Geometry Of Complex Monge Ampere Equations

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Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics

Author : Vincent Guedj
Publisher : Springer Science & Business Media
ISBN 13 : 3642236685
Total Pages : 315 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics by : Vincent Guedj

Download or read book Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics written by Vincent Guedj and published by Springer Science & Business Media. This book was released on 2012-01-06 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Geometry of Complex Monge-Ampere Equations

Author : Valentino Tosatti
Publisher :
ISBN 13 :
Total Pages : 312 pages
Book Rating : 4.:/5 (477 download)

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Book Synopsis Geometry of Complex Monge-Ampere Equations by : Valentino Tosatti

Download or read book Geometry of Complex Monge-Ampere Equations written by Valentino Tosatti and published by . This book was released on 2009 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Kähler-Ricci flow is studied on compact Kähler manifolds with positive first Chern class, where it reduces to a parabolic complex Monge-Ampere equation. It is shown that the flow converges to a Kähler-Einstein metric if the curvature remains bounded along the flow, and if the manifold is stable in an algebro-geometric sense. On a compact Calabi-Yau manifold there is a unique Ricci-flat Kähler metric in each Kähler cohomology class, produced by Yau solving a complex Monge-Ampere equation. The behaviour of these metrics when the class degenerates to the boundary of the Kähler cone is studied. The problem splits into two cases, according to whether the total volume goes to zero or not. On a compact symplectic four-manifold Donaldson has proposed an analog of the complex Monge-Ampère equation, the Calabi-Yau equation. If solved, it would lead to new results in symplectic topology. We solve the equation when the manifold is nonnegatively curved, and reduce the general case to bounding an integral of a scalar function.

Complex Monge-Ampère Equation and Its Applications in Complex Geometry

Author : Xiangwen Zhang
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (841 download)

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Book Synopsis Complex Monge-Ampère Equation and Its Applications in Complex Geometry by : Xiangwen Zhang

Download or read book Complex Monge-Ampère Equation and Its Applications in Complex Geometry written by Xiangwen Zhang and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The main threads of this thesis are related by the theme of the complex Monge-Ampère type equations. It consists of some analysis results from the partial differential equation aspect and several geometric consequences as applications.In the first part, we study the a priori estimates for complex Hessian type equations on Hermitian manifolds. These estimates are the key ingredients for the solvability of the corresponding equations by virtue of the continuity method. In particular, we establish the first and second order derivative estimates for complex Monge-Ampère equations which are analogous to Yau's estimates on Kãhler manifolds. In Chapter 3, we investigate the interior Schauder estimates of the solutions to complex Monge-Ampère equations. Moreover, aiming to extend such regularity results to more general geometric setting, we also establish the classical Bedford-Taylor's interior second order estimate and a local version of Calabi's third order ...

Degenerate Complex Monge-Ampère Equations

Author : Vincent Guedj
Publisher :
ISBN 13 : 9783037191675
Total Pages : 472 pages
Book Rating : 4.1/5 (916 download)

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Book Synopsis Degenerate Complex Monge-Ampère Equations by : Vincent Guedj

Download or read book Degenerate Complex Monge-Ampère Equations written by Vincent Guedj and published by . This book was released on with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Complex Monge-Ampere Equation and Application on Kähler Geometry

Author : Gang Tian
Publisher : Springer Verlag
ISBN 13 : 9783540440529
Total Pages : 15 pages
Book Rating : 4.4/5 (45 download)

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Book Synopsis Complex Monge-Ampere Equation and Application on Kähler Geometry by : Gang Tian

Download or read book Complex Monge-Ampere Equation and Application on Kähler Geometry written by Gang Tian and published by Springer Verlag. This book was released on 2005-01-01 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics

Author : Vincent Guedj
Publisher : Springer
ISBN 13 : 9783642236709
Total Pages : 310 pages
Book Rating : 4.2/5 (367 download)

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Book Synopsis Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics by : Vincent Guedj

Download or read book Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics written by Vincent Guedj and published by Springer. This book was released on 2012-01-26 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Nonlinear Analysis on Manifolds. Monge-Ampère Equations

Author : Thierry Aubin
Publisher : Springer Science & Business Media
ISBN 13 : 1461257344
Total Pages : 215 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Nonlinear Analysis on Manifolds. Monge-Ampère Equations by : Thierry Aubin

Download or read book Nonlinear Analysis on Manifolds. Monge-Ampère Equations written by Thierry Aubin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis.

Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations

Author : Hiroyoshi Mitake
Publisher : Springer
ISBN 13 : 3319542087
Total Pages : 233 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations by : Hiroyoshi Mitake

Download or read book Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations written by Hiroyoshi Mitake and published by Springer. This book was released on 2017-06-14 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Consisting of two parts, the first part of this volume is an essentially self-contained exposition of the geometric aspects of local and global regularity theory for the Monge–Ampère and linearized Monge–Ampère equations. As an application, we solve the second boundary value problem of the prescribed affine mean curvature equation, which can be viewed as a coupling of the latter two equations. Of interest in its own right, the linearized Monge–Ampère equation also has deep connections and applications in analysis, fluid mechanics and geometry, including the semi-geostrophic equations in atmospheric flows, the affine maximal surface equation in affine geometry and the problem of finding Kahler metrics of constant scalar curvature in complex geometry. Among other topics, the second part provides a thorough exposition of the large time behavior and discounted approximation of Hamilton–Jacobi equations, which have received much attention in the last two decades, and a new approach to the subject, the nonlinear adjoint method, is introduced. The appendix offers a short introduction to the theory of viscosity solutions of first-order Hamilton–Jacobi equations.

Geometry of the complex homogeneous Monge-Ampère equation

Author : Pit-Mann Wong
Publisher :
ISBN 13 :
Total Pages : 28 pages
Book Rating : 4.:/5 (46 download)

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Book Synopsis Geometry of the complex homogeneous Monge-Ampère equation by : Pit-Mann Wong

Download or read book Geometry of the complex homogeneous Monge-Ampère equation written by Pit-Mann Wong and published by . This book was released on 1981 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Higher Asymptotics of the Complex Monge-Ampère Equation and Geometry of CR-manifolds

Author : John Marshall Lee
Publisher :
ISBN 13 :
Total Pages : 158 pages
Book Rating : 4.:/5 (11 download)

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Book Synopsis Higher Asymptotics of the Complex Monge-Ampère Equation and Geometry of CR-manifolds by : John Marshall Lee

Download or read book Higher Asymptotics of the Complex Monge-Ampère Equation and Geometry of CR-manifolds written by John Marshall Lee and published by . This book was released on 1982 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Complex Monge-Ampere Equation and Pluripotential Theory

Author : Sławomir Kołodziej
Publisher : American Mathematical Soc.
ISBN 13 : 082183763X
Total Pages : 82 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Complex Monge-Ampere Equation and Pluripotential Theory by : Sławomir Kołodziej

Download or read book The Complex Monge-Ampere Equation and Pluripotential Theory written by Sławomir Kołodziej and published by American Mathematical Soc.. This book was released on 2005 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

An Introduction to the Kähler-Ricci Flow

Author : Sebastien Boucksom
Publisher : Springer
ISBN 13 : 3319008196
Total Pages : 342 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis An Introduction to the Kähler-Ricci Flow by : Sebastien Boucksom

Download or read book An Introduction to the Kähler-Ricci Flow written by Sebastien Boucksom and published by Springer. This book was released on 2013-10-02 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research. The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman’s celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation). As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman’s ideas: the Kähler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman’s surgeries.

The Monge—Ampère Equation

Author : Cristian E. Gutierrez
Publisher : Springer Science & Business Media
ISBN 13 : 9780817641771
Total Pages : 148 pages
Book Rating : 4.6/5 (417 download)

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Book Synopsis The Monge—Ampère Equation by : Cristian E. Gutierrez

Download or read book The Monge—Ampère Equation written by Cristian E. Gutierrez and published by Springer Science & Business Media. This book was released on 2001-05-11 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Monge-Ampère equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampère type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis – covering lemmas and set decompositions.

