Geometric flow | Differential geometry

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 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. (Wikipedia).

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Related pages

Mean curvature flow | Variational principle | Riemannian Penrose inequality | Separation of variables | Maximum principle | Mean curvature | Closed manifold | Geometric flow | Elliptic partial differential equation | Submanifold | Geometric analysis | Ordinary differential equation | Riemannian manifold | Isometry | Euclidean space | Weak solution | Differential geometry | Pseudo-Riemannian manifold | Scalar curvature