On the uniqueness of Sasaki-Einstein metrics

TL;DR

This paper proves that on a compact Sasakian manifold without non-trivial Hamiltonian holomorphic vector fields, any Einstein-Sasakian metric is unique, establishing a key uniqueness result in Sasakian geometry.

Contribution

It establishes the uniqueness of Einstein-Sasakian metrics on certain compact Sasakian manifolds, under the absence of non-trivial Hamiltonian holomorphic vector fields.

Findings

Uniqueness of Einstein-Sasakian metrics under specified conditions

No non-trivial Hamiltonian holomorphic vector fields implies metric uniqueness

Provides a fundamental result in the study of Sasakian geometry

Abstract

Let $S$ be a compact Sasakian manifold which does not admit non-trivial Hamiltonian holomorphic vector fields. If there exists an Einstein-Sasakian metric on $S$, then it is unique.