Mean curvature flow
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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). Intuitively, a family of surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean curvature of the surface. 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".
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[edit] Physical examples
The most familiar example of mean curvature flow is in the evolution of soap films.
A similar 2 dimensional phenomenon is oil drops on the surface of water, which evolve into disks (circular boundary).
[edit] 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 symplectic 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.
Related flows are:
- the surface tension flow
- the Lagrangian mean curvature flow
- the inverse mean curvature flow
[edit] References
- Ecker, Klaus. "Regularity Theory for Mean Curvature Flow", Progress in nonlinear differential equations and their applications, 75, Birkhauser, Boston, 2004.
