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".

Physical examples

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.

Properties

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:

• 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 $z=S(x,y)$ is given by

$\frac{\partial S}{\partial t} = 2D\ H(x,y) \sqrt{1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2}$

with $D$ being a constant relating the curvature and the speed of the surface normal, and the mean curvature being

$\begin{align} H(x,y) & = \frac{1}{2}\frac{ \left(1 + \left(\frac{\partial S}{\partial x}\right)^2\right) \frac{\partial^2 S}{\partial y^2} - 2 \frac{\partial S}{\partial x} \frac{\partial S}{\partial y} \frac{\partial^2 S}{\partial x \partial y} + \left(1 + \left(\frac{\partial S}{\partial y}\right)^2\right) \frac{\partial^2 S}{\partial x^2} }{\left(1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2\right)^{3/2}}. \end{align}$

In the limits $|\frac{\partial S}{\partial x}| \ll 1$ and $|\frac{\partial S}{\partial y}| \ll 1$, so that the surface is nearly planar with its normal nearly parallel to the z axis, this reduces to a diffusion equation

$\frac{\partial S}{\partial t} = D\ \nabla^2 S$

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;^[1] 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.^[2]

References

1. Huisken, Gerhard (1990), "Asymptotic behavior for singularities of the mean curvature flow", Journal of Differential Geometry 31 (1): 285–299, MR 1030675.
2. 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.