The theory of real-valued Sobolev functions is a classical part of
analysis and has a wide range of applications in pure and applied
mathematics. By contrast, the study of manifold-valued Sobolev maps is
relatively new. The incentive to explore these spaces arose in the last
forty years from geometry and physics. 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.
Many surprising connections to other areas of mathematics are explored,
including the Monge-Kantorovich theory in optimal transport, items in
geometric measure theory, Fourier series, and non-local functionals
occurring, for example, as denoising filters in image processing.
Numerous digressions provide a glimpse of the theory of sphere-valued
Sobolev maps.
Each chapter focuses on a single topic and starts with a detailed
overview, followed by the most significant results, and rather complete
proofs. The "Complements and Open Problems" sections provide short
introductions to various subsequent developments or related topics, and
suggest newdirections of research. Historical perspectives and a
comprehensive list of references close out each chapter. Topics covered
include lifting, point and line singularities, minimal connections and
minimal surfaces, uniqueness spaces, factorization, density, Dirichlet
problems, trace theory, and gap phenomena.
Sobolev Maps to the Circle will appeal to mathematicians working in
various areas, such as nonlinear analysis, 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.
