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Concept
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".
The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.
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 .
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.
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Curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, and arc length evolution. As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth.