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Abstract: Inverse mean curvature flow (IMCF) is a geometric flow that expands hypersurfaces by mean curvature. IMCF has many geometric applications, but a key obstacle to these is the formation of finite-time singularities. To deal with this, Huisken and Ilmanen developed a theory of weak solutions of IMCF which flow beyond these singularities and out to infinity. The flow surfaces of weak IMCF may not be smooth and may vary discontinuously in the time variable, a phenomenon known as a "jump". Furthermore, the asymptotic behavior of weak solutions is poorly understood in manifolds with negative curvature, e.g. hyperbolic space.

 

In this talk, I will show that a weak IMCF in hyperbolic space eventually becomes a classical IMCF for arbitrary closed initial data-- that is, the flow surfaces become smooth and the jumps cease after an explicit time depending on the initial hypersurface. The proof is based on an Alexandrov reflection method in the Poincare ball. Then, I will apply this regularity result to prove two Minkowski inequalities in hyperbolic space. One of these implies a Penrose-type inequality in general relativity.