On Weak Inverse Mean Curvature Flow and Minkowski-type Inequalities in Hyperbolic Space

Abstract We prove that a proper weak solution $$\{ \Omega _{t} \}_{0 \le t < \infty }$$ { Ω t } 0 ≤ t < ∞ to inverse mean curvature flow in hyperbolic space $$\mathbb {H}^{n}$$ H n , $$3 \le n \le 7$$ 3 ≤ n ≤ 7 , is eventually smooth and star-shaped for an arbitrary initial domain $$\Omega _{0}$$ Ω 0 . In fact, this happens by the time $$\begin{aligned} T= (n-1) \log \left( \frac{\text {sinh} \left( r_{+} \right) }{ \text {sinh} \left( r_{-} \right) } \right) , \end{aligned}$$ T = ( n - 1 ) log sinh r + sinh r - , where $$r_{+}$$ r + and $$r_{-}$$ r - are the geodesic out-radius and in-radius of $$\Omega _{0}$$ Ω 0 . The approach is based on an Alexandrov reflection method for extrinsic curvature flows originally introduced by Chow-Gulliver [9]. In addition, our methods characterize expanding spheres as proper weak IMCF on $$\mathbb {H}^{n} \setminus \{ 0 \}$$ H n \ { 0 } for arbitrary n , thereby implying a result for ancient smooth solutions. As applications of the regularity theorem, we derive optimal Minkowski inequalities for arbitrary smooth domains of $$\mathbb {H}^{n}$$ H n ,