An Introduction to the Theory of Wave Maps and Related Geometric Problems

The wave maps system is one of the most beautiful and challenging nonlinear hyperbolic systems, which has captured the attention of mathematicians for more than thirty years now. In the study of its various issues, such as the well-posedness theory, the formation of singularities, and the stability of the solitons, in order to obtain optimal results, one has to use intricate tools coming not only from analysis, but also from geometry and topology. Moreover, the wave maps system is nothing other than the Euler–Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory.

One of the goals of our book is to give an up-to-date and almost self-contained overview of the main regularity results proved for wave maps.

Another one is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave maps system (e.g., the Skyrme model), which are extremely interesting and difficult in their own right.

Readership: Advanced graduate students in mathematics, professional mathematicians and theoretical physicists.

"The book offers a unified view on a very active and exciting area of current research, leading the reader to the frontier of the field. The book is a valuable source of reference for some of the most interesting recent research results on geometric nonlinear wave equations and the technical tools that they rely on. It may be highly recommended for any researcher wishing to become acquainted with wave maps and related field theories, or as background material for an advanced seminar on these topics." -- Jahresbericht der Deutsche Mathematiker-Vereinigung