Some Nonlinear Problems in Riemannian Geometry

ISBN-13: 9783642082368 / Angielski / Miękka / 2010 / 397 str.

Thierry Aubin

Opis

During the last few years, the field of nonlinear problems has undergone great development. This book consisting of the updated Grundlehren volume 252 by the author and of a newly written part, deals with some important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved. Each problem is explained, up-to-date results are given and proofs are presented. Thus, the reader is given access, for each specific problem, to its present status of solution as well as to the most up-to-date methods for approaching it. The main objective of the book is to explain some methods and new techniques, and to apply them. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber.

Spis treści

1 Riemannian Geometry.- 2 Sobolev Spaces.- 3 Background Material.- 4 Complementary Material.- 5 The Yamabe Problem.- 6 Prescribed Scalar Curvature.- 7 Einstein—Kähler Metrics.- 8 Monge—Ampère Equations.- 9 The Ricci Curvature.- 10 Harmonic Maps.- Bibliography*.- Notation.

Blurb

During the last few years, the field of nonlinear problems has undergone great development.This book, the core of which is the content of the author's earlier book (Springer-Verlag 1983), updated and extended in each chapter, and augmented by several completely new chapters, deals with some important geometric problems that have only recently been solved or partially been solved. Each problem is explained with the present status of its solution and the most recent methods of approaching the proofs. The main aim is to explain some methods and new techniques, and to apply them to problems coming from geometry or from physics. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, topological methods. ..........