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Isoperimetric profile function comparisons with Integral Ricci curvature bounds

Jihye Lee, Fabio Ricci

Abstract

We prove comparison results for the Isoperimetric profile function in the setting of manifolds with integral bounds on the Ricci curvature. We extend previous work of Ni and Wang and Bayle and Rosales under the usual pointwise bounds for the Ricci curvature.

Isoperimetric profile function comparisons with Integral Ricci curvature bounds

Abstract

We prove comparison results for the Isoperimetric profile function in the setting of manifolds with integral bounds on the Ricci curvature. We extend previous work of Ni and Wang and Bayle and Rosales under the usual pointwise bounds for the Ricci curvature.
Paper Structure (6 sections, 10 theorems, 80 equations)

This paper contains 6 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements:
  3. Comparisons for $h_2$ isoperimetric profile
  4. Proof of Theorem \ref{['them:1.1negative']} and Theorem \ref{['them:1.1']}
  5. Relative isoperimetric profile function comparison
  6. Comparisons for $h_1$ isoperimetric profile

Key Result

Theorem 1

Let $(M^n,g)$ be a complete Riemannian manifold. For a given $k\in\mathbb{R}$ assume that $\mathrm{Ric}\geq (n-1)k$. Then for $\beta \in (0,|M|)$ where $(\mathbb{M}^n_k,g_k)$ is a complete simply connected space of constant curvature $k$. If the equality ever holds somewhere, $(M^n,g)$ must be isometric to $(\mathbb{M}^n_k,g_k)$.

Theorems & Definitions (19)

  • Theorem : Morgan and Johnson Morgan, Ni and Wang Ni
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 9 more