2022/05/25 by Cui, Jingshi, Zhao, Peibiao
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.12582
Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature (K ≥ 0), and the proof relies on the Alexandrov-Bakelman-Pucci method. Nevertheless, this result cannot be extended to hyperbolic space ℍn+1 (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 \citeB21 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 ℍn+1 and prove its longtime existence and exponential convergence. Then, the sharp Michael-Simon type inequality for mean curvature of starshaped hypersurfaces in ℍn+1 is confirmed through the flow (1.9). For k ≥ 2, the sharp Michael-Simon inequality for k-th mean curvatures of starshaped, strictly k-convex hypersurfaces in ℍn+1 is proven using the locally constrained inverse curvature flow (1.11) introduced by Scheuer and Xia [31].