Author: Vincent GuedjPublisher:
ISBN: 9783037191675
Category :
Languages : en
Pages : 472
Book Description
Degenerate Complex Monge-Ampère Equations
Author: Vincent GuedjPublisher:
ISBN: 9783037191675
Category :
Languages : en
Pages : 472
Book Description
The Degenerate Complex Monge-ampere Equation
Author: Roberto MoriyónThe Complex Monge-Ampere Equation and Pluripotential Theory
Author: Sławomir KołodziejPublisher: American Mathematical Soc.
ISBN: 082183763X
Category : Mathematics
Languages : en
Pages : 82
Book Description
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.
A Priori Estimates of the Degenerate Monge-Ampère Equation on Compact Kähler Manifolds
Author: Sebastien PicardPublisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
"The regularity theory of the degenerate complex Monge-Ampère equation is studied. First, the equation is considered on a compact Kahler manifold without boundary. Accordingly, some background information on Kahler geometry is presented. Given a solution of the degenerate complex Monge-Ampère equation, it is shown that its oscillation and gradient can be bounded. The Laplacian of the solution is also estimated. There is a slight improvement from the literature on the conditions required in order to obtain the estimate on the Laplacian of the solution, however the estimates developed only hold in the case of manifolds with non-negative bisectional curvature. As an application, a Dirichlet problem in complex space is considered. The obtained estimates are used to show existence and uniqueness of pluri-subharmonic solutions to the degenerate complex Monge-Ampere equation in a domain in complex space." --
The Monge—Ampère Equation
Author: Cristian E. GutierrezPublisher: Springer Science & Business Media
ISBN: 1461201950
Category : Mathematics
Languages : en
Pages : 140
Book Description
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.
Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics
Author: Vincent GuedjPublisher: Springer Science & Business Media
ISBN: 3642236685
Category : Mathematics
Languages : en
Pages : 315
Book Description
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-Ampère Equation and Pluripotential Theory
Author: Sławomir KołodziejPublisher: American Mathematical Soc.
ISBN: 9781470404413
Category : Mathematics
Languages : en
Pages : 64
Book Description
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.
An Introduction to the Kähler-Ricci Flow
Author: Sebastien BoucksomPublisher: Springer
ISBN: 3319008196
Category : Mathematics
Languages : en
Pages : 342
Book Description
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.
Author: G. Patrizio
Author: Giorgio Patrizio