- Inverse mean curvature flow
In the field of

differential geometry inmathematics ,**inverse mean curvature flow (IMCF)**is an example of a geometric flow of hypersurfaces aRiemannian 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 themean 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 ingeneral relativity .**Example: a round sphere**Consider a two-dimensional

sphere of radius $R(t)$ evolving under IMCF in 3-dimensionalEuclidean space , where $t$ is the time parameter of the flow. (By symmetry considerations, a round sphere will remain round under this flow, so that the radius at time $t$ determines the surface at time $t$.) The outward speed under the flow is the derivative, $R\text{'}(t)$, and the mean curvature equals $frac\{2\}\{R(t)\}$. (This may be computed from thefirst variation of area formula .) Setting the speed equal to the reciprocal of the mean curvature, we have theordinary differential equation :$frac\{dR\}\{dt\}=frac\{R(t)\}\{2\},$

which possesses a unique, smooth solution given by

:$R(t)\; =\; R\_0\; e^\{t/2\},$

where $R\_0$ is the radius of the sphere at time $t=0$. 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 G. 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, theHawking 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

* prove theRiemannian Penrose inequality for the case of a single black hole (due to Huisken and Ilmanen)

* compute theYamabe invariant of three-dimensionalreal projective space (due to H. Bray and A. Neves)**See also***

mean curvature flow **References*** Huisken, G., and Ilmanen, T. "The inverse mean curvature flow and the Riemannian Penrose inequality",

*Journal of Differential Geometry*,**59**, (2001), 353-437.

* Bray, H., and Neves A. "Classification of prime 3-manifolds with Yamabe invariant greater than RP3."*Annals of Mathematics*,**159**, (2004), 407-424.

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