Related papers: Extremal metrics on toric surfaces

We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.

We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.

We shall study stability conditions and Fourier-Mukai transforms on an elliptic surface. In particular we shall explain duality of elliptic surfaces by Fourier-Mukai transforms.

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

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…

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round…

We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic $u$-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special…

In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real…

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature K\"ahler metrics to the case of extremal metrics. Namely, we show that a given…

We construct new explicit toric scalar-flat K{\"a}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{\"a}hler metrics on appropriate resolutions of orbifolds. In particular, we…

In this short note, we investigate the existence of orbifold K\"ahler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold K\"ahler-Einstein metric.…

We point out how some recent developments in the theory of constant scalar curvature K\"ahler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.

A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface. We deal with results obtained in this last…

We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.

We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity…

We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains…

The aim of this paper is to solve a uniform version of the Yau-Tian-Donaldson conjecture for polarized toric manifolds. Also, we show a combinatorial sufficient condition for uniform relative K-polystability.

We study Laplace eigenvalues $\lambda_k$ on K\"ahler manifolds as functionals on the space of K\"ahler metrics with cohomologous K\"ahler forms. We introduce a natural notion of a $\lambda_k$-extremal K\"ahler metric and obtain necessary…

In an algebro-geometric way, we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic…