Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

TL;DR

This paper proves convergence of the Sasaki-Ricci flow to unique solitons on certain Sasakian manifolds of dimension up to seven, revealing specific solitons on three-spheres and conditions for triviality in higher dimensions.

Contribution

It establishes convergence results for the Sasaki-Ricci flow on compact quasi-regular Sasakian manifolds of dimension up to seven, including classification of solitons on three-spheres.

Findings

Flow converges to unique singular orbifold Sasaki-Ricci solitons.

Only two nontrivial solitons on three-spheres with specific leaf spaces.

Trivial solitons occur if the manifold is transverse K-stable.

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 transverse Fano Sasakian (2n+1)-manifold M. When M is dimension up to seven and the space of leaves of the characteristic foliation is well-formed, we first show that any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton on the limit space which is a S^1-orbibundle over the unique singular Kaehler-Ricci soliton on a normal projective variety with codimension two orbifold singularities. Secondly, for n=1, we show that there are only two nontrivial Sasaki-Ricci solitons on a compact quasi-regular Fano Sasakian three-sphere with its leave space a teardrop-like and football-like space, respectively. For n=2,3, we show that the Sasaki-Ricci soliton is trivial one if M is transverse K-stable.