Complete gradient shrinking Ricci solitons with pinched curvature

TL;DR

This paper classifies complete gradient Ricci solitons with pinched Weyl curvature, showing they are quotients of standard spaces without requiring local conformal flatness.

Contribution

It proves a classification result for gradient Ricci solitons with pinched Weyl curvature, removing the need for local conformal flatness assumptions.

Findings

Such solitons are quotients of Euclidean space, cylinders, or spheres.

The classification holds in all dimensions without local conformal flatness.

The result extends previous classifications by relaxing curvature conditions.

Abstract

We prove that any $n$--dimensional complete gradient Ricci soliton with pinched Weyl curvature is a finite quotient of $\RR^{n}$, $\RR \times \SS^{n-1}$ or $\SS^{n}$. In particular, we do not need to assume the metric to be locally conformally flat.