The Sasaki–Ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature

Abstract

We show that Perelman’s \({\mathcal{W}}\) functional on Kahler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kahler–Ricci flow (the first Chern class is positive) can be generalized to Sasaki–Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectional curvature is preserved along Sasaki–Ricci flow, using Bando and Mok’s methods and results in Kahler–Ricci flow. In particular, we show that the Sasaki–Ricci flow converges to a Sasaki–Ricci soliton when the initial metric has nonnegative transverse bisectional curvature.