, the mean curvature
of a surface
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.
Let be a point on the surface . Consider all curves on passing through the point on the surface. Every such has an associated curvature given at . Of those curvatures , at least one is characterized as maximal and one as minimal , and these two curvatures are known as the principal curvatures of .
The mean curvature at is the average of curvatures , hence the name:
More generally , for a hypersurface the mean curvature is given as
More abstractly, the mean curvature is ( times) the trace of the second fundamental form (or equivalently, the shape operator).
Additionally, the mean curvature may be written in terms of the covariant derivative as
using the Gauss-Weingarten relations,
is a family of smoothly embedded hypersurfaces,
a unit normal vector, and
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 , 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:
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 , and using downward pointing normal the (doubled) mean curvature expression is