Paper · Nov 2, 2022

Michael-Simon type inequalities in hyperbolic space via Brendle-Guan-Li's flows

Jingshi Cui, Peibiao Zhao

math.DGWider flowsarXiv:2211.00855v3
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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 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that is -convex and is a positive smooth function, where . In particular, when 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 -th mean curvatures in by virtue of the Brendle-Guan-Li's flow, provided that is -convex and is the domain enclosed by . In particular, when is of constant and is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei.

Comments: 14 pages

Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsGeometric inequalities

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)