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Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature

Carlo Morpurgo, Stefano Nardulli, Liuyu Qin

TL;DR

The authors prove a sharp Sobolev embedding on complete noncompact manifolds with bounded Ricci curvature and positive injectivity radius: for all $1\le p<n$ and $q=\frac{np}{n-p}$ there exists $B\ge0$ such that $\|u\|_q \le K(n,p)\|\nabla u\|_p + B\|u\|_p$ for every $u\in W^{1,p}(M)$. The core idea is to reduce the global inequality to a small-volume setting and to derive the small-volume estimate from a first-order expansion of the isoperimetric profile $I_M(v)$ at small $v$, which itself follows from a local uniform Sobolev inequality for $W^{1,1}$ with small diameter. A first-order isoperimetric expansion is established under the geometric bounds, ensuring the needed control of $I_M(v)$ for small volumes. The approach blends local geometric analysis (harmonic coordinates, harmonic radius) with a global isoperimetric and rearrangement framework to obtain the sharp constant $K(n,p)$ and a finite additive term, with potential implications for AB-program-type rigidity questions on noncompact spaces.

Abstract

Given a smooth, complete Riemannian manifold $M$ with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of $W^{1,p}(M)$ into $L^{\frac{np}{n-p}}(M)$, when $1\le p< n$. We will first reduce the inequality to functions having support with small enough volume. In turn, we will show that the inequality for small volumes is implied by a first order uniform asymptotic expansion of the isoperimetric profile for $M$, for small volumes. We will then show that such an expansion follows from a local, uniform Sobolev inequality for functions in $W^{1,1}$, having support with small enough diameter.

Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature

TL;DR

The authors prove a sharp Sobolev embedding on complete noncompact manifolds with bounded Ricci curvature and positive injectivity radius: for all and there exists such that for every . The core idea is to reduce the global inequality to a small-volume setting and to derive the small-volume estimate from a first-order expansion of the isoperimetric profile at small , which itself follows from a local uniform Sobolev inequality for with small diameter. A first-order isoperimetric expansion is established under the geometric bounds, ensuring the needed control of for small volumes. The approach blends local geometric analysis (harmonic coordinates, harmonic radius) with a global isoperimetric and rearrangement framework to obtain the sharp constant and a finite additive term, with potential implications for AB-program-type rigidity questions on noncompact spaces.

Abstract

Given a smooth, complete Riemannian manifold with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of into , when . We will first reduce the inequality to functions having support with small enough volume. In turn, we will show that the inequality for small volumes is implied by a first order uniform asymptotic expansion of the isoperimetric profile for , for small volumes. We will then show that such an expansion follows from a local, uniform Sobolev inequality for functions in , having support with small enough diameter.
Paper Structure (8 sections, 9 theorems, 209 equations, 1 table)

This paper contains 8 sections, 9 theorems, 209 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev inequalities: from small volumes to global
  4. From isoperimetric profile expansions to Sobolev inequalities for small volumes
  5. A local Sobolev inequality for $W^{1,1}$
  6. A first order isoperimetric profile expansion
  7. Proof of Theorem \ref{['thm']}
  8. Appendix

Key Result

Theorem 1

On a complete, smooth Riemannian $n-$dimensional manifold $(M,g)$, suppose that for some $K\ge0.$ Then, there exists $B$ such that for any $p\in[1,n)$ where $K(n,p)$ denotes the best constant in the Sobolev embedding $W^{1,p}(\mathbb{R}^n)\hookrightarrow L^q(\mathbb{R}^n)$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 4
  • Remark 3
  • Corollary 1
  • Remark 4
  • ...and 7 more