Stability and convergence of the Sasaki–Ricci flow

Abstract

Abstract We introduce a holomorphic sheaf ℰ ${\mathcal{E}}$ on a Sasakian manifold S and study two new notions of stability for ℰ ${\mathcal{E}}$ along the Sasaki–Ricci flow related to the ‘jumping up’ of the number of global holomorphic sections of ℰ ${\mathcal{E}}$ at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bounded in C 0 ${C^{0}}$ along the flow, and the C ∞ ${C^{\infty}}$ closure of the Sasaki structure on S under the diffeomorphism group does not contain a Sasaki structure with strictly more global holomorphic sections of ℰ ${\mathcal{E}}$ , 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 ∂ ¯ ${\bar{\partial}}$ Laplacian on global sections of ℰ ${\mathcal{E}}$ is bounded away from zero uniformly along the flow, then the Sasaki–Ricci flow converges exponentially fast to a Sasaki–Einstein metric.