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Sharp estimates and inequalities on Riemannian manifolds with Euclidean volume growth

Luigi Fontana, Carlo Morpurgo, Liuyu Qin

TL;DR

The paper advances the analysis on complete noncompact manifolds with nonnegative Ricci curvature and Euclidean volume growth by deriving sharp, uniform heat-kernel and Green-function estimates, including large-distance asymptotics controlled by the asymptotic volume ratio AVR. Building on these estimates, it proves sharp Moser–Trudinger inequalities for the Bessel potential spaces $H^{\alpha,\frac{n}{\alpha}}(M)$ under bounded-geometry hypotheses, with exponential constants scaled by AVR and verified to be optimal. A key technical contribution is the development of Talenti-type rearrangement bounds for Green functions on EVG manifolds, enabling an Adams-type framework to extend classic Euclidean results to this noncompact setting. The results illuminate how volume growth and geometry at infinity influence sharp functional inequalities and provide tools for potential theory on EVG spaces. The work thus bridges heat-kernel/Green-function analysis with sharp nonlinear functional inequalities in a broad geometric context.

Abstract

We obtain sharp estimates for heat kernels and Green's functions on complete noncompact Riemannian manifolds with Euclidean volume growth and nonnegative Ricci curvature. We will then apply these estimates to obtain sharp Moser-Trudinger inequalities on such manifolds.

Sharp estimates and inequalities on Riemannian manifolds with Euclidean volume growth

TL;DR

The paper advances the analysis on complete noncompact manifolds with nonnegative Ricci curvature and Euclidean volume growth by deriving sharp, uniform heat-kernel and Green-function estimates, including large-distance asymptotics controlled by the asymptotic volume ratio AVR. Building on these estimates, it proves sharp Moser–Trudinger inequalities for the Bessel potential spaces under bounded-geometry hypotheses, with exponential constants scaled by AVR and verified to be optimal. A key technical contribution is the development of Talenti-type rearrangement bounds for Green functions on EVG manifolds, enabling an Adams-type framework to extend classic Euclidean results to this noncompact setting. The results illuminate how volume growth and geometry at infinity influence sharp functional inequalities and provide tools for potential theory on EVG spaces. The work thus bridges heat-kernel/Green-function analysis with sharp nonlinear functional inequalities in a broad geometric context.

Abstract

We obtain sharp estimates for heat kernels and Green's functions on complete noncompact Riemannian manifolds with Euclidean volume growth and nonnegative Ricci curvature. We will then apply these estimates to obtain sharp Moser-Trudinger inequalities on such manifolds.

Paper Structure

This paper contains 13 sections, 11 theorems, 251 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background notation
  3. Uniform heat kernel estimates
  4. Uniform global bounds.
  5. Uniform asymptotic bounds: small times, small distances, bounded geometry.
  6. Uniform asymptotic bounds: large distances from a fixed point
  7. Uniform Green functions estimates
  8. Uniform asymptotic bounds: small distances
  9. Uniform asymptotic bounds: large distances from a fixed point
  10. Uniform global rearrangement bounds: Talenti's type estimates
  11. Proofs of Theorems \ref{['main1']} and Theorem \ref{['main2']}: inequalities
  12. Proofs of Theorem \ref{['main1']} and Theorem \ref{['main2']}: sharpness
  13. Proof of \ref{['reg1']}-\ref{['reg4']}

Key Result

Theorem 1

Let $(M,g)$ be a complete, connected, noncompact Riemannian manifold with nonnegative Ricci curvature and Euclidean volume growth. For $\alpha$ even and $\alpha<n$, or for $\alpha$ odd and $\alpha\le n/2$, there exists $C>0$ such that for all $u\in H^{\alpha,\frac{n}{\alpha}}(M)$ with we have If $\alpha=1$, $\alpha=2$, $4\le \alpha<\frac{n}{2}$ and $\alpha$ even, $3\le \alpha<\frac{n}{2}+1$ and

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Lemma 1
  • Definition 1
  • Proposition 2
  • Lemma 2
  • ...and 18 more