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

• Published in: Advanced nonlinear studies 2024-07, Vol.24 (3), p.720-733
• Main Authors: Cui, Jingshi, Zhao, Peibiao
• Format: Article
• Language: English
• Subjects: 35K96; locally constrained curvature flow; Michael-Simon type inequality; Primary 53E99; Secondary 52A20; th mean curvatures
• ISSN: 2169-0375; 2169-0375
• DOI: 10.1515/ans-2023-0127

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 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in ,” (Preprint)) as follows provided that is -convex and is a positive smooth function, where ′( ) = cosh . In particular, when is of constant, coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” , vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the th mean curvatures in by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in ,” (Preprint)) as below provided that is -convex and Ω is the domain enclosed by , ) = ′) , , ′( ) = cosh , , the area for a geodesic sphere of radius , and is the inverse function of . In particular, when is of constant and is odd, is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” , vol. 382, nos. 3–4, pp. 1425–1474, 2022).