Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space
Jingshi Cui, Peibiao Zhao
TL;DR
This work advances sharp Michael-Simon type inequalities in hyperbolic space by developing two curvature flows under weakened convexity: a locally constrained mean curvature flow for mean curvature and a locally constrained inverse curvature flow for higher-order curvatures. The first flow yields a sharp MS inequality for mean curvature on starshaped hypersurfaces in $\mathbb{H}^{n+1}$ with exponential convergence to geodesic spheres, while the second flow, under Assumption 1.14, delivers sharp inequalities for the $k$-th mean curvatures on strictly $k$-convex hypersurfaces, with equality characterized by geodesic spheres. Together, these results extend Brendle–type inequalities to hyperbolic space and broaden applicability to weaker convexity classes through new flow techniques and monotonicity arguments.
Abstract
Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature ($\mathcal{K} \geq 0$), and the proof relies on the Alexandrov-Bakelman-Pucci method. Nevertheless, this result cannot be extended to hyperbolic space $\mathbb{H}^{n+1}$ ($\mathcal{K} = -1$), as demonstrated by Counterexample 1.7. In the present paper, we propose Conjectures 1.8 and 1.9 concerning the hyperbolic version of the sharp Michael-Simon type inequality for $k$-th mean curvatures. However, the proof method in \cite{B21} failed to verify the validity of these conjectures. Recently, the authors [12] proved Conjectures 1.8 and 1.9 only for $h$-convex hypersurfaces by means of the Brendle-Guan-Li's flow. This paper aims to utilize other types of curvature flows to prove Conjectures 1.8 and 1.9 for hypersurfaces with weaker convexity conditions. For $k = 1$, we first investigate a new locally constrained mean curvature flow (1.9) in $\mathbb{H}^{n+1}$ and prove its longtime existence and exponential convergence. Then, the sharp Michael-Simon type inequality for mean curvature of starshaped hypersurfaces in $\mathbb{H}^{n+1}$ is confirmed through the flow (1.9). For $k \geq 2$, the sharp Michael-Simon inequality for $k$-th mean curvatures of starshaped, strictly $k$-convex hypersurfaces in $\mathbb{H}^{n+1}$ is proven using the locally constrained inverse curvature flow (1.11) introduced by Scheuer and Xia [31].