = ∇ f(0, t0) − h(0, t0) = ∇f(0, t0) − ∇h(0, t0), 0 ≤ ∆ f(0, t0) − h(0, t0) = ∆f(0, t0) − ∆h(0, t0). Moreover, from our choice of the set of coordinates, we also have;
∇f(0, t0) = ∇h(0, t0) = 0.
Using (5.1), and Df(0, t0) = ∇f(0, t0) = 0, we have Df H(0, t ) = div (0, t ) = 0 1 + |Df|2 0 2 ∆f DifDijfDjf = − (0, t0) = ∆f(0, t0). p1 + |Df|2 (1 + |Df|2)3/2 Again, all we got for f is also valid for h. If we denote as νϕ and νψ the unit normal fields associated to the ϕ and ψ, respectively, we can write ϕ ϕ H (p0, t0)ν (p0, t0) = 0, ∆f(0, t0) ,
ψ ψ H (q0, t0)ν (q0, t0) = 0, ∆h(0, t0) ,
If we multiply by en+1, then we obtain the following equalities: ϕ ϕ ∆f(0, t0) = H (p0, t0)hν (p0, t0), en+1i,
ψ ψ ∆h(0, t0) = H (q0, t0)hν (q0, t0), en+1i. 48 FRANCISCO MARTIN AND JESUS PEREZ
Hence,
ϕ ϕ ψ ψ (5.10) hH (p0, t0)ν (p0, t0) − H (q0, t0)ν (q0, t0), en+1i =
= ∆f(0, t0) − ∆h(0, t0) ≥ 0.
∂ Now, we can compute ∂t u(p0, q0, t0), taking into account that ϕ and ψ are solutions of the MCF equation: ∂ ∂ u(p , q , t) = hϕ(p , t) − ψ(q , t), ϕ(p , t) − ψ(q , t)i = ∂t 0 0 ∂t 0 0 0 0 ∂ϕ(p , t) ∂ψ(q , t) = 2 0 − 0 , ϕ(p , t) − ψ(q , t) = ∂t ∂t 0 0 ϕ ϕ ψ ψ = 2hH (p0, t)ν (p0, t) − H (q0, t)ν (q0, t), ϕ(p0, t) − ψ(q0, t)i = ϕ ϕ ψ ψ = 2 H (p0, t)ν (p0, t) − H (q0, t)ν (q0, t), en+1 |ϕ(p0, t0) − ψ(q0, t0)|,
ϕ(p0,t0)−ψ(q0,t0) where we have used that en+1 = . |ϕ(p0,t0)−ψ(q0,t0))| Evaluating the last equality at t = t0 and taking (5.10) into account, ∂ we obtain that ∂t u(p0, q0, t0) ≥ 0. As we noted at the beginning of the proof, the second step proves that d(t) is non-decreasing, which completes the demonstration. Remark 5.9 ([MSHS14]). We would like to point out that the proper- ness assumption cannot be relaxed with that of completeness. Indeed, 3 3 take as f : M1 → R be the unit euclidean sphere and as g : M2 → R a complete minimal surface lying inside the unit ball. Such examples were first constructed by Nadirashvili [Nad96]. Obviously f and g do not have intersection points. However, under the mean curvature flow, f shrinks to a point in finite time while g remains stationary.
The following result is an immediate consequence of the previous the- orem, but it is useful to state it independently. It has a very geometric meaning and it will be used in several arguments.
n+1 Corollary 5.10 (Inclusion principle). Let ϕ: M1 × [0,T ) → R n+1 and ψ : M2 ×[0,T ) → R two compact hypersurfaces moving by mean curvature. Assume that the inner domain of ϕ(M1, 0) strictly contains ψ(M2, 0). Then, ψ(M2, t) stays strictly inside of ϕ(M1, t) for any t ∈ [0,T ).
Among other things, the above corollary has an interesting conse- quence, which has a decisive influence on the study of mean curvature MEAN CURVATURE FLOW 49
flow; the existence of singularities in finite time for the flow of a com- pact hypersurface. Corollary 5.11 (Existence of singularities in finite time). Let M n+1 be a compact hypersurface in R . If Mt represents its evolution by the mean curvature flow, then Mt must develop singularities in finite time. Moreover, if we denote this maximal time as Tmax, then we have that: 2 n+1 2 n Tmax ≤ (diamR (M)) .
