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

Abstract

We prove that a proper weak solution $\{\hat{I}{\text{c◯}}_t{\}}_{0\le t<\infty }$ to inverse mean curvature flow in ${\mathbb{H}}^n$, $3\le n\le 7$, 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 $\hat{I}{\text{c◯}}_0$. 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 $\hat{I}{\text{c◯}}_0\subset {\mathbb{H}}^n$ in dimensions $3\le n\le 7$. 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.

Extended versions of the Minkowski inequalities to arbtirary domains in hyperbolic space