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

Abstract

We prove that a proper weak solution $\{ Ω_{t} \}_{0 \leq t < \infty}$ to inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n \leq 7$, is smooth and star-shaped by the time \begin{equation*} T= (n-1) \log \left( \frac{\text{sinh} \left( r_{+} \right)}{ \text{sinh} \left( r_{-} \right)} \right), \end{equation*} where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the initial domain $Ω_{0}$. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in $\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on $\mathbb{H}^{n} \setminus \{ 0 \}$ in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains $Ω_{0} \subset \mathbb{H}^{n}$ in dimensions $3 \leq n \leq 7$. From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of $\mathbb{H}^{n}$ in these dimensions.