Inverse mean curvature flow

In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the Riemannian Penrose inequality, which is of interest in general relativity.

Formally, given a pseudo-Riemannian manifold (M, g) and a smooth manifold S, an inverse mean curvature flow consists of an open interval I and a smooth map F from I × S into M such that

${\displaystyle {\frac {\partial F}{\partial t}}={\frac {-\mathbf {H} }{|\mathbf {H} |^{2}}},}$

where H is the mean curvature vector of the immersion F(t, ⋅).

If g is Riemannian, if S is closed with dim(M) = dim(S) + 1, and if a given smooth immersion f of S into M has mean curvature which is nowhere zero, then there exists a unique inverse mean curvature flow whose "initial data" is f.^[1]