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A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature

Adam Rudnik

Abstract

In this work we establish a sharp geometric inequality for closed hypersurfaces in complete noncompact Riemannian manifolds with asymptotically nonnegative curvature using standard comparison methods in Riemannian Geometry. These methods have been applied in a recent work by Xiaodong Wang to greatly simplify the proof of a Willmore-type inequality in complete noncompact Riemannian manifolds of nonnegative Ricci curvature, which was first proved by Agostiniani, Fagagnolo and Mazzieri.

A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature

Abstract

In this work we establish a sharp geometric inequality for closed hypersurfaces in complete noncompact Riemannian manifolds with asymptotically nonnegative curvature using standard comparison methods in Riemannian Geometry. These methods have been applied in a recent work by Xiaodong Wang to greatly simplify the proof of a Willmore-type inequality in complete noncompact Riemannian manifolds of nonnegative Ricci curvature, which was first proved by Agostiniani, Fagagnolo and Mazzieri.
Paper Structure (13 sections, 10 theorems, 127 equations)

This paper contains 13 sections, 10 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Notions and notations
  3. Elementary inequalities
  4. A sharp geometric inequality, 1 case
  5. The decay of $\lambda$
  6. A sharp geometric inequality, 2 case
  7. Application to warped product spaces
  8. The converse of Theorems \ref{['main_theo']} and \ref{['sec_main_theo']}
  9. Application to the Schwarzschild and Reissner-Nordstrom manifolds
  10. Annexes
  11. The geometry of $\Omega$
  12. A weaker Willmore-type inequality
  13. Acknowledgements

Key Result

Theorem 1.1

$[$Willmore-type inequality in asymptotic nonnegatively curved spaces$]$ Let $(\textsl{M},\textsl{g})$ be a noncompact, complete n-dimensional Riemannian manifold with asymptotically nonnegative Ricci curvature with base point $\textit{p}\space_{\textit{0}}$ and associated function $\lambda$. Let $\ where $AVR(\textsl{g})$ is the asymptotic volume ratio of $\textsl{g}$. In the following, we distin

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Example 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 5 more