Sobolev Maps to the Circle: From the Perspective of Analysis, Geometry, and Topology

The theory of real-valued Sobolev functions is a classical part of analysis with a wide range of applications, while the study of manifold-valued Sobolev maps is relatively new. This monograph is the first to provide a unified, comprehensive treatment of Sobolev maps to the circle, presenting numerous results obtained by the authors and others. It explores surprising connections to other areas of mathematics, such as optimal transport, geometric measure theory, Fourier series, and non-local functionals. Each chapter focuses on a single topic and provides complete proofs. The book will appeal to mathematicians and physicists working in various areas, including nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology.

Format: Hardback
Length: 530 pages
Publication date: 02 December 2021
Publisher: Springer-Verlag New York Inc.

The theory of real-valued Sobolev functions is a fundamental and versatile branch of analysis, with applications spanning both pure and applied mathematics. In contrast, the study of manifold-valued Sobolev maps is relatively recent, having gained momentum in the past forty years due to its connections to geometry and physics. This monograph is a pioneering effort in presenting a comprehensive treatment of Sobolev maps to the circle, encompassing a wide range of results obtained by the authors and other scholars.

Throughout the book, numerous surprising connections to other branches of mathematics are explored, including the Monge-Kantorovich theory in optimal transport, geometric measure theory, Fourier series, and non-local functionals, such as denoising filters in image processing. These connections shed light on the broader implications and applications of Sobolev maps.

To enhance the understanding of the subject, numerous digressions are included, offering insights into the theory of sphere-valued Sobolev maps. Each chapter in the book focuses on a specific topic, providing a detailed overview at the beginning, followed by the most significant results and comprehensive proofs. The "Complements and Open Problems" sections offer concise introductions to subsequent developments or related topics, suggesting new avenues for research.

To conclude each chapter, historical perspectives are provided, offering insights into the development of Sobolev maps and their significance in mathematics and physics. A comprehensive list of references is also included, providing further resources for readers interested in exploring this field further.

The topics covered in this monograph are diverse and encompass a wide range of subjects within Sobolev maps. These include lifting, point and line singularities, minimal connections and minimal surfaces, uniqueness spaces, factorization, density, Dirichlet problems, trace theory, and gap phenomena. Each topic is approached with clarity and rigor, making it accessible to both advanced students and researchers in the field.

Sobolev Maps to the Circle is a valuable resource for mathematicians working in various areas, including nonlinear analysis, partial differential equations (PDEs), geometric analysis, minimal surfaces, optimal transport, and topology. It will also be of interest to physicists working on liquid crystals and the Ginzburg-Landau theory of superconductors, as Sobolev maps have profound implications in these fields as well.

In conclusion, this monograph is a groundbreaking contribution to the field of Sobolev maps, providing a comprehensive and unified treatment of Sobolev maps to the circle. It offers a wealth of results, connections, and insights, making it an essential resource for mathematicians and physicists alike. With its clear and concise writing style, it is accessible to a wide audience, and its applications in various fields make it a valuable tool for researchers and practitioners.

Weight: 1004g
Dimension: 235 x 155 (mm)
ISBN-13: 9781071615102
Edition number: 1st ed. 2021