Rigidity of manifolds admitting stable solutions of an elliptic problem

2018/02/10 by Márcio Batista, Batista, Marcio, J. I. Santos +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1802.03614

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.

Related

• Stable solutions of elliptic equations on Riemannian manifolds
• Some elliptic PDEs on Riemannian manifolds with boundary
• Rigidity results for elliptic boundary value problems: stable solutions for quasilinear equations with Neumann or Robin boundary conditions
• Rigidity phenomena in manifolds with boundary under a lower weighted Ricci curvature bound
• Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds