Geometry of Shrinking Sasaki-Ricci Solitons I: Fundamental Equations and Characterization of 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.