Relative K-stability of extremal metrics

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...

Extremal Almost-kähler Metrics

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain example...

Relative K-stability for Kähler Manifolds

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general Lp lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a ques...

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-...