Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

TL;DR

This paper investigates properties of Sasaki-Ricci solitons, establishing fundamental equations, criteria for transverse rigidity, and characterizing low-dimensional and harmonic Weyl tensor cases.

Contribution

It introduces new criteria for transverse rigidity and characterizes certain Sasaki-Ricci solitons as Sasaki-Einstein or quotients of the sphere.

Findings

Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein.

Sasaki-Ricci solitons with harmonic Weyl tensor are finite quotients of the sphere.

Fundamental equations for Sasaki-Ricci solitons enable potential estimates and positivity results.

Abstract

In this paper, we study several properties of Sasaki-Ricci solitons as singularity models of the Sasaki-Ricci flow. First, we establish several fundamental equations for Sasaki-Ricci solitons, which enable us to derive potential estimates and prove the positivity of the scalar curvature. Then we present two criteria for the transverse rigidity of Sasaki-Ricci solitons. As essential applications, we prove that any low-dimensional Sasaki-Ricci soliton with constant scalar curvature must be Sasaki-Einstein, and that any Sasaki-Ricci soliton with harmonic Weyl tensor is a finite quotient of the sphere.