Geometry of Shrinking Sasaki-Ricci Solitons I: Fundamental Equations and Characterization of Rigidity
Shu-Cheng Chang, Fengjiang Li, Chien Lin
TL;DR
This work studies shrinking Sasaki-Ricci solitons as transverse solitons on Sasaki manifolds, deriving fundamental equations and potential estimates that yield positivity of the scalar curvature. It establishes two equivalent criteria for transverse rigidity and derives a quantization of possible constant-scalar-curvature values, then shows that in low dimensions (up to seven) any complete Sasaki-Ricci soliton with constant scalar curvature is Sasaki-Einstein. The results extend rigidity phenomena from Riemannian and Kähler settings to Sasakian geometry and provide a route toward classifying Sasaki-Ricci solitons in dimension five. The techniques combine transverse geometry, weighted Laplacians, and isoparametric-type arguments to relate curvature, potential functions, and rigidity.
Abstract
In this paper, we study some properties of Sasaki-Ricci soltions as the singularity models of Sasaki-Ricci flows. First, we establish some fundamental equations about the Sasaki-Ricci soltions which enable us to obtain the potential estimate and the positivity of the scalar curvature. Subsequently, two criteria about the transverse rigidity of Sasaki-Ricci soltions are given; and then, as an essential application, we prove that any Sasaki-Ricci soltion of low dimension with constant scalar curvature must be Sasaki-Einstein.