minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young–Laplace equation
.
Definition
Let be a point on the surface inside the three dimensional Euclidean space R3. Each plane through containing the normal line to cuts in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle (always containing the normal line) that curvature can vary. The
The sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is weakened to "immersed surface".[3]
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 "towards" 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. Mean Curvature may also be calculated
where I and II denote first and second quadratic form matrices, respectively.
If is a parametrization of the surface and are two linearly independent vectors in parameter space then the mean curvature can be written in terms of the first and second fundamental forms as
For the special case of a surface defined as a function of two coordinates, e.g. , and using the upward pointing normal the (doubled) mean curvature expression is
In particular at a point where , the mean curvature is half the trace of the Hessian matrix of .
If the surface is additionally known to be
axisymmetric
with ,
where comes from the derivative of .
Implicit form of mean curvature
The mean curvature of a surface specified by an equation can be calculated by using the gradient and the Hessian matrix
Another form is as the divergence of the unit normal. A unit normal is given by and the mean curvature is
Mean curvature in fluid mechanics
An alternate definition is occasionally used in fluid mechanics to avoid factors of two:
.
This results in the pressure according to the Young–Laplace equation inside an equilibrium spherical droplet being surface tension times ; the two curvatures are equal to the reciprocal of the droplet's radius
An extension of the idea of a minimal surface are surfaces of constant mean curvature. The surfaces of unit constant mean curvature in hyperbolic space are called Bryant surfaces.[7]