Rigidity of weighted manifolds via classification results for semilinear equations

TL;DR

This paper classifies positive solutions to semilinear equations on weighted manifolds with non-negative Bakry-Émery Ricci curvature, establishing rigidity results and non-existence theorems, thus linking geometric properties with solution behavior.

Contribution

It provides a classification of solutions at the Sobolev-critical exponent and demonstrates how solution existence implies manifold rigidity and trivial weights, extending previous results to infinite-dimensional curvature cases.

Findings

Classification of positive solutions at critical exponent

Rigidity results for weighted manifolds with non-negative curvature

Non-existence of solutions under sub-critical conditions

Abstract

We study model semilinear equations on complete and non-compact weighted Riemannian manifolds with non-negative Bakry-\'Emery Ricci curvature. Our main goal is to classify positive solutions of the equation at the Sobolev-critical exponent, and furthermore to prove that the existence of such solutions implies rigidity of the manifold and triviality of the weight. This is possible when the weighted manifold has non-negative finite dimensional Bakry-\'Emery Ricci curvature, and even under the weaker condition of non-negative infinite dimensional Bakry-\'Emery Ricci curvature, up to imposing some additional conditions in the latter case. To exhibit the sharpness of these additional conditions, we construct a non-trivial positive solution of the critical problem on a weighted manifold with positive infinite dimensional curvature. We also obtain a corresponding rigidity result for solutions of the Liouville equation on weighted Riemannian surfaces. Finally, we prove some non-existence theorems when the nonlinearity is sub-critical or simply under certain volume growth conditions. In particular, the latter rules out all positive solutions on shrinking gradient Ricci solitons.