Sasaki-Einstein metrics and K-stability

TL;DR

This paper proves that a polarized affine variety admits a Ricci flat Kähler cone metric if and only if it is K-stable, extending the Yau-Tian-Donaldson conjecture to Kähler cones and Sasakian manifolds, with applications to five-spheres.

Contribution

It generalizes the Yau-Tian-Donaldson conjecture to Kähler cones and Sasakian manifolds, establishing a criterion for Ricci flat Kähler cone metrics based on K-stability.

Findings

A polarized affine variety admits a Ricci flat Kähler cone metric iff it is K-stable.

The five-sphere admits infinitely many Sasaki-Einstein metrics.

Extension of the Yau-Tian-Donaldson conjecture to Kähler cones.

Abstract

We show that a polarized affine variety admits a Ricci flat K\"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.