Ricci curvature and monotonicity for harmonic functions

TL;DR

This paper extends monotonicity formulas for harmonic functions on manifolds with nonnegative Ricci curvature, showing that Green's function level sets become asymptotically umbilic, with implications for geometric analysis.

Contribution

It generalizes existing monotonicity formulas to broader classes of manifolds, enhancing understanding of harmonic functions in geometric contexts.

Findings

Level sets of Green's functions are asymptotically umbilic.

Monotone quantities are crucial for analysis and geometry applications.

Generalization of monotonicity formulas to manifolds with nonnegative Ricci curvature.

Abstract

In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.