Sharp gradient estimates and monotonicity in positive Ricci curvature

TL;DR

This paper establishes a sharp gradient estimate for Green's functions on closed manifolds with positive Ricci curvature, linking it to monotonicity formulas and providing new geometric insights including a novel proof of Bishop's volume comparison theorem.

Contribution

It introduces a new sharp gradient estimate for Green's functions on closed manifolds with positive Ricci curvature and connects it to monotonicity formulas, extending prior open manifold results.

Findings

Sharp gradient estimate for Green's function

Monotonicity formulas related to Ricci curvature

New proof of Bishop's volume comparison in dimension four

Abstract

We prove a sharp gradient estimate for the natural Green's function of a closed manifold with positive Ricci curvature. We also show that this estimate is closely related to a family of monotonicity formulae. These results extend those previously obtained by Colding and Minicozzi for open manifolds with non-negative Ricci curvature. We further obtain several geometric applications, including a new proof of Bishop's volume comparison theorem in dimension four.