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Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below

Xiaoshang Jin, Jiabin Yin

Abstract

In this paper, we establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension $n+1$ with ${\rm Ric}\geq-ng$. It extends the classic result of Argostianiani, Fogagnolo, and Mazzieri in [1] to the Riemannian manifold of negative curvature. As an application, we construct a Willmore-type inequality for closed hypersurfaces in hyperbolic space and obtain the characterization of geodesic sphere.

Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below

Abstract

In this paper, we establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension with . It extends the classic result of Argostianiani, Fogagnolo, and Mazzieri in [1] to the Riemannian manifold of negative curvature. As an application, we construct a Willmore-type inequality for closed hypersurfaces in hyperbolic space and obtain the characterization of geodesic sphere.
Paper Structure (6 sections, 8 theorems, 61 equations)

This paper contains 6 sections, 8 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. A preliminary analysis in Riemannian geometry
  3. Proof of the main theorem
  4. Rigidity
  5. A Willmore typer inequality in hyperbolic space
  6. Some examples

Key Result

Theorem 1.3

Let $(M,g)$ be a complete noncompact Riemannian manifold of $n+1$-dimension $(n\geq 2)$ with $Ric\geq -ng$ and $\Omega\subseteq M$ is a bounded open subset with smooth boundary $\partial\Omega.$ Then where $H$ is the mean curvature of $\partial\Omega$ with respect to the outer normal. Moreover, if $H>-n$ and $K=(\frac{{\rm RV}(\Omega)\cdot\omega_n}{V(\partial\Omega)})^{\frac{1}{n}}-1>-1,$ then th

Theorems & Definitions (20)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 10 more