Stability and Convergence of the Sasaki-Ricci Flow

TL;DR

This paper investigates conditions under which the Sasaki-Ricci flow converges exponentially to a Sasaki-Einstein metric, introducing new stability notions related to holomorphic sections and analyzing flow behavior.

Contribution

It introduces two new stability concepts for holomorphic sheaves on Sasaki manifolds and links these to exponential convergence of the Sasaki-Ricci flow.

Findings

Flow converges exponentially under bounded Mabuchi K-energy and stability conditions.

Flow converges exponentially if Futaki invariant vanishes and eigenvalues are bounded.

Provides criteria for stability and convergence in Sasaki geometry.

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.