Constant Scalar Curvature Metrics on Toric Surfaces

By Simon Donaldson. 2008

179 citations recorded by the source corpus.

Abstract.The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0fef8eaf743d0e1fff75588fae24aa44e82e6129

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On the constant scalar curvature Kähler metrics, general automorphism group (2018, background): “However, if one examines carefully the original papers of Donaldson [27], [28], [29] on this subject, it is already known through work there that L1 stability implies both filtrated stability and uniform stability, from which properness of K-energy follows.”
Extremal Kähler metrics on projective bundles over a curve (2009, background): “ion of this paper to study projective bundles over a curve, the theory of rigid semisimple toric bundles which we reviewed in Sect. 3 extends the theory of extremal Ka¨hler metrics on toric manifolds [21, 22, 24, 66, 74, 75] to this more general context. To recast the leading conjectures [21, 66] in”
The Kähler-Ricci flow and K-polystability (2008, background): “In the converse direction little progress has been made, except in the case of toric surfaces, where Donaldson [4] has proved Conjecture 1.”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “This was made more precise as the Yau-Tian-Donaldson conjecture (see [46]).”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “imates. W next turn to a dual version of Theorem 1.1 formulated directly on the convex body P.It concerns the “Kähler-Einstein case” when g= 1 and is motivated by the works of Abreu [1] and Donaldson [30]. First recall that a Kähler metric ωon a complex complex manifold Xsatisﬁes the KählerEinstein ”
The complex Monge–Ampère equation on compact Kähler manifolds (2010, background): “Recently, Donaldson solved the constant scalar curvature equation on toric Kähler surfaces with K-stability in a series of paper [23, 25, 26, 27, 28].”
Kahler geometry on toric manifolds, and some other manifolds with large symmetry (2008, background): “ishes if fis aﬃne linear and LA(f) >0 if fis a convex function which is not aﬃne linear, then there is an admissible symplectic potential satisfying the equation uij ij= −A. We refer to [9], [10], [11] for more information about this, particularly in the case when n= 2. 19 3 Toric Fano manifolds ”