Markus Wolff: On the evolution of hypersurfaces along their inverse space-time mean curvature

We introduce a new inverse curvature flow on asymptotically flat initial data sets \((M,g,K)\). In time-symmetry (\(K=0\)), Huisken—Ilmanen used a weak notion of inverse mean curvature flow to resolve the Riemannian Penrose Inequality for connected, outermost minimal surfaces. In the context of General Relativity, it thus appears natural to generalize this flow adapted to general initial data sets.

A first attempt at such a generalization was introduced by Moore, proving existence of weak solutions to inverse null mean curvature flow. Motivated by a recent construction of an asymptotic foliation of constant spacetime mean curvature surfaces by Cederbaum--Sakovich, we propose inverse spacetime mean curvature flow as another such generalization, and develop the theory of both classical and weak soltions, where the latter are defined as solutions of a comparison principle. The main result presented in this talk is the existence of weak solutions to inverse spacetime mean curvature flow in maximal initial data sets. We further study the development of jump times in the interior region. Unlike null mean curvature flow studied by Moore, inverse space-time mean curvature flow is always comparable to inverse mean curvature flow, in particular in its jumping behavior. This is joint work with Gerhard Huisken.