On Sasaki-Ricci solitons and their deformations

TL;DR

This paper extends a key decomposition result to Sasakian geometry involving Sasaki-Ricci solitons and demonstrates their stability under deformations, broadening understanding in geometric analysis.

Contribution

It generalizes Tian and Zhu's decomposition result to Sasakian manifolds and establishes stability of generalized Sasaki-Ricci solitons through deformation techniques.

Findings

Decomposition of holomorphic vector fields extended to Sasakian setting.

Stability of generalized Sasaki-Ricci solitons shown under deformations.

Broader class of Sasaki-Ricci solitons analyzed for stability.

Abstract

We extend to the Sasakian setting a result of Tian and Zhu about the decomposition of the Lie algebra of holomorphic vector fields on a K\"ahler manifold in the presence of a K\"ahler-Ricci soliton. Furthermore we apply known deformations of Sasakian structures to a Sasaki-Ricci soliton to obtain a stability result concerning \emph{generalized} Sasaki-Ricci solitons, generalizing Li in the K\"ahler setting and also He and Song by relaxing some of their assumptions.