  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.
