On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

TL;DR

This paper proves the existence of conic Sasaki-Einstein metrics on five-dimensional log Fano Sasakian manifolds by analyzing the conic Sasaki-Ricci flow and establishing convergence to singular orbifold solitons.

Contribution

It demonstrates the convergence of the conic Sasaki-Ricci flow to singular orbifold solitons and establishes conditions for the existence of Sasaki-Einstein metrics on these manifolds.

Findings

Uniform L^4-bound of transverse conic Ricci curvature established

Convergence of the flow to a unique singular orbifold soliton shown

Existence of Sasaki-Einstein metrics under transverse log K-polystability proven

Abstract

In this paper, we derive the uniform L^{4}-bound of the transverse conic Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log Fano Sasakian manifold M of dimension five and the space of leaves of the characteristic foliation is not well-formed. Then we first show that any solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold conic Sasaki-Ricci soliton on M_{infinite} which is a S^{1}-orbibundle over the unique singular conic Keahler-Ricci soliton on a log del Pezzo orbifold surface. As a consequence, there exists a Keahler-Ricci soliton orbifold metric on its leave space which is a log del Pezzo orbifold surface. Second, we show that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if M is transverse log K-polystable. In summary, we have the existence theorems of orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-regular Sasakian manifold of dimension five.