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Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities

Debora Impera, Michele Rimoldi, Giona Veronelli

TL;DR

The paper advances isoperimetric inequalities on complete non-compact manifolds by allowing a controlled amount of negative Ricci curvature, quantified via the elliptic Kato constant k_∞. It leverages a conformal, weighted-geometry reformulation to apply Brendle–Cabré ABP-type arguments in the presence of small negative curvature, yielding sharp isoperimetric constants in the limit as curvature decay improves. The authors establish sufficient conditions—either polynomial curvature decay with the (VC) relative volume condition or asymptotically non-negative sectional curvature with one end—that guarantee k_∞ is small enough to apply the theory, and they develop detailed Green function estimates to support these results. The work unifies gauge-theoretic, weighted-geometry, and Green-function techniques to produce sharp, asymptotically Euclidean isoperimetric bounds with explicit constants and optimal limiting behavior. These results extend the reach of ABP-based isoperimetry to broader geometric contexts and provide new tools for studying isoperimetric, Sobolev, and parabolic-type inequalities under near-nonnegative curvature.

Abstract

The ABP method for proving isoperimetric inequalities has been first employed by Cabré in $\mathbb{R}^n$, then developed by Brendle, notably in the context of non-compact Riemannian manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part $\mathrm{Ric}_-$ of the Ricci curvature in terms of its elliptic Kato constant. Indeed, the Kato constant turns out to control the non-negativity of the ($\infty$-)Bakry-Émery Ricci-tensor of a suitable conformal deformation of the manifold, and the ABP method can be implemented in this setting. Secondly, we show that the smallness of the Kato constant is ensured provided that the asymptotic volume ratio is positive and either $M$ has one end and asymptotically non-negative sectional curvature, or there is a suitable polynomial decay of $\mathrm{Ric}_{-}$, and the relative volume comparison condition known as $\textbf{(VC)}$ holds. To show this latter fact, we enhance techniques elaborated by Li-Tam and Kasue to obtain new estimates of the Green function valid on the whole manifold.

Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities

TL;DR

The paper advances isoperimetric inequalities on complete non-compact manifolds by allowing a controlled amount of negative Ricci curvature, quantified via the elliptic Kato constant k_∞. It leverages a conformal, weighted-geometry reformulation to apply Brendle–Cabré ABP-type arguments in the presence of small negative curvature, yielding sharp isoperimetric constants in the limit as curvature decay improves. The authors establish sufficient conditions—either polynomial curvature decay with the (VC) relative volume condition or asymptotically non-negative sectional curvature with one end—that guarantee k_∞ is small enough to apply the theory, and they develop detailed Green function estimates to support these results. The work unifies gauge-theoretic, weighted-geometry, and Green-function techniques to produce sharp, asymptotically Euclidean isoperimetric bounds with explicit constants and optimal limiting behavior. These results extend the reach of ABP-based isoperimetry to broader geometric contexts and provide new tools for studying isoperimetric, Sobolev, and parabolic-type inequalities under near-nonnegative curvature.

Abstract

The ABP method for proving isoperimetric inequalities has been first employed by Cabré in , then developed by Brendle, notably in the context of non-compact Riemannian manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part of the Ricci curvature in terms of its elliptic Kato constant. Indeed, the Kato constant turns out to control the non-negativity of the (-)Bakry-Émery Ricci-tensor of a suitable conformal deformation of the manifold, and the ABP method can be implemented in this setting. Secondly, we show that the smallness of the Kato constant is ensured provided that the asymptotic volume ratio is positive and either has one end and asymptotically non-negative sectional curvature, or there is a suitable polynomial decay of , and the relative volume comparison condition known as holds. To show this latter fact, we enhance techniques elaborated by Li-Tam and Kasue to obtain new estimates of the Green function valid on the whole manifold.
Paper Structure (10 sections, 23 theorems, 149 equations)

This paper contains 10 sections, 23 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Isoperimetric inequalities beyond non-negative Ricci curvature
  3. About the smallness of the Elliptic Kato constant
  4. A key step: isoperimetry in the weighted setting via ABP
  5. Structure of the paper
  6. Proof of Theorem \ref{['hbounded']}
  7. A remark about the case $|\nabla f|$ bounded
  8. Elliptic Kato constant and the proof of Theorem \ref{['isoKato']}
  9. Kato constant of manifolds with asymptotically non-negative Ricci curvature
  10. Estimates on the Green function on manifolds with asymptotically non-negative Ricci curvature

Key Result

Theorem 1.1

Let $(M^n, g)$, $n\geq3$, be a complete non-parabolic Riemannian manifold satisfying for some (hence any) $y\in M$. Assume that $k_\infty < \frac{1}{n-2}$. Let $\Omega$ be a compact domain with smooth boundary $\partial \Omega$ and let $h$ be a positive smooth function on $\Omega$. Then where $C(k_\infty,n,\beta)=n \left(1-(n-2)k_\infty\right)^{\frac{4(n-1)}{n(n-2)}}\beta^{\frac{1}{n}}$. In par

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1: cf. Lemma 2.1 in Johne-arxiv
  • proof
  • Lemma 2.2: Lemma 2.2 in brendle
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 33 more