Bounds on volume growth of geodesic balls under Ricci flow

TL;DR

This paper establishes an upper bound on the volume growth of geodesic balls under Ricci flow, complementing known non-collapsing results, and deriving volume doubling without Ricci curvature bounds.

Contribution

It introduces a $$ non-inflating property for Ricci flow, providing new volume growth bounds that complement Perelman's non-collapsing theorem.

Findings

Proves an upper bound for volume ratio of geodesic balls under Ricci flow.

Derives volume doubling property without assuming Ricci curvature lower bounds.

Establishes a $$ non-inflating property as a counterpart to non-collapsing.

Abstract

We prove a so called $\kappa$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's $\kappa$ non-collapsing property for Ricci flow. These two results together imply volume doubling property for Ricci flow without assuming Ricci curvature lower bound.