Quantitative Rigidity Using Colding's Monotonicity Formulas for Ricci Curvature

TL;DR

This paper explores how pinching of Colding's monotone functionals provides quantitative control over manifold splitting and cone approximation, extending Green function estimates to geometric structure analysis.

Contribution

It introduces a method to quantitatively control manifold splitting and cone approximation using pinched Colding's monotone functionals derived from Green functions.

Findings

Pinching of monotone functionals yields quantitative splitting estimates.

Constructs $k$-splitting functions with regularity controlled by pinching.

Pinching at independent points bounds the distance to Euclidean cone models.

Abstract

In \cite{Colding}, Colding proved monotonicity formulas for the Green function on manifolds with nonnegative Ricci curvature. Inspired by the sharp estimates relating the pinching of monotone quantities to the splitting function in \cite{cjn}, in this paper we investigate quantitative control obtained from pinching of Colding's monotone functionals. From the Green functions with poles at $(k+1)$-many independent points, $k$-splitting functions are constructed with regularity quantitatively controlled by the pinching. Moreover, the pinching at these independent points controls the distance to the nearest cone of the form $\mathbb{R}^k \times C(X)$.