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.
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 .
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:
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.
The mean curvature flow extremalizes surface area, and minimal surfaces are the critical points for the mean curvature flow; minima solve the isoperimetric problem.
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; 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.
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 ,
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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.
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