Interior estimates for solutions of Abreu's equation

By Simon Donaldson. 2004

155 citations recorded by the source corpus.

This paper develops various estimates for solutions of a nonlinear, fourth order PDE which corresponds to prescribing the scalar curvature of a toric K?ahler metric. The results combine techniques from Riemannian geometry and from the theory of MongeAmp` ere equations.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/29e06fc98aee6b52a86ee885edaed215973590a7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Constant Scalar Curvature Metrics on Toric Surfaces (2008, methodology): “Much the same issue arose in [4], deriving estimates in the interior of the polygon.”
Extremal metrics and K‐stability (2004, unknown): “For a torus T in G with Lie algebra t, define two subalgebras of g: gT := {α ∈ g | [α, β] = 0 for all β ∈ t}; gT⊥ := {α ∈ gT | 〈α, β〉 = 0 for all β ∈ t} ⊂ gT .”
Extremal Kähler metrics on projective bundles over a curve (2009, background): “[2, 22]: (i) U is smooth and strictly convex(7) on the interior, F 0, of each face F of ∆; (ii) if F = ⋂ i Fi, where Fi is a codimension one face of ∆ on which 〈ui, z〉 + ci = 0, then, in some neighbourhood of F 0 in ∆, U is equal to 12 ∑ i(〈ui, z〉 + ci) log(〈ui, z〉 + ci) up to a smooth function.”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “This remains a challenging problem, but when dimX = 2, the following progresses have been made by Donaldson: (a) It has been shown in [47] that K-stability implies that the K-energy is bounded from below; (b) Interior a priori estimates for Abreu’s equation have been obtained in [48]; (c) A method o”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, background): “Readers are encouraged to compare the results on lower bound of Calabi energy to [18] (In particular, Theorem 1.”
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].”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, result): “Thomas [45] and G. Sz´ ekelyhidi [50, 51] independently obtained similar results for rational Kahler classes on ruled surfaces, by using a different approach relying on recent works of S. Donaldson [15, 17 , 18].”