Mean curvature flow
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 normal component of 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".
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).
Mean curvature flow was originally proposed as a model for the formation of grain boundaries in the annealing of pure metal.
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.
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 three-dimensional surface
The differential equation for mean-curvature flow of a surface given by is given by
with being a constant relating the curvature and the speed of the surface normal, and the mean curvature being
In the limits and , 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.
- Ecker, Klaus. "Regularity Theory for Mean Curvature Flow", Progress in nonlinear differential equations and their applications, 75, Birkhauser, Boston, 2004.
- Mantegazza, Carlo. " Lecture Notes on Mean Curvature Flow", Progress in Mathematics, 290, Birkhauser, Basel, 2011.
- Equations 3a and 3b of C. Lu, Y. Cao, and D. Mumford. "Surface Evolution under Curvature Flows", Journal of Visual Communication and Image Representation, 13, pp. 65-81, 2002.