New monotonicity formulas for Ricci curvature and applications; I

TL;DR

This paper introduces three new monotonicity formulas for manifolds with Ricci curvature bounds, linking geometric convergence rates to tangent cones and establishing connections with classical comparison theorems and gradient estimates.

Contribution

The paper presents novel monotonicity formulas for Ricci curvature manifolds, connecting convergence rates to tangent cones and relating to classical theorems and gradient estimates.

Findings

Monotonicity formulas relate to tangent cone convergence.

Bounded derivatives linked to Gromov-Hausdorff distance.

New sharp gradient estimate for Green's function.

Abstract

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov-Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop-Gromov volume comparison theorem and Perelman's celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green's function that we prove. This is parallel to that Perelman's monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li-Yau. In [CM4] we will use the monotonicity formulas we prove here to show uniqueness of certain tangent cones of Einstein manifolds and in [CM3] we will prove a number of related monotonicity formulas. Finally, there are obvious parallels between our monotonicity and the positive mass theorem of Schoen-Yau and Witten.