A necessary and sufficient condition for Ricci shrinkers to have positive AVR

TL;DR

This paper establishes a precise condition under which complete noncompact shrinking gradient Ricci solitons possess positive asymptotic volume ratio, linking recent and prior results in Ricci flow geometry.

Contribution

It provides a necessary and sufficient criterion for Ricci shrinkers to have positive AVR, unifying recent findings with earlier work.

Findings

Identifies a condition for positive AVR in Ricci shrinkers

Connects recent and classical results in Ricci soliton theory

Clarifies the geometric implications of Ricci shrinkers' volume growth

Abstract

In this short note we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang to give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.