The Complex Monge Ampere Equation And Pluripotential Theory
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The Complex Monge Ampere Equation and Pluripotential Theory
Author | : Sławomir Kołodziej |
Publsiher | : American Mathematical Soc. |
Total Pages | : 82 |
Release | : 2005 |
Genre | : Monge-Ampère equations |
ISBN | : 9780821837634 |
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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.
The Complex Monge Amp re Equation and Pluripotential Theory
![The Complex Monge Amp re Equation and Pluripotential Theory](https://youbookinc.com/wp-content/uploads/2024/06/cover.jpg)
Author | : Sławomir Kołodziej |
Publsiher | : American Mathematical Soc. |
Total Pages | : 64 |
Release | : 2005 |
Genre | : Mathematics |
ISBN | : 1470404419 |
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This is a collection of 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. Firstly 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
Author | : Maciej Klimek |
Publsiher | : Unknown |
Total Pages | : 296 |
Release | : 1991 |
Genre | : Mathematics |
ISBN | : UCAL:B4406070 |
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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 concepts provides the focal point of this monograph, which contains an up-to-date account of the developments from the large volume of recent work in this area. A substantial proportion of the work is devoted to classical properties of subharmonic and plurisubharmonic functions, which makes the pluripotential theory available for the first time to a wide audience of analysts.
Degenerate Complex Monge Amp re Equations
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Author | : Vincent Guedj,Ahmed Zeriahi |
Publsiher | : Unknown |
Total Pages | : 472 |
Release | : 2024 |
Genre | : Electronic Book |
ISBN | : 3037191678 |
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Pluripotential Theory
Author | : Giorgio Patrizio,Zbigniew Błocki,Francois Berteloot,Jean Pierre Demailly |
Publsiher | : Springer |
Total Pages | : 328 |
Release | : 2013-05-16 |
Genre | : Mathematics |
ISBN | : 9783642364211 |
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Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. This volume brings together the lectures held at the 2011 CIME session on "pluripotential theory" in Cetraro, Italy. This CIME course focused on complex Monge-Ampére equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics. The contributions provide an extensive description of the theory and its very recent developments, starting from basic introductory materials and concluding with open questions in current research.
Complex Monge Amp re Equations and Geodesics in the Space of K hler Metrics
Author | : Vincent Guedj |
Publsiher | : Springer Science & Business Media |
Total Pages | : 315 |
Release | : 2012-01-06 |
Genre | : Mathematics |
ISBN | : 9783642236686 |
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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.
Complex Non K hler Geometry
Author | : Sławomir Dinew,Sebastien Picard,Andrei Teleman,Alberto Verjovsky |
Publsiher | : Springer Nature |
Total Pages | : 242 |
Release | : 2019-11-05 |
Genre | : Mathematics |
ISBN | : 9783030258832 |
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Collecting together the lecture notes of the CIME Summer School held in Cetraro in July 2018, the aim of the book is to introduce a vast range of techniques which are useful in the investigation of complex manifolds. The school consisted of four courses, focusing on both the construction of non-Kähler manifolds and the understanding of a possible classification of complex non-Kähler manifolds. In particular, the courses by Alberto Verjovsky and Andrei Teleman introduced tools in the theory of foliations and analytic techniques for the classification of compact complex surfaces and compact Kähler manifolds, respectively. The courses by Sebastien Picard and Sławomir Dinew focused on analytic techniques in Hermitian geometry, more precisely, on special Hermitian metrics and geometric flows, and on pluripotential theory in complex non-Kähler geometry.
Logarithmic Potentials with External Fields
Author | : Edward B. Saff,Vilmos Totik |
Publsiher | : Springer Science & Business Media |
Total Pages | : 517 |
Release | : 2013-11-11 |
Genre | : Mathematics |
ISBN | : 9783662033296 |
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In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials.