Rigidity of manifolds admitting stable solutions of an elliptic problem

TL;DR

This paper investigates the geometric rigidity of Riemannian manifolds that admit stable solutions to elliptic problems, providing characterizations and splitting results under curvature conditions.

Contribution

It offers new characterizations of manifolds with stable solutions and establishes splitting theorems under non-negative weighted Ricci curvature.

Findings

Characterization of manifolds admitting stable solutions.

Splitting results under non-negative weighted Ricci curvature.

Data about stable solutions in the context of geometric rigidity.

Abstract

In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian manifold which admits a stable solution. Furthermore, under the non-negativity of the weighted Ricci curvature, we deduce several data about the stable solution and a splitting result for the manifold.