In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space).
For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward).
Except in special cases, the mean curvature flow develops singularities.
Under the constraint that volume enclosed is constant, this is called surface tension flow.
It is a parabolic partial differential equation, and can be interpreted as "smoothing".
The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.
be a complete smooth Riemannian manifold, and let
as the (maximally extended) mean curvature flow with initial data
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:[3] Note that if
is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map
and, from elementary differential topology, that all immersions considered above are embeddings.
Gage and Hamilton extended Huisken's result to the case
is any smooth embedding, then the mean curvature flow with initial data
eventually consists exclusively of embeddings with strictly positive curvature, at which point Gage and Hamilton's result applies.
[4] In summary: 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: For a 2D surface embedded in
, the differential equation for mean-curvature flow is given by with
, so that the surface is nearly planar with its normal nearly parallel to the z axis, this reduces to a diffusion equation 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;[5] 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.
[6] A simple example of mean curvature flow is given by a family of concentric round hyperspheres in
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
reduces to the ordinary differential equation, for an initial sphere of radius
, The solution of this ODE (obtained, e.g., by separation of variables) is which exists for