An alternative proof of a rigidity theorem for the sharp Sobolev constant

TL;DR

This paper offers a geometric proof of a rigidity theorem related to Sobolev inequalities on manifolds with non-negative Ricci curvature, extending the result to asymptotically non-negative curvature cases.

Contribution

It provides a new geometric proof of Ledoux and Xia's rigidity theorem and generalizes it to manifolds with asymptotically non-negative curvature.

Findings

Rigidity theorem holds for manifolds with non-negative Ricci curvature.

Extension of the theorem to asymptotically non-negative curvature manifolds.

New geometric proof technique for Sobolev inequality-related rigidity results.

Abstract

We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using the same technique we also generalize Ledoux-Xia result to complete manifolds with asymptotically non-negative curvature.