Monotonicity formulas and Hessian of the Green function

TL;DR

This paper derives monotonicity formulas for nonparabolic manifolds based on a Hessian assumption of the Green function, providing explicit examples and conditions under which the assumption holds.

Contribution

It introduces a new Hessian-based assumption leading to monotonicity formulas and verifies this on specific warped product manifolds.

Findings

Monotonicity formulas derived under the Hessian assumption

Explicit examples of manifolds satisfying the assumption

Conditions involving curvature bounds ensuring the assumption holds

Abstract

Based on an assumption on the Hessian of the Green function, we derive some monotonicity formulas on nonparabolic manifolds. This assumption is satisfied on manifolds that meet certain conditions including bounds on the sectional curvature and covariant derivative of the Ricci curvature, as shown in the author's previous work \cite{P}. We also give explicit examples of warped product manifolds on which this assumption holds.