Definitions

Mean curvature

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

The concept was introduced by Sophie Germain in her work on elasticity theory.

Definition

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 $kappa_1$ and one as minimal $kappa_2$, and these two curvatures $kappa_1,kappa_2$ are known as the principal curvatures of $S$.

The mean curvature at $pin S$ is the average of curvatures , hence the name:

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

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

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

More abstractly, the mean curvature is ($frac\left\{1\right\}\left\{n\right\}$ times) the trace of the second fundamental form (or equivalently, the shape operator).

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

$Hvec\left\{n\right\} = g^\left\{ij\right\}nabla_inabla_j X,$
using the Gauss-Weingarten relations, where $X\left(x,t\right)$ is a family of smoothly embedded hypersurfaces, $vec\left\{n\right\}$ a unit normal vector, and $g_\left\{ij\right\}$ 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.

Surfaces in 3D space

For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface:

$2 H = nabla cdot hat n$

where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "away" from the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated.

For the special case of a surface defined as a function of two coordinates, eg $z = S\left(x, y\right)$, and using downward pointing normal the (doubled) mean curvature expression is

begin\left\{align\right\}2 H & = nabla cdot left\left(frac\left\{nabla\left(S - z\right)\right\}
>right)
& = nabla cdot left(frac{nabla S} {sqrt{1 + (nabla S)^2}}right) & = frac{ left(1 + left(frac{partial S}{partial x}right)^2right) 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)^2right) frac{partial^2 S}{partial x^2} }{left(1 + left(frac{partial S}{partial x}right)^2 + left(frac{partial S}{partial y}right)^2right)^{3/2}}. end{align}

If the surface is additionally known to be axisymmetric with $z = S\left(r\right)$,

$2 H = frac\left\{frac\left\{partial^2 S\right\}\left\{partial r^2\right\}\right\}\left\{left\left(1 + left\left(frac\left\{partial S\right\}\left\{partial r\right\}right\right)^2right\right)^\left\{3/2\right\}\right\} + frac\left\{frac\left\{partial S\right\}\left\{partial r\right\}\right\}\left\{r left\left(1 + left\left(frac\left\{partial S\right\}\left\{partial r\right\}right\right)^2right\right)^\left\{1/2\right\}\right\}.$

Mean curvature in fluid mechanics

An alternate definition is occasionally used in fluid mechanics to avoid factors of two:

$H_f = \left(kappa_1 + kappa_2\right).$

This results in the pressure according to the Young-Laplace equation inside an equilibrium spherical droplet being surface tension times $H_f$; the two curvatures are equal to the reciprocal of the droplet's radius: $kappa_1 = kappa_2 = r^\left\{-1\right\}$.

Minimal surfaces

A minimal surface is a surface which has zero mean curvature at all points. Classic examples include the catenoid, helicoid and Enneper surface. Recent discoveries include Costa's minimal surface and the Gyroid.

An extension of the idea of a minimal surface are surfaces of constant mean curvature.