Monotonicity of the p-Green Functions.

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 Munteanu-Wang [ 43 ] in the case of |$p=2$|⁠. No smoothness assumption is made on the |$p$| -Green function when |$1<p\leq 2$|⁠. Several rigidity results are also proven. [ABSTRACT FROM AUTHOR]