Michael-Simon type inequalities in hyperbolic space

Abstract

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n+1 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li [4] as followsprovided that M is h-convex and f is a positive smooth function, where λ ′ (r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in [5].Further, we also establish and confirm a new sharp Michael-Simon inequality for the k-th mean curvatures in H n+1 by virtue of the Brendle-Guan-Li's flow [4] as below), the area for a geodesic sphere of radius r, and q −1 1 is the inverse function of q1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei in [17].