Mean curvature flow

The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows. Let M be a compact smooth manifold, let (M',g) be a complete smooth Riemannian manifold, and let f:M\to M' be a smooth immersion. Then there is a positive number T, which could be infinite, and a map F:[0,T)\times M\to M' with the following properties: • F(0,\cdot)=f • F(t,\cdot):M\to M' is a smooth immersion for any t\in[0,T) • as t\searrow 0, one has F(t,\cdot)\to f in C^\infty • for any (t_0,p)\in(0,T)\times M, the derivative of the curve t\mapsto F(t,p) at t_0 is equal to the mean curvature vector of F(t_0,\cdot) at p. • if \widetilde{F}:[0,\widetilde{T})\times M\to M' is any other map with the four properties above, then \widetilde{T}\leq T and \widetilde{F}(t,p)=F(t,p) for any (t,p)\in [0,\widetilde{T})\times M. Necessarily, the restriction of F to (0,T)\times M is C^\infty. One refers to F as the (maximally extended) mean curvature flow with initial data f. ==Convex solutions==

Following Hamilton's epochal 1982 work on the Ricci flow, in 1984 Gerhard Huisken employed the same methods for the mean curvature flow to produce the following analogous result: • If (M',g) is the Euclidean space \mathbb{R}^{n+1}, where n\geq 2 denotes the dimension of M, then T is necessarily finite. If the second fundamental form of the 'initial immersion' f is strictly positive, then the second fundamental form of the immersion F(t,\cdot) is also strictly positive for every t\in(0,T), and furthermore if one choose the function c:(0,T)\to(0,\infty) such that the volume of the Riemannian manifold (M,(c(t)F(t,\cdot))^\ast g_{\text{Euc}}) is independent of t, then as t\nearrow T the immersions c(t)F(t,\cdot):M\to\mathbb{R}^{n+1} smoothly converge to an immersion whose image in \mathbb{R}^{n+1} is a round sphere. Note that if n\geq 2 and f:M\to\mathbb{R}^{n+1} is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map \nu:M\to S^n is a diffeomorphism, and so one knows from the start that M is diffeomorphic to S^n and, from elementary differential topology, that all immersions considered above are embeddings. Gage and Hamilton extended Huisken's result to the case n=1. Matthew Grayson (1987) showed that if f:S^1\to\mathbb{R}^2 is any smooth embedding, then the mean curvature flow with initial data f eventually consists exclusively of embeddings with strictly positive curvature, at which point Gage and Hamilton's result applies. In summary: • If f:S^1\to\mathbb{R}^2 is a smooth embedding, then consider the mean curvature flow F:[0,T)\times S^1\to\mathbb{R}^2 with initial data f. Then F(t,\cdot):S^1\to\mathbb{R}^2 is a smooth embedding for every t\in(0,T) and there exists t_0\in(0,T) such that F(t,\cdot):S^1\to\mathbb{R}^2 has positive (extrinsic) curvature for every t\in(t_0,T). If one selects the function c as in Huisken's result, then as t\nearrow T the embeddings c(t)F(t,\cdot):S^1\to\mathbb{R}^2 converge smoothly to an embedding whose image is a round circle. ==Properties==

The mean curvature flow extremalizes surface area, and minimal surfaces are the critical points for the mean curvature flow; minima solve the isoperimetric problem. For manifolds embedded in a Kähler–Einstein manifold, if the surface is a Lagrangian submanifold, the mean curvature flow is of Lagrangian type, so the surface evolves within the class of Lagrangian submanifolds. Huisken's monotonicity formula gives a monotonicity property of the convolution of a time-reversed heat kernel with a surface undergoing the mean curvature flow. Related flows are: • Curve-shortening flow, the one-dimensional case of mean curvature flow • the surface tension flow • the Lagrangian mean curvature flow • the inverse mean curvature flow ==Mean curvature flow of a 2D surface==

For a 2D surface embedded in \mathbb{R}^3 as z=S(x,y) , the differential equation for mean-curvature flow is given by :\frac{\partial S}{\partial t} = 2D\ H(x,y) \sqrt{1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2} with D being a constant relating the curvature and the speed of the surface normal, and the mean curvature being : \begin{align} H(x,y) & = \frac{1}{2}\frac{ \left(1 + \left(\frac{\partial S}{\partial x}\right)^2\right) \frac{\partial^2 S}{\partial y^2} - 2 \frac{\partial S}{\partial x} \frac{\partial S}{\partial y} \frac{\partial^2 S}{\partial x \partial y} + \left(1 + \left(\frac{\partial S}{\partial y}\right)^2\right) \frac{\partial^2 S}{\partial x^2} }{\left(1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2\right)^{3/2}}. \end{align} In the limits \left|\frac{\partial S}{\partial x}\right| \ll 1 and \left|\frac{\partial S}{\partial y}\right| \ll 1 , so that the surface is nearly planar with its normal nearly parallel to the z axis, this reduces to a diffusion equation :\frac{\partial S}{\partial t} = D\ \nabla^2 S While the conventional diffusion equation is a linear parabolic partial differential equation and does not develop singularities (when run forward in time), mean curvature flow may develop singularities because it is a nonlinear parabolic equation. In general additional constraints need to be put on a surface to prevent singularities under mean curvature flows. Every smooth convex surface collapses to a point under the mean-curvature flow, without other singularities, and converges to the shape of a sphere as it does so. For surfaces of dimension two or more this is a theorem of Gerhard Huisken; for the one-dimensional curve-shortening flow it is the Gage–Hamilton–Grayson theorem. However, there exist embedded surfaces of two or more dimensions other than the sphere that stay self-similar as they contract to a point under the mean-curvature flow, including the Angenent torus. ==Example: mean curvature flow of m-dimensional spheres==

A simple example of mean curvature flow is given by a family of concentric round hyperspheres in \mathbb{R}^{m+1}. The mean curvature of an m-dimensional sphere of radius R is H = m/R. Due to the rotational symmetry of the sphere (or in general, due to the invariance of mean curvature under isometries) the mean curvature flow equation \partial_t F = - H \nu reduces to the ordinary differential equation, for an initial sphere of radius R_0, :\begin{align} \frac{\text{d}}{\text{d}t}R(t) & = - \frac{m}{R(t)} , \\ R(0) & = R_0 . \end{align} The solution of this ODE (obtained, e.g., by separation of variables) is :R(t) = \sqrt{R_0^2 - 2 m t}, which exists for t \in (-\infty,R_0^2/2m). ==See also==