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

TL;DR

This paper extends Perelman's W-functional to Sasaki-Ricci flow, establishing bounds and preservation of curvature positivity, leading to convergence results for compact Sasakian manifolds with positive transverse holomorphic bisectional curvature.

Contribution

It generalizes Perelman's results from Kähler to Sasakian geometry, proving bounds and curvature preservation under Sasaki-Ricci flow.

Findings

Uniform bounds on scalar curvature, diameter, and Ricci potential along the flow.

Positivity of transverse holomorphic bisectional curvature is preserved.

Flow converges to a Sasaki-Ricci soliton under certain curvature conditions.

Abstract

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 the scalar curvature, the diameter and a uniform $C^1$ bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler-Ricci flow to the Sasakian setting. We also show that the positivity of transverse of holomorphic bisectional curvature is preserved along the flow, using the methods and the results proved by Bando and Mok in Kahler setting. In particular, we show that the Sasaki-Ricci flow would converge to a Sasaki-Ricci soliton if the initial metric has nonnegative transverse holomorphic bisectional curvature.