By Vincent Guedj

The objective of those lecture notes is to supply an advent to the speculation of advanced Monge–Ampère operators (definition, regularity concerns, geometric homes of ideas, approximation) on compact Kähler manifolds (with or with out boundary).

These operators are of primary use in different basic difficulties of advanced differential geometry (Kähler–Einstein equation, area of expertise of continuing scalar curvature metrics), complicated research and dynamics. the themes coated contain, the Dirichlet challenge (after Bedford–Taylor), Monge–Ampère foliations and laminated currents, polynomial hulls and Perron envelopes without analytic constitution, a self-contained presentation of Krylov regularity effects, a modernized facts of the Calabi–Yau theorem (after Yau and Kolodziej), an creation to countless dimensional riemannian geometry, geometric constructions on areas of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity thought of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).

Each bankruptcy will be learn independently and relies on a chain of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, dropped at non-experts. The ebook is hence addressed to any mathematician with a few curiosity in a single of the next fields, advanced differential geometry, advanced research, advanced dynamics, absolutely non-linear PDE's and stochastic analysis.

**Read Online or Download Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics (Lecture Notes in Mathematics) PDF**

**Similar differential equations books**

Incorporating a couple of improvements, resolution suggestions for ordinary Partial Differential Equations, moment variation offers essentially the most very important and familiar equipment for fixing partial differential equations (PDEs). The recommendations lined comprise separation of variables, approach to features, eigenfunction enlargement, Fourier and Laplace variations, Green’s features, perturbation tools, and asymptotic research.

**The Universality of the Radon Transform (Oxford Mathematical Monographs)**

Written via a number one student in arithmetic, this monograph discusses the Radon rework, a box that has huge ranging functions to X-ray know-how, partial differential equations, nuclear magnetic resonance scanning, and tomography. during this e-book, Ehrenpreis makes a speciality of contemporary examine and highlights the robust courting among high-level natural arithmetic and purposes of the Radon remodel to components equivalent to clinical imaging.

**Abstract Parabolic Evolution Equations and their Applications (Springer Monographs in Mathematics)**

This monograph is meant to give the basics of the speculation of summary parabolic evolution equations and to teach tips to practice to numerous nonlinear dif- sion equations and platforms bobbing up in technological know-how. the idea supplies us a uni? ed and s- tematic remedy for concrete nonlinear diffusion versions.

**Scientific Computing and Differential Equations: An Introduction to Numerical Methods**

Clinical Computing and Differential Equations: An advent to Numerical equipment, is a superb supplement to creation to Numerical equipment by way of Ortega and Poole. The e-book emphasizes the significance of fixing differential equations on a working laptop or computer, which includes a wide a part of what has turn out to be referred to as medical computing.

- Integral and Discrete Inequalities and Their Applications: Volume II: Nonlinear Inequalities: 2
- Introduction to Partial Differential Equations for Scientists and Engineers Using Mathematica
- Solution Techniques for Elementary Partial Differential Equations, Third Edition
- Applied Non-Linear Dynamical Systems (Springer Proceedings in Mathematics & Statistics)

**Additional resources for Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics (Lecture Notes in Mathematics)**

**Example text**