Geometry of Kähler metrics and foliations by holomorphic discs

By Xiuxiong Chen, G. Tian, Yves Laszlo, Franccois Dahmani, Martin Olsson, D. Groves. 2005

251 citations recorded by the source corpus.

The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, we will prove the uniqueness of extremal K\"ahler metrics and give an necessary condition for existence of extremal K\"ahler metrics. Further applications will be discussed in our forthcoming papers.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/319bf6ece1baae703ffe61dd2cb2a025e37fa55a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Blowing up Kähler manifolds with constant scalar curvature, II (2005, methodology): “A fundamental result concerning Kähler constant scalar curvature metrics is their uniqueness up to automorphisms in their Kähler class, as recently proved (even for the more general class of extremal metrics) by Chen-Tian [7].”
CM Stability and the Generalized Futaki Invariant I (2006, background): “Tian [12]) Assume that ω admits a metric of csc.”
The Dirichlet problem for degenerate complex Monge-Ampere equations (2009, unknown): “3) has been provided by [11].”
On the $C^{2,α}$-regularity of the complex Monge–Ampère equation (2012, background): “Our result settles one of the regularity problems left open in the paper [12] (see also [13]).”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “This settled a regularity issue in complex geometry related to [12] (see also [19]).”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, methodology): “z) has zero discriminant. Since it is analytic in Ω, it is identically zero, a contradiction. HAMILTONIAN 2-FORMS IN KAHLER GEOMETRY, III 19¨ In the second case we apply instead the uniqueness result [10, 11] for extremal Ka¨hler metrics, as in [46]. Let Ωk be a sequence of admissible Ka¨hler classe”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, methodology): “The uniqueness part corresponding to the poly-stable case is known by Theorem 1.1 (Donaldson [19], Chen-Tian [15]).”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, background): “Let φ0, φ1 ∈ KT ω be two smooth Kähler potentials and φt the weak geodesic connecting φ0 and φ1 (see [11, 23] and the references therein for the definition of a weak geodesic).”