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.
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.
- Huisken, Gerhard (1990), "Asymptotic behavior for singularities of the mean curvature flow", Journal of Differential Geometry, 31 (1): 285–299, MR 1030675.
- Angenent, Sigurd B. (1992), "Shrinking doughnuts" (PDF), Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Progress in Nonlinear Differential Equations and their Applications, 7, Boston, MA: Birkhäuser, pp. 21–38, MR 1167827.
- Ecker, Klaus (2004), Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications, 57, Boston, MA: Birkhäuser, ISBN 0-8176-3243-3, MR 2024995, doi:10.1007/978-0-8176-8210-1.
- Mantegazza, Carlo (2011), Lecture Notes on Mean Curvature Flow, Progress in Mathematics, 290, Basel: Birkhäuser/Springer, ISBN 978-3-0348-0144-7, MR 2815949, doi:10.1007/978-3-0348-0145-4.
- Lu, Conglin; Cao, Yan; Mumford, David (2002), "Surface evolution under curvature flows", Journal of Visual Communication and Image Representation, 13 (1-2): 65–81, doi:10.1006/jvci.2001.0476. See in particular Equations 3a and 3b.