The volume growth of complete gradient shrinking Ricci solitons

TL;DR

This paper proves that complete gradient shrinking Ricci solitons have at most Euclidean volume growth, removing previous scalar curvature growth conditions, thus advancing understanding of their geometric properties.

Contribution

It establishes a sharper volume growth bound for gradient shrinking Ricci solitons without scalar curvature growth assumptions.

Findings

Gradient shrinking Ricci solitons have at most Euclidean volume growth.

Removes scalar curvature growth condition from previous results.

Enhances understanding of the geometric structure of Ricci solitons.

Abstract

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature.