The Sasaki-ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature

We show that Perelman’s 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.