# 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.

Related flows are:

## 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.

## References

• Ecker, Klaus. "Regularity Theory for Mean Curvature Flow", Progress in nonlinear differential equations and their applications, 75, Birkhauser, Boston, 2004.
• Mantegazza, Carlo. " Lecture Notes on Mean Curvature Flow", Progress in Mathematics, 290, Birkhauser, Basel, 2011.
• Equations 3a and 3b of C. Lu, Y. Cao, and D. Mumford. "Surface Evolution under Curvature Flows", Journal of Visual Communication and Image Representation, 13, pp. 65-81, 2002.