A variational approach to $S^1$-harmonic maps and applications

TL;DR

This paper introduces a renormalization method for Dirichlet energy of maps into $S^1$, enabling analysis of $S^1$-harmonic maps with singularities and applications to geometric energies.

Contribution

It develops a novel renormalization scheme for the Dirichlet Lagrangian for $S^1$-valued maps, facilitating applications in geometric analysis.

Findings

Renormalization of Willmore energy for Lagrangian singular immersions.

Application to frame energies in Euclidean surface immersions.

Characterization of $S^1$-harmonic maps with isolated singularities.

Abstract

We present a renormalization procedure of the Dirichlet Lagrangian for maps from surfaces with or without boundary into $S^1$ and whose finite energy critical points are the $S^1-$harmonic maps with isolated singularities. We give some applications of this renormalization scheme in two different frameworks. The first application has to do with the renormalization of the Willmore energy for Lagrangian singular immersions into K\"ahler-Einstein Surfaces while the second application is dealing with frame energies for surfaces immersions into Euclidian spaces.