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We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow.When the basic first Chern class is positive, we prove a uniform bound on thescalar curvature, the diameter and a uniform $C^1$ bound for the transverseRicci potential along the Sasaki-Ricci flow, which generalizes Perelman'sresults Kahler-Ricci flow to the Sasakian setting. We also show that thepositivity of transverse of holomorphic bisectional curvature is preservedalong the flow, using the methods and the results proved by Bando and Mok inKahler setting. In particular, we show that the Sasaki-Ricci flow wouldconverge to a Sasaki-Ricci soliton if the initial metric has nonnegativetransverse holomorphic bisectional curvature.

The Sasaki-Ricci flow and compact Sasakian manifolds of positive transverse holomorphic bisectional curvature