Nonlinear PDEs, Their Geometry, and Applications

Author : Radosław A. Kycia
Publisher : Springer
ISBN 13 : 3030170314
Total Pages : 279 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Nonlinear PDEs, Their Geometry, and Applications by : Radosław A. Kycia

Download or read book Nonlinear PDEs, Their Geometry, and Applications written by Radosław A. Kycia and published by Springer. This book was released on 2019-05-18 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.

Convex Analysis and Nonlinear Geometric Elliptic Equations

Author : Ilya J. Bakelman
Publisher : Springer Science & Business Media
ISBN 13 : 3642698816
Total Pages : 524 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Convex Analysis and Nonlinear Geometric Elliptic Equations by : Ilya J. Bakelman

Download or read book Convex Analysis and Nonlinear Geometric Elliptic Equations written by Ilya J. Bakelman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: Investigations in modem nonlinear analysis rely on ideas, methods and prob lems from various fields of mathematics, mechanics, physics and other applied sciences. In the second half of the twentieth century many prominent, ex emplary problems in nonlinear analysis were subject to intensive study and examination. The united ideas and methods of differential geometry, topology, differential equations and functional analysis as well as other areas of research in mathematics were successfully applied towards the complete solution of com plex problems in nonlinear analysis. It is not possible to encompass in the scope of one book all concepts, ideas, methods and results related to nonlinear analysis. Therefore, we shall restrict ourselves in this monograph to nonlinear elliptic boundary value problems as well as global geometric problems. In order that we may examine these prob lems, we are provided with a fundamental vehicle: The theory of convex bodies and hypersurfaces. In this book we systematically present a series of centrally significant results obtained in the second half of the twentieth century up to the present time. Particular attention is given to profound interconnections between various divisions in nonlinear analysis. The theory of convex functions and bodies plays a crucial role because the ellipticity of differential equations is closely connected with the local and global convexity properties of their solutions. Therefore it is necessary to have a sufficiently large amount of material devoted to the theory of convex bodies and functions and their connections with partial differential equations.

Contact Geometry and Nonlinear Differential Equations

Author : Alexei Kushner
Publisher : Cambridge University Press
ISBN 13 : 0521824761
Total Pages : 472 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis Contact Geometry and Nonlinear Differential Equations by : Alexei Kushner

Download or read book Contact Geometry and Nonlinear Differential Equations written by Alexei Kushner and published by Cambridge University Press. This book was released on 2007 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shows novel and modern ways of solving differential equations using methods from contact and symplectic geometry.

Nonarchimedean and Tropical Geometry

Author : Matthew Baker
Publisher : Springer
ISBN 13 : 3319309455
Total Pages : 534 pages
Book Rating : 4.3/5 (193 download)

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Book Synopsis Nonarchimedean and Tropical Geometry by : Matthew Baker

Download or read book Nonarchimedean and Tropical Geometry written by Matthew Baker and published by Springer. This book was released on 2016-08-18 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume grew out of two Simons Symposia on "Nonarchimedean and tropical geometry" which took place on the island of St. John in April 2013 and in Puerto Rico in February 2015. Each meeting gathered a small group of experts working near the interface between tropical geometry and nonarchimedean analytic spaces for a series of inspiring and provocative lectures on cutting edge research, interspersed with lively discussions and collaborative work in small groups. The articles collected here, which include high-level surveys as well as original research, mirror the main themes of the two Symposia. Topics covered in this volume include: Differential forms and currents, and solutions of Monge-Ampere type differential equations on Berkovich spaces and their skeletons; The homotopy types of nonarchimedean analytifications; The existence of "faithful tropicalizations" which encode the topology and geometry of analytifications; Relations between nonarchimedean analytic spaces and algebraic geometry, including logarithmic schemes, birational geometry, and the geometry of algebraic curves; Extended notions of tropical varieties which relate to Huber's theory of adic spaces analogously to the way that usual tropical varieties relate to Berkovich spaces; and Relations between nonarchimedean geometry and combinatorics, including deep and fascinating connections between matroid theory, tropical geometry, and Hodge theory.