In the field of differential geometry in mathematics, inverse mean curvature flow is an example of a geometric flow of hypersurfaces of a Riemannian manifold. 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. In general, this flow does not exist, 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.
Example: a round sphere
Consider a two-dimensional sphere of radius evolving under IMCF in 3-dimensional Euclidean space, where is the time parameter of the flow. The outward speed under the flow is the derivative,, and the mean curvature equals. Setting the speed equal to the reciprocal of the mean curvature, we have the ordinary differential equation which possesses a unique, smooth solution given by where is the radius of the sphere at time. Thus, in this case we see that a round sphere evolves under IMCF by uniformly expanding outward with an exponentially increasing radius.
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.
It was observed by Geroch, Jang, and Wald that if a closed, connected surface evolves smoothly under IMCF in a 3-manifold with nonnegativescalar 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
