Monotonicity of the $p$-Green functions

Abstract

On a complete p-nonparabolic 3-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper p-Green function along its level sets for 1 < p < 3. This can be viewed as a generalization of the recent result by in the case of p = 2. No smoothness assumption is made on the p-Green function when 1 < p ≤ 2. Several rigidity results are also proven.