Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

TL;DR

This paper proves that the Sasaki-Ricci flow on certain 5-dimensional Sasakian manifolds converges to a singular Einstein metric, with specific geometric contractions occurring over time.

Contribution

It establishes uniform L^4 bounds for the transverse Ricci curvature and demonstrates convergence of the flow to a unique singular Sasaki-{\

Findings

Flow converges to a singular Sasaki-{\

Floating foliation curves contract to orbifold points

Convergence results hold for manifolds of dimension up to 5

Abstract

In this paper, we show that the uniform L^{4}-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular Sasakian (2n+1)-manifold M of general type. As an application, any solution of the normalized Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular Sasaki {\eta}-Einstein metric on the transverse canonical model M_{can} of M if n is less than or equal to 3. In particular for n equal to 2, M_{can} is a S^{1}-orbibundle over the unique Keahler-Einstein orbifold surface (Z_{can},{\omega}_{KE}) with finite point orbifold singularities. The floating foliation (-2)-curves in M will be contracted to orbifold points by the Sasaki-Ricci flow as t goes to infinite.