Sobolev Maps to the Circle: From the Perspective of Analysis, Geometry, and Topology (Hardcover, 2021)

Lifting in $W^{1,p}$.- The Geometry of $J(u)$ and $\Sigma(u)$ in 2D; Point Singularities and Minimal Connections.- The Geometry of $J(u)$ and $\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces.- A Digression: Sphere-Valued Maps.- Lifting in Fractional Sobolev Spaces and in $VMO$.- Uniqueness of Lifting and Beyond.- Factorization.- Applications of the Factorization.- Estimates of Phases: Positive and Negative Results.- Density.- Traces.- Degree.- Dirichlet Problems, Gaps, Infinite Energies.- Domains with Topology.- Appendices.