Related papers: Extremal metrics on toric surfaces
The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.
In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.
In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified…
This paper consists of real and complex affine techniques for studying the Abreu equation on toric surfaces. In particular, an interior estimate for Ricci tensor is given.
We prove that an extremal metric on a polarised smooth complex projective variety exists if it is $\mathbb{G}$-uniformly $K$-stable relative to the extremal torus over models, extending a result due to Chi Li for constant scalar curvature…
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\"ahler-Einstein and…
We give some detailed numerical information about extremal metrics on four different toric surfaces. These are sample of many other cases which can be treated using a computer programme outlined in the paper.
We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.
In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we…
We prove that the existence of extremal metrics implies asymptotically relative Chow stability. An application of this is the uniqueness, up to automorphisms, of extremal metrics in any polarization.
This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we…
We investigate the relationship between stability and the existence of extremal K\"ahler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a…
We show that if a compact K\"ahler manifold admits a weighted extremal metric for the action of a torus, so too does its blowup at a relatively stable point that is fixed by both the torus action and the extremal field. This generalises…
The second author has shown that existence of extremal K\"ahler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to…
Conformally K\"{a}hler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in K\"{a}hler geometry. We introduce uniform K-stability for toric K\"{a}hler manifolds, and show that uniform K-stability is…
We develop a general theory for the existence of extremal K\"ahler metrics of Poincar\'e type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we…
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…
We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar…
We construct extremal metrics on the total space of certain destabilising test configurations for strictly semistable K\"ahler manifolds. This produces infinitely many new examples of manifolds admitting extremal K\"ahler metrics. It also…
In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic…