Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

TL;DR

This paper improves a gap theorem for nonnegative Ricci curvature manifolds with positive volume ratio by establishing a linear relationship between curvature concentration and volume ratio, using Ricci flow techniques.

Contribution

It refines the existing gap theorem by showing curvature concentration depends linearly on the asymptotic volume ratio, employing a novel Ricci flow approach.

Findings

Curvature concentration depends linearly on volume ratio.

Constructs a Ricci flow with curvature decay faster than 1/t.

Provides explicit bounds on the size of the curvature gap.

Abstract

We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than $1/t$ curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.