The Space of Kähler Metrics

By Xiuxiong Chen. 2000

554 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/98476c4aab19b3a98819b891be824b75ac2c3c3a

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In Applied Mathematics

In Astronomy and Astrophysics

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Citations in the corpus

Long time existence of Minimizing Movement solutions of Calabi flow (2012, methodology): “([6] §3) For a smooth map φ0 : ∂Σ → H, there exists a unique C(1,1) solution φ of (3.”
On Einstein, Hermitian 4-Manifolds (2010, background): “In fact, every extremal Kähler metric turns out to be an absolute minimizer for the Calabi problem, and the proof of this [7] moreover implies that any Kähler metric g with Kähler class [ω] = Ω satisfies the sharp estimate”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, background): “This established the existence of C1,1 solutions of the homogeneous complex Monge–Ampère equation when ∂M is weakly pseudoconcave, and settled the long standing problem of C1,1 regularity of Chen’s weak geodesics [14] in the space of Kähler potentials [17].”
Lectures on Stability and Constant Scalar Curvature (2008, background): “The main interest lies in its formulation, otherwise it is a generalization, requiring no new estimates, of results in Chen [26].”
Bergman kernels and equilibrium measures for line bundles over projective manifolds (2007, result): “The result should also be compared to various C1,1-results for boundary value problems for complex Monge-Ampere equations on manifolds with boundary [13], [14], intimately related to the study of the geometry of the space of Kähler metrics on a Kähler manifold (see also [31], [9] for other relations”
Geometric pluripotential theory on Kähler manifolds (2019, background): “Chen [34] and our proof in case of Finsler metrics with smooth weight follows his ideas and a careful differential analysis of Orlicz norms.”
Test configurations for K-stability and geodesic rays (2006, unknown): “D G ]”
Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds (2010, background): “Related work can be found in [2], [5], [6], [9], [14], [15], [19], [20], and [21].”
Complex Monge-Ampere equations and totally real submanifolds (2009, background): “[17] X.-X. Chen, G. Tian, Geometry of Kähler metrics and foliations by holomorphic discs, Publ.”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, background): “Our main result here is Theorem 5 which extends the Chen–Tian formula for the Mabuchi functional to the general (v,w)-case.”
Liouville and Calabi-Yau type theorems for complex Hessian equations (2012, methodology): “ such an estimate by exploiting blow-up analysis. This approach was advocated by Hou-Ma-Wu in [HMW]. In the setting of complex fully nonlinear equations the blow-up analysis was introduced by Chen in [Ch1]. In essence it boils down to proving a Liouville type theorem for a suitable class of entire f”
Scalar Curvature and Projective Embeddings, I (2001, background): “Chen has proved the uniqueness result of Corollary 5 [2].”
Bifurcation currents in holomorphic dynamics on ${\bf P}^k$ (2005, background): “Endowing Hω with a suitable metric it turns out that Hω, as Riemannian manifold, is an infinite dimensional symmetric space and there is a (unique) Levi-Civita connection whose curvature is covariant constant (see [25] and [5]).”