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

We prove that a proper weak solution \ Ωₜ \0 ≤ t < ∞ to inverse mean curvature flow in Hⁿ, 3≤ n ≤ 7, is smooth and star-shaped by the time {equation*} T= (n-1) log ( {sinh ( r₊ )}{ sinh ( r₋ )} ), {equation*} where r₊ and r₋ are the geodesic out-radius and in-radius of the initial domain Ω₀. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in Rⁿ 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 Ω₀ ⊂ Hⁿ 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 Hⁿ, 3 ≤ n ≤ 7.