Rigidity of Bach-flat gradient Schouten solitons

TL;DR

This paper proves rigidity results for Bach-flat gradient Schouten solitons, classifying their structure under specific eigenvalue and divergence-free conditions, especially in dimensions three and higher.

Contribution

It introduces new rigidity theorems for Bach-flat Schouten solitons, extending classification results to cases with limited Ricci eigenvalues and divergence-free Bach tensor.

Findings

Complete Schouten solitons with Ricci tensor having at most two eigenvalues are rigid.

Classification of shrinking and expanding Bach-flat Schouten solitons for dimensions n ≥ 4.

Rigidity of locally conformally flat Schouten solitons for n ≥ 3.

Abstract

In this paper we show that a complete Schouten soliton whose Ricci tensor has at most two eigenvalues at each point is rigid. This allows the classification of both shrinking and expanding Bach-flat Schouten solitons for $n\geq$ 4. When $n=3$ we are able to conclude rigidity under a more general condition, namely when the Bach tensor is divergence free. These results imply rigidity of locally conformally flat Schouten solitons for $n\geq3$.