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 $C i$ on $S$ passing through the point $p$ on the surface. Every such $C i$ has an associated curvature $K i$ given at $p$ . Of those curvatures $K i$ , at least one is characterized as maximal $κ 1$ and one as minimal $κ 2$ , and these two curvatures $κ 1,κ 2$ comprise the principal curvatures of $S$ . The product $κ 1 κ 2$ 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 $g i j$ 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.

The sphere is the only surface of constant positive mean curvature without boundary or singularities.