On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian $5$-manifold

TL;DR

This paper proves the Hamilton-Tian conjecture for certain 5-dimensional compact transverse Fano Sasakian manifolds, establishing conditions under which Sasaki-Ricci solitons are Sasaki-Einstein.

Contribution

It confirms the Hamilton-Tian conjecture for compact transverse Fano Sasakian 5-manifolds and derives related compactness theorems for Sasaki-Ricci solitons.

Findings

Confirmed the Hamilton-Tian conjecture in the specified setting.

Derived the compactness theorem for Sasaki-Ricci solitons.

Showed Sasaki-Einstein metrics under transverse K-stability.

Abstract

In this paper, we first confirm the Hamilton-Tian conjecture for the Sasaki-Ricci flow in a compact transverse Fano quasi-regular Sasakian $5$-manifold with klt foliation singularities. Secondly, we derive the compactness theorem of Sasaki-Ricci solitons on transverse Fano quasi-regular Sasakian $5$-manifolds. Then,by the second Sasakian structure theorem, we confirm the Hamilton-Tian conjecture for a compact transverse Fano Sasakian $5$-manifold. With its applications, we show that the gradient Sasaki-Ricci soliton orbifold metric on a compact Sasakian $5$-manifold is Sasaki-Einstein if $M$ is transverse $K$-stable.