Extremal metrics on toric surfaces, I
نویسنده
چکیده
The datum σ yields a measure dσ on the boundary ∂P—on each edge E we take dσ to be a constant multiple of the standard Lebesgue measure with the constant normalised so that the mass of the edge is σ(E). Equally, the datum σ specifies an affine-linear defining function λE for each edge E, i.e. the edge lies in the hyperplane λ E (0). We choose an inward-pointing normal vector v at a point of E with |iV dμ| = dσE where dμ is the fixed standard area form on R and we specify λE by the condition that ∇vλE = 1. For a continuous function f on P we set
منابع مشابه
Numerical approximations to extremal metrics on toric surfaces
This is a report on numerical work on extremal Kahler metrics on toric surfaces, using the ideas developed in [3]. These are special Riemannian metrics on manifolds of 4 real, or 2 complex dimensions. They can be regarded as solutions of a fourth order partial differential equation for a convex function on a polygon. We refer to the preceding article [4] for further background. Motivation for t...
متن کاملToric Geometry of Convex Quadrilaterals
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 Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler–Einstein ...
متن کاملA Note on the K−stability on Toric Manifolds
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 K-energy is proper in the space of G0invariant Kähler metrics for the case of toric surfaces which admit the extremal metrics. 0. Introduction Around the exist...
متن کاملMinimizing Weak Solutions for Calabi’s Extremal Metrics on Toric Manifolds
The existence of extremal metrics has been recently studied extensively on Kähler manifolds. The goal is to establish a sufficient and necessary condition for the existence of extremal metrics in the sense of Geometric Invariant Theory. There are many important works related to the necessary part ([Ti], [D1], [M1],[M2]). The sufficient part seems more difficult than the necessary part since the...
متن کاملKähler Geometry of Toric Varieties and Extremal Metrics
A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope ∆ ⊂ Rn. Recently Guillemin gave an explicit combinatorial way of constructing “toric” Kähler metrics on X, using only data on ∆. In this paper, differential geometric properties of these metrics are investigated using Guillemin’s construction. In particular, a nice combinatorial formula for the...
متن کاملJa n 20 09 Kähler Ricci Flow on Fano Surfaces ( I )
We show the properties of the blowup limits of Kähler Ricci flow solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that Kähler Ricci flow converges to a Kähler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existence of Kähler Ricci soliton metrics on toric surfaces.
متن کامل