The Degenerate Complex Monge Ampere Equation
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Degenerate Complex Monge Amp Re Equations
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Author : Vincent Guedj
language : en
Publisher:
Release Date :
Degenerate Complex Monge Amp Re Equations written by Vincent Guedj and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
The Degenerate Complex Monge Ampere Equation
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Author : Roberto Moriyón
language : en
Publisher:
Release Date : 1979
The Degenerate Complex Monge Ampere Equation written by Roberto Moriyón and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1979 with categories.
Complex Monge Amp Re Equations And Geodesics In The Space Of K Hler Metrics
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Author : Vincent Guedj
language : en
Publisher: Springer
Release Date : 2012-01-05
Complex Monge Amp Re Equations And Geodesics In The Space Of K Hler Metrics written by Vincent Guedj and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-05 with Mathematics categories.
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 Complex Monge Ampere Equation And Pluripotential Theory
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Author : Sławomir Kołodziej
language : en
Publisher: American Mathematical Soc.
Release Date : 2005
The Complex Monge Ampere Equation And Pluripotential Theory written by Sławomir Kołodziej and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
We collect here results on the existence and stability of weak solutions of complex Monge-Ampere equation proved by applying pluripotential theory methods and obtained in past three decades. First we set the stage introducing basic concepts and theorems of pluripotential theory. Then the Dirichlet problem for the complex Monge-Ampere equation is studied. The main goal is to give possibly detailed description of the nonnegative Borel measures which on the right hand side of the equation give rise to plurisubharmonic solutions satisfying additional requirements such as continuity, boundedness or some weaker ones. In the last part, the methods of pluripotential theory are implemented to prove the existence and stability of weak solutions of the complex Monge-Ampere equation on compact Kahler manifolds. This is a generalization of the Calabi-Yau theorem.
Pluripotential Theory
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Author : Maciej Klimek
language : en
Publisher:
Release Date : 1991
Pluripotential Theory written by Maciej Klimek and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991 with Mathematics categories.
Pluripotential theory is a recently developed non-linear complex counterpart of classical potential theory. Its main area of application is multidimensional complex analysis. The central part of the pluripotential theory is occupied by maximal plurisubharmonic functions and the generalized complex Monge-Ampere operator. The interplay between these two notions provides the focal point of this monography, which contains an account of the developments from the large volume of recent work in this area. The substantial proportion of this volume devoted to classical properties of subharmonic and plurisubharmonic functions makes the pluripotential theory available to a wide audience of analysts.
An Introduction To Extremal Kahler Metrics
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Author : Gábor Székelyhidi
language : en
Publisher: American Mathematical Soc.
Release Date : 2014-06-19
An Introduction To Extremal Kahler Metrics written by Gábor Székelyhidi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-19 with Mathematics categories.
A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result, in the setting of Kähler geometry. This book gives an introduction to the study of extremal Kähler metrics and in particular to the conjectural picture relating the existence of extremal metrics on projective manifolds to the stability of the underlying manifold in the sense of algebraic geometry. The book addresses some of the basic ideas on both the analytic and the algebraic sides of this picture. An overview is given of much of the necessary background material, such as basic Kähler geometry, moment maps, and geometric invariant theory. Beyond the basic definitions and properties of extremal metrics, several highlights of the theory are discussed at a level accessible to graduate students: Yau's theorem on the existence of Kähler-Einstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and Arezzo-Pacard's existence theorem for constant scalar curvature Kähler metrics on blow-ups.
Local Boundedness Maximum Principles And Continuity Of Solutions To Infinitely Degenerate Elliptic Equations With Rough Coefficients
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Author : Lyudmila Korobenko
language : en
Publisher: American Mathematical Soc.
Release Date : 2021-06-21
Local Boundedness Maximum Principles And Continuity Of Solutions To Infinitely Degenerate Elliptic Equations With Rough Coefficients written by Lyudmila Korobenko and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-21 with Education categories.
View the abstract: https://bookstore.ams.org/memo-269-1311/
The Bergman Kernel And Related Topics
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Author : Kengo Hirachi
language : en
Publisher: Springer Nature
Release Date : 2024-03-19
The Bergman Kernel And Related Topics written by Kengo Hirachi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-19 with Mathematics categories.
This volume consists of 15 papers contributing to the Hayama Symposium on Complex Analysis in Several Variables XXIII, which was dedicated to the 100th anniversary of the creation of the Bergman kernel. The symposium took place in Hayama and Tokyo in July 2022. Each article is closely related to the Bergman kernel, covering topics in complex analysis, differential geometry, representation theory, PDE, operator theory, and complex algebraic geometry. Specifically, some papers address the L2 extension operators from a newly opened viewpoint after solving Suita's conjecture for the logarithmic capacity. They are also continuations of quantitative solutions to the openness conjecture for the multiplier ideal sheaves. The study involves estimates for the solutions of the d-bar equations, focusing on the existence of compact Levi-flat hypersurfaces in complex manifolds. The collection also reports progress on various topics, including the existence of extremal Kähler metrics on compact manifolds, Lp variants of the Bergman kernel, Wehrl-type inequalities, homogeneous Kähler metrics on bounded homogeneous domains, asymptotics of the Bergman kernels, and harmonic Szegő kernels and operators on the Bergman spaces and Segal-Bargmann spaces. Some of the papers are written in an easily accessible way for beginners. Overall, this collection updates how a basic notion provides strong insights into the internal relationships between independently found phenomena.
Analysis Of Monge Amp Re Equations
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Author : Nam Q. Le
language : en
Publisher: American Mathematical Society
Release Date : 2024-03-07
Analysis Of Monge Amp Re Equations written by Nam Q. Le and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-07 with Mathematics categories.
This book presents a systematic analysis of the Monge–Ampère equation, the linearized Monge–Ampère equation, and their applications, with emphasis on both interior and boundary theories. Starting from scratch, it gives an extensive survey of fundamental results, essential techniques, and intriguing phenomena in the solvability, geometry, and regularity of Monge–Ampère equations. It describes in depth diverse applications arising in geometry, fluid mechanics, meteorology, economics, and the calculus of variations. The modern treatment of boundary behaviors of solutions to Monge–Ampère equations, a very important topic of the theory, is thoroughly discussed. The book synthesizes many important recent advances, including Savin's boundary localization theorem, spectral theory, and interior and boundary regularity in Sobolev and Hölder spaces with optimal assumptions. It highlights geometric aspects of the theory and connections with adjacent research areas. This self-contained book provides the necessary background and techniques in convex geometry, real analysis, and partial differential equations, presents detailed proofs of all theorems, explains subtle constructions, and includes well over a hundred exercises. It can serve as an accessible text for graduate students as well as researchers interested in this subject.