Vol. 269, No. 2, 2014

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

David Brander, Jun-ichi Inoguchi and Shimpei Kobayashi

Vol. 269 (2014), No. 2, 281–303
Abstract

In this paper we study constant positive Gauss curvature $K$ surfