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".
Existence and uniqueness
Let be a compact smooth manifold, let be a complete smooth Riemannian manifold, and let be a smooth immersion. Then there is a positive number , which could be infinite, and a map with the following properties:
- is a smooth immersion for any
- as one has in
- for any , the derivative of the curve at is equal to the mean curvature vector of at .
- if is any other map with the four properties above, then and for any
Necessarily, the restriction of to is .
One refers to as the (maximally extended) mean curvature flow with initial data .
- If is the Euclidean space , where denotes the dimension of , then is necessarily finite. If the second fundamental form of the 'initial immersion' is strictly positive, then the second fundamental form of the immersion is also strictly positive for every , and furthermore if one choose the function such that the volume of the Riemannian manifold is independent of , then as the immersions smoothly converge to an immersion whose image in is a round sphere.
Note that if and is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map is a diffeomorphism, and so one knows from the start that is diffeomorphic to and, from elementary differential topology, that all immersions considered above are embeddings.
Gage and Hamilton extended Huisken's result to the case . Matthew Grayson (1987) showed that if 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. In summary:
- If is a smooth embedding, then consider the mean curvature flow with initial data . Then is a smooth embedding for every and there exists such that has positive (extrinsic) curvature for every . If one selects the function as in Huisken's result, then as the embeddings converge smoothly to an embedding whose image is a round circle.
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.
- Gage, M.; Hamilton, R.S. (1986). "The heat equation shrinking convex plane curves". J. Differential Geom. 23 (1): 69–96.
- Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature". J. Differential Geometry. 17 (2): 255–306.
- Huisken, Gerhard (1984). "Flow by mean curvature of convex surfaces into spheres". J. Differential Geom. 20 (1): 237–266.
- Grayson, Matthew A. (1987). "The heat equation shrinks embedded plane curves to round points". J. Differential Geom. 26 (2): 285–314.
- 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, doi:10.1007/978-0-8176-8210-1, ISBN 0-8176-3243-3, MR 2024995.
- Mantegazza, Carlo (2011), Lecture Notes on Mean Curvature Flow, Progress in Mathematics, 290, Basel: Birkhäuser/Springer, doi:10.1007/978-3-0348-0145-4, ISBN 978-3-0348-0144-7, MR 2815949.
- 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.