Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows

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 $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that $M$ is $h$-convex and $f$ is a positive smooth function, where $λ^{'}(r)=\rm{cosh}$$r$. In particular, when $f$ is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the $k$-th mean curvatures in $\mathbb{H}^{n+1}$ by virtue of the Brendle-Guan-Li's flow, provided that $M$ is $h$-convex and $Ω$ is the domain enclosed by $M$. 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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Conjecture 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.9
• Corollary 1.10
• Remark 1.12