Relative K-stability and extremal Sasaki metrics

The Sasaki Cone and Extremal Sasakian Metrics

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest L2-norm among all Sasakian...

Extremal Kähler metrics and K-stability

Extremal Metrics and Geometric Stability

This paper grew out of my lectures at Nankai Institute as well as a few other conferences in the last few years. The purpose of this paper is to describe some of my works on extremal Kähler metrics in the last fifteen years in a more unified way. In [Ti4], [Ti2], the author developed a method of relating certain stability of underlying manifolds to Kähler-Einstein metrics. A necessary and new c...

On the uniqueness of Sasaki-Einstein metrics

Let S be a compact Sasakian manifold which does not admit nontrivial Hamiltonian holomorphic vector fields. If there exists an EinsteinSasakian metric on S, then it is unique.

Relative K-stability and Modified K-energy on Toric Manifolds

Abstract. In this paper, we discuss the relative K-stability and the modified K-energy associated to the Calabi’s extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative Kstability and the properness of modified K-energy. In particular, our results hold for toric Fano manifolds with vanishing Futaki-...