Mean curvature

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In mathematics, the mean curvature H of a surface S is a measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space, e.g. Euclidean space.

That is, let p be a point on the surface S. Consider all curves Ci on S passing through the point p on the surface. Every such Ci has an associated curvature Ki given at p. Of those curvatures Ki, at least one is characterized as maximal κ1 and one as minimal κ2, and these two curvatures κ1,κ2 comprise the principal curvatures of S. The product κ1κ2 is called the Gaussian curvature at p\in S. The corresponding average is what is known as the mean curvature at p\in S. That is, the mean curvature H is given by the equation

H = {1 \over 2} (\kappa_1 + \kappa_2).

More generally, for a hypersurface T the mean curvature is given as

H=\frac{1}{n}\sum_{i=1}^{n} \kappa_{i}.

Additionally, the mean curvature H may be written in terms of the covariant derivative \nabla as

H\vec{n} = g^{ij}\nabla_i\nabla_j X,

using the Gauss-Weingarten relations, where X(x,t) is a family of smoothly embedded hypersurfaces, \vec{n} a unit normal vector, and gij the metric tensor.

A surface is a minimal surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface S, is said to obey a heat-type equation called the mean curvature flow equation.

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