Locally constrained flows and Michael–Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$

Abstract

Brendle [ Sobolev inequalities in manifolds with nonnegative curvature , Comm. Pure Appl. Math. 76 (2022), no. 9, 2192–2218] successfully establishes the sharp Michael–Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature

via the Alexandrov–Bakelman–Pucci method. Nevertheless, this result cannot be extended to the hyperbolic space

H n + 1 ${\mathbb{H}}^{n+1}$

( K = − 1 ) $\left(\mathcal{K}=-1\right)$

, as a geodesic sphere provides a counterexample. In the present paper, we propose two conjectures concerning the hyperbolic version of the sharp Michael–Simon type inequality for k -th mean curvatures. However, the proof method posed by S. Brendle is not suitable for verifying the validity of these conjectures. This paper aims to utilize the curvature flow argument to prove the two conjectures for hypersurfaces with weaker convexity conditions. For k = 1, we first investigate a new locally constrained mean curvature flow in

H n + 1 ${\mathbb{H}}^{n+1}$

and prove its longtime existence and exponential convergence. Then, the sharp Michael–Simon type inequality for the mean curvature of starshaped hypersurfaces in

H n + 1 ${\mathbb{H}}^{n+1}$

is confirmed through the flow introduced in this paper. For k ≥ 2, the sharp Michael–Simon inequality for k -th mean curvatures of starshaped, strictly k -convex hypersurfaces in

H n + 1 ${\mathbb{H}}^{n+1}$

is proven using the locally constrained inverse curvature flow introduced by Scheuer and Xia [ Locally constrained inverse curvature flows , Trans. Amer. Math. Soc. 372 (2019), no. 10, 6771–6803].