相关论文: Extremal metrics on toric surfaces, I

In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.

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 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 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 solve Abreu's equation with periodic right hand side, in any dimension. This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety.

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature K\"ahler surfaces. This is related to finding constant scalar curvature K\"ahler (cscK) metrics on…

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.

We give a qualitative description of extremals for Morrey's inequality. Our theory is based on exploiting the invariances of this inequality, studying the equation satisfied by extremals and the observation that extremals are optimal for a…

We give the proof of the real analyticity of the accessory parameters in Liouville field theory as a function of the position of the sources in the case in which in addition to elliptic sources, parabolic sources are present. The method is…

In this paper we characterize the extremal points of the unit ball of the Benamou--Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points…

For any closed hyperbolic Riemann surface $X$, we show that the extremal length of the Liouville current is determined solely by the topology of \(X\). This confirms a conjecture of Mart\'inez-Granado and Thurston. We also obtain an upper…

The purpose of [1] was as follows. ?We consider special sets of continuants which occur in applications. For these sets we solve the problem of finding maximal and minimal continuants. There are several methods for finding extremum such as…

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…

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…

In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.

We consider a fourth order partial differential equation in n-dimensional space introduced by Abreu in the context of K\"{a}hler metrics on toric orbifolds. Similarity solutions depending only on the radial coordinate in R^n are determined…

We study the solvability of the second boundary value problem of a class of highly singular, fully nonlinear fourth order equations of Abreu type in higher dimensions under either a smallness condition or radial symmetry.

The paper develops various estimates for solutions of a fourth order nonlinear PDE, which corresponds to prescribing the scalar curvature of a toric Kahler metric.

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit…

We apply the Le Roy-Poincar\'e continuation method to prove the real analytic dependence of the accessory parameters on the position of the sources in Liouville theory in presence of any number of elliptic sources. The treatment is easily…