Wave maps and constant curvature surfaces: singularities and bifurcations

TL;DR

This paper explores the singularities and bifurcations of wave maps and constant curvature surfaces, introducing methods to construct and analyze Lorentzian-harmonic maps and pseudospherical surfaces with prescribed singularities.

Contribution

It presents a novel method for constructing Lorentzian-harmonic maps from their jets and analyzes the singularities and bifurcations of pseudospherical surfaces.

Findings

Constructed germs of Lorentzian-harmonic maps from their jets.

Developed a method to create pseudospherical surfaces with specific singularities.

Analyzed bifurcations of singularities in generic 1-parameter families.

Abstract

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surfaces and obtain their bifurcations in generic 1-parameter families of such surfaces.