Rigidity of four-dimensional K\"ahler-Ricci solitons

TL;DR

This paper studies four-dimensional gradient shrinking Ricci solitons near K"ahler models, establishing rigidity results and conditions under which these solitons are either half-conformally flat or K"ahler, advancing understanding of their geometric structure.

Contribution

It provides new rigidity theorems for four-dimensional K"ahler-Ricci solitons, characterizing their structure under proximity conditions to K"ahler models.

Findings

Rigidity result for K"ahler-Ricci soliton on 

Conditions under which solitons are half-conformally flat or K"ahler

Characterization of solitons close to  models

Abstract

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a K\"ahler model. The first theorem could be considered as a rigidity result for the K\"ahler-Ricci soliton structure on $\mathbb{S}^2\times \mathbb{R}^2$ (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a K\"ahler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally K\"ahler.