On Bach-flat gradient shrinking Ricci solitons

TL;DR

This paper classifies complete Bach-flat gradient shrinking Ricci solitons in higher dimensions, showing they are either Einstein or locally conformally flat, generalizing known results to dimensions greater than three.

Contribution

It provides a comprehensive classification of Bach-flat gradient shrinking Ricci solitons in dimensions greater than three, extending previous understanding to higher dimensions.

Findings

4D solitons are either Einstein or locally conformally flat.

Higher-dimensional solitons are either Einstein or quotients of Gaussian or product manifolds.

The classification applies to all dimensions greater than three.

Abstract

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton $R^4$ or the round cylinder $S^3\times R$. More generally, for n>4, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton $R^n$ or the product $N^{n-1}\times R$, where $N^{n-1}$ is Einstein.