Stability and Convergence of the Sasaki-Ricci Flow

Abstract

We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bounded in C^{0} along the flow, and the C -infinity closure of the Sasaki structure under the diffeomorphism group does not contain a Sasaki structure with strictly more global holomorphic sections of E, then the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric. Secondly, we show that if the Futaki invariant vanishes, and the lowest positive eigenvalue of the d-bar Laplacian on global sections of E is bounded away from zero uniformly along the flow, then the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric.