The Monge-Ampère operator and geodesics in the space of Kähler potentials

Abstract

It is shown that geodesics in the space of Kähler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for Kähler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to Kähler manifolds.

25 pages, no figure, minor misprints corrected

References in corpus (3)

• The Space of Kähler metrics (II)
• Intrinsic capacities on compact Kähler manifolds
• The Futaki invariant and the Mabuchi energy of a complete intersection

Cited by in corpus (35)

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• Space of Kähler metrics (IV)--On the lower bound of the K-energy
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• Random Kähler Metrics
• Probability measures related to geodesics in the space of Kähler metrics
• Geodesics in the space of Kähler cone metrics
• The Dirichlet problem for degenerate complex Monge-Ampere equations
• The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform
• Space of Kähler metrics III--On the lower bound of the Calabi energy and geodesic distance
• Quantisation and the Hessian of Mabuchi energy
• Convergence of Bergman geodesics on CP^1
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
• Space of Kähler metrics (V)-- Kähler quantization
• Complex Monge-Ampere equations and totally real submanifolds
• Test configurations, large deviations and geodesic rays on toric varieties
• Off-diagonal decay of toric Bergman kernels
• Stability and integration over Bergman metrics
• On the regularity of geodesic rays associated to test configurations
• On Pointwise Gradient Estimates for the Complex Monge-Ampere Equation
• Curvatures of direct image sheaves of vector bundles and applications I
• Perspectives on geometric analysis
• $C^{1,1}$ regularity of geodesics of singular Kähler metrics
• Deligne pairings and the Knudsen-Mumford expansion
• Bernstein polynomials, Bergman kernels and toric Kähler varieties
• A Wess--Zumino--Witten type equation in the space of Kähler potentials in terms of Hermitian--Yang--Mills metrics
• The Gursky-Streets equations
• Regularity of the geodesic equation in the space of Sasakian metrics
• Regularity of geodesic rays and Monge-Ampere equations
• Bergman metrics and geodesics in the space of Kähler metrics on principally polarized Abelian varieties
• Test Configurations for K-Stability and Geodesic Rays
• Convergence properties of Donaldson's $T$-iterations on the Riemann sphere
• On large deviation principles and the Monge--Ampère equation (following Berman, Hultgren)