Mean curvature flow

Summary

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 uniquenessEdit

The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.[1][2]

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  .

Convergence theoremsEdit

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]

  • 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.[4] 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.

Physical examplesEdit

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.

PropertiesEdit

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:

Mean curvature flow of a three-dimensional surfaceEdit

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;[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]

Example: mean curvature flow of m-dimensional spheresEdit

A simple example of mean curvature flow is given by a family of concentric round hyperspheres in  . The mean curvature of an  -dimensional sphere of radius   is  .

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  .[7]

ReferencesEdit

  1. ^ Gage, M.; Hamilton, R.S. (1986). "The heat equation shrinking convex plane curves". J. Differential Geom. 23 (1): 69–96. doi:10.4310/jdg/1214439902.
  2. ^ Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry. 17 (2): 255–306. doi:10.4310/jdg/1214436922.
  3. ^ Huisken, Gerhard (1984). "Flow by mean curvature of convex surfaces into spheres". J. Differential Geom. 20 (1): 237–266. doi:10.4310/jdg/1214438998.
  4. ^ Grayson, Matthew A. (1987). "The heat equation shrinks embedded plane curves to round points". J. Differential Geom. 26 (2): 285–314. doi:10.4310/jdg/1214441371.
  5. ^ Huisken, Gerhard (1990), "Asymptotic behavior for singularities of the mean curvature flow", Journal of Differential Geometry, 31 (1): 285–299, doi:10.4310/jdg/1214444099, hdl:11858/00-001M-0000-0013-5CFD-5, MR 1030675.
  6. ^ 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, vol. 7, Boston, MA: Birkhäuser, pp. 21–38, MR 1167827.
  7. ^ Ecker, Klaus (2004), Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Boston, MA: Birkhäuser, doi:10.1007/978-0-8176-8210-1, ISBN 0-8176-3243-3, MR 2024995.
  • Ecker, Klaus (2004), Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications, vol. 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, vol. 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.