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

Abstract

In the present paper, we first investigate a new locally constrained mean curvature flow (1.7) for starshaped hypersurfaces in hyperbolic space H n+1 and prove its longtime existence, exponential convergence. As an application, we establish a new sharp Michael-Simon inequality for mean curvatures inprovided that M is starshaped and f is a positive smooth function, where λ ′ (r) = cosh r. In particular, when f is of constant, (0.1) is exactly the Minkowski type inequality established by Brendle, Hung and Wang in [7] for mean convex and starshaped hypersurfaces.In the second part of this paper, we use a locally constrained inverse curvature flow (1.10) in H n+1 , which was introduced by Scheuer and Xia [30] to establish a new sharp Michael-Simon inequality for k-th mean curvatures of starshaped and strictly k-convex domain

Michael-Simon type inequalities in hyperbolic space

“…We will investigate and confirm the new geometric inequalities (1.7) and (1.7 ′ ) for h-convex hypersurfaces in this section. Proof of Theorem 1.9: First, we reduce the inequality (1.7 ′ ) in briefly by scaling (See e.g., the proof of Theorem 1.13 in [9]), and obtain the following inequality…”

“…To this day, as far as we know, the Michael-Simon type inequality in Riemannian manifolds with negative sectional curvatures is still open. A counterexample in [9] shows that the Michael-Simon inequality (1.3) does not hold in Riemannian manifolds with negative sectional curvatures. One exciting thing is that Cui and Zhao in [9] proposed a hyperbolic version of the Michael-Simon type inequality with respect to the k-th mean curvatures, which is the following conjecture for the Michael-Simon type inequality for the k-th mean curvatures inequality in hyperbolic space H n+1 .…”

“…Conjecture 1.5. [9] Let M be a compact hypersurface in H n+1 (possibly with boundary ∂M ) and let f be a positive smooth function on M . For 1 ≤ k ≤ n, there holds…”

“…Remark 1.6. Conjecture 1.5 is partially resolved after adding appropriate constraints to f (one can see [9] for details).…”

“…The conclusions in [9] are not particularly excellent in terms of applications, that is, it is difficult to extract or infer certain well-known outcomes or connections from the conclusions in [9]. They cannot, for example, contain the scenario where f is constant, which is required for the application of the Michael-Simon type inequality.…”