On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space

We prove that a proper weak solution \ ₓ \₀ ₓ < to inverse mean curvature flow in H^n, 3 n 7, is smooth and star-shaped by the time equation* T= (n-1) (sinh (r+) sinh (r-) ), equation* where r+ and r- are the geodesic out-radius and in-radius of the initial domain ₀. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in R^n due to Chow-Gulliver and uses a result of Li-Wei. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains ₀ H^n in these dimensions. From this, we also extend the asymptotically hyperbolic Riemannian Penrose inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of H^n, 3 n 7.