In the field of differential geometry in mathematics, inverse mean curvature flow (IMCF) is an example of a geometric flow of hypersurfaces of a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under IMCF if the outward normal speed at which a point on the surface moves is given by the reciprocal of the mean curvature of the surface. For example, a round sphere evolves under IMCF by expanding outward uniformly at an exponentially growing rate (see below). In general, this flow does not exist (for example, if a point on the surface has zero mean curvature), and even if it does, it generally develops singularities. Nevertheless, it has recently been an important tool in differential geometry and mathematical problems in general relativity.
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Example: a round sphere
Consider a two-dimensional sphere of radius
which possesses a unique, smooth solution given by
where
Generalization: weak IMCF
In 1997 Gerhard Huisken and T. Ilmanen showed that it makes sense to define a weak solution to IMCF. Geometrically, this means that the flow can be continued past singularities if the surface is allowed to "jump" outward at certain times.
Monotonicity of the Hawking mass
It was observed by Geroch, Jang, and Wald that if a closed, connected surface evolves smoothly under IMCF in a 3-manifold with nonnegative scalar curvature, then a certain geometric quantity associated to the surface, the Hawking mass, is non-decreasing under the flow. Amazingly, the Hawking mass is non-decreasing even under IMCF in the sense of Huisken and Ilmanen. This fact is at the heart of the geometric applications of IMCF.
Applications
In the late 1990s and early 2000s, weak IMCF has been used to