Rigidity results for shrinking and expanding Ricci solitons

TL;DR

This paper establishes new rigidity results for shrinking and expanding Ricci solitons, showing conditions under which these solitons must be Einstein or have specific geometric properties.

Contribution

It provides novel rigidity theorems for both compact and non-compact Ricci solitons under curvature and asymptotic conditions.

Findings

Compact shrinking Ricci solitons are Einstein under potential function bounds.

Rigidity results for non-compact solitons with pinched Ricci and scalar curvatures.

Asymptotic scalar curvature conditions imply geometric rigidity.

Abstract

In this paper, we prove some rigidity results for both shrinking and expanding Ricci solitons. First, we prove that compact shrinking Ricci solitons are Einstein if we control the maximum value of the potential function. Then, we prove some rigidity results for non-compact gradient expanding and shrinking Ricci solitons with pinched Ricci and scalar curvatures, assuming an asymptotic condition on the scalar curvature at infinity.