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

In this paper we study constant positive Gauss curvature K surfaces in the 3-sphere S3 with 0<K<1, as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in S3 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≠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.