Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature

TL;DR

This paper establishes sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature, linking isoperimetric profiles and local Sobolev inequalities to derive the main result.

Contribution

It introduces a novel approach connecting local Sobolev inequalities and isoperimetric profile expansions to obtain sharp Sobolev inequalities on manifolds.

Findings

Derived a sharp Sobolev inequality for manifolds with bounded Ricci curvature.

Linked the inequality to the asymptotic expansion of the isoperimetric profile.

Showed that local Sobolev inequalities imply global inequalities for small volumes.

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.