Paper · May 25, 2022

Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space

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Abstract

Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature ( $K \geq 0$ ), and the proof relies on the Alexandrov-Bakelman-Pucci method. Nevertheless, this result cannot be extended to hyperbolic space $H^{n + 1}$ ( $K = - 1$ ), 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 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 $H^{n + 1}$ and prove its longtime existence and exponential convergence. Then, the sharp Michael-Simon type inequality for mean curvature of starshaped hypersurfaces in $H^{n + 1}$ is confirmed through the flow (1.9). For $k \geq 2$ , the sharp Michael-Simon inequality for $k$ -th mean curvatures of starshaped, strictly $k$ -convex hypersurfaces in $H^{n + 1}$ is proven using the locally constrained inverse curvature flow (1.11) introduced by Scheuer and Xia [31].

Comments: 22 pages

Mean curvature flow Other curvature flows Inverse mean curvature flow Volume-preserving & constrained flows Long-time existence & convergence

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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)

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Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space

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