Proof. As M in compact, then it can be included inside an open ball n B(p, ρ). So, M must develop a singularity before the flow of Sp collapses at the point p, otherwise we would contradict Corollary 5.10. The upper bound of Tmax is just a consequence of (5.8).
A natural question is: What can we say when M is not compact? In this case, we can have long time existence. A trivial example is the case of a complete, properly embedded minimal hypersurface M in Rn+1. Under the mean curvature flow, M remains stationary, so the flow exists for any value of t. If we are looking for non-stationary examples, then we can consider the following example: Example 5.12 (Grim hyperplanes). Consider the euclidean prod- uct M = Γ × Rn−1, where Γ is the grim reaper in R2 represented by the immersion f :(−π/2, π/2) → R2 given by f(x) = (x, 1 − log cos x).
If we move M by mean curvature we get Mt = φt(M) + t · en+1, where n+1 again {e1,..., en+1} represents the canonical basis of R and φt : M → M is a (tangent) diffeomorphism. In other words, M moves by vertical translations, that do not have singularities for any value of t. By definition, we say that M is a translating soliton in the direction of en+1. More generally, any translator in the direction of en+1 which is a Riemannian product of a planar curve and an euclidean space Rn−1 can be obtained from this example by a suitable combination of a rotation and a dilation (see [MSHS14] for further details.) Each of these translators will be called a grim hyperplane.
The ideas introduced in the proof of Theorem 5.8 can be used to get a very interesting result. Theorem 5.13 (Embeddedness principle). Let F : M ×[0,T ) → Rn+1 n+1 a MCF, with M compact. If F0 : M → R is an embedding, then Ft is an embedding for any t ∈ [0,T ). 50 FRANCISCO MARTIN AND JESUS PEREZ
Figure 3. A grim hyperplane
Proof. Let us define
A := {t ∈ [0,T ): Fs is an embedding for all s ∈ [0, t]}. Following a classical connectedness argument, we are going to prove that A is open, closed and non-empty, so A = [0,T ). Notice that A is not empty, since 0 ∈ A . Moreover A is open. Indeed, take t ∈ A and assume his set is not open. That would mean the existence of a sequence {tj}j∈N & t and sequences of points {pj}j∈N and {qj}j∈N in M such that F (pj, tj) = F (qj, tj), pj 6= qj, for all j ∈ N. As M is compact, then (up to a subsequence) we can assume that {pj}j∈N → p and {qj}j∈N → q. We have two possibilities; either p = q or p 6= q.
If p 6= q, then we would conclude that Ft is not an embedding which id contrary to t ∈ A .
If p = q, we can take U a neighborhood of p in M such that Ftj |U is an embedding for all j ∈ N. This is possible since Ftj are immersions and they converge to Ft which is an embedding. If j is large enough, then we have that pj and qj belong to U, which is absurd because F (pj, tj) = F (qj, tj). This contradiction proves the openness of A . MEAN CURVATURE FLOW 51
Note that so far we have not used that F : M × [0,T ) → Rn+1 moves by its mean curvature. Finally, we are going to prove that A is closed, or equivalently that sup A = T. We proceed again by contradiction. Suppose t0 = sup A < T. So, we consider W ⊂ M × M,
W := {(p, q) ∈ M × M : F (p, t0) = F (q, t0), p 6= q}. Then, W is a closed subset disjoint from the diagonal ∆ = {(p, p): p ∈ M}. Indeed, if (p, q) is a point in W , we only have to check that p 6= q to guarantee that (p, q) ∈ W (the other condition is closed.) Let {(p,qn)} ⊂ W such that {(p,qn)} → (p, q). If p = q, then take a chart (V, φ) around p in M. There exists n0 ∈ N such that, for any n ≥ n0, one has pn, qn ∈ V . Up to a subsequence, we have that φ(pn) − φ(qn) vn := |φ(pn) − φ(qn)| converges to some unitary vector v ∈ Sn.. Then