Families of spherical surfaces and harmonic maps

TL;DR

This paper investigates singularities and bifurcations of constant positive Gaussian curvature surfaces, their relation to harmonic maps, and classifies certain map-germs representable by harmonic maps.

Contribution

It explicitly constructs bifurcations of these surfaces using loop group methods and analyzes the connection between surface singularities and harmonic maps.

Findings

Explicit bifurcation constructions using loop groups

Classification of finitely A-determined map-germs by harmonic maps

Analysis of singularity types in constant positive Gaussian curvature surfaces

Abstract

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A-determined map-germs from the plane to the plane can be represented by harmonic maps.