Constant Gaussian curvature surfaces in the 3-sphere via loop groups

TL;DR

This paper develops a loop group framework to study constant Gaussian curvature surfaces in the 3-sphere, providing a unified method to construct and relate surfaces with positive and negative curvature.

Contribution

It introduces a uniform loop group formulation for constant curvature surfaces in S^3 with K≠0 and establishes a correspondence between surfaces with negative and positive curvature.

Findings

Normal Gauss map is Lorentz harmonic iff the surface has constant curvature.

A generalized d'Alembert method is used to construct explicit examples.

The framework links surfaces with K<0 and 0<K<1 in a natural way.

Abstract

In this paper we study constant positive Gauss curvature $K$ surfaces in the 3-sphere $S^3$ with $0<K<1$ as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in $S^3$ with Gauss curvature $K<1$ is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if $K$ is constant. We give a uniform loop group formulation for all such surfaces with $K\neq 0$, and use the generalized d'Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with $K<0$ and those with $0<K<1$.