Ebook: Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics
- Tags: Several Complex Variables and Analytic Spaces, Differential Geometry, Partial Differential Equations, Algebraic Geometry
- Series: Lecture Notes in Mathematics 2038
- Year: 2012
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
The purpose of these lecture notes is to provide an introduction to the theory of complex Monge–Ampère operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kähler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (Kähler–Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford–Taylor), Monge–Ampère foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi–Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).
Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.
The purpose of these lecture notes is to provide an introduction to the theory of complex Monge–Amp?re operators (definition, regularity issues, geometric properties of solutions, approximation) on compact K?hler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (K?hler–Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford–Taylor), Monge–Amp?re foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi–Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of K?hler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).
Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.
The purpose of these lecture notes is to provide an introduction to the theory of complex Monge–Amp?re operators (definition, regularity issues, geometric properties of solutions, approximation) on compact K?hler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (K?hler–Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford–Taylor), Monge–Amp?re foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi–Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of K?hler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).
Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.
Content:
Front Matter....Pages i-viii
Front Matter....Pages 11-11
Introduction....Pages 1-10
Dirichlet Problem in Domains of ? n ....Pages 13-32
Geometric Properties of Maximal psh Functions....Pages 33-52
Front Matter....Pages 53-53
Probabilistic Approach to Regularity....Pages 55-198
Front Matter....Pages 199-199
The Calabi–Yau Theorem....Pages 201-227
Front Matter....Pages 229-229
The Riemannian Space of K?hler Metrics....Pages 231-255
Monge–Amp?re Equations on Complex Manifolds with Boundary....Pages 257-282
Bergman Geodesics....Pages 283-302
Back Matter....Pages 303-310
The purpose of these lecture notes is to provide an introduction to the theory of complex Monge–Amp?re operators (definition, regularity issues, geometric properties of solutions, approximation) on compact K?hler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (K?hler–Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford–Taylor), Monge–Amp?re foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi–Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of K?hler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).
Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.
Content:
Front Matter....Pages i-viii
Front Matter....Pages 11-11
Introduction....Pages 1-10
Dirichlet Problem in Domains of ? n ....Pages 13-32
Geometric Properties of Maximal psh Functions....Pages 33-52
Front Matter....Pages 53-53
Probabilistic Approach to Regularity....Pages 55-198
Front Matter....Pages 199-199
The Calabi–Yau Theorem....Pages 201-227
Front Matter....Pages 229-229
The Riemannian Space of K?hler Metrics....Pages 231-255
Monge–Amp?re Equations on Complex Manifolds with Boundary....Pages 257-282
Bergman Geodesics....Pages 283-302
Back Matter....Pages 303-310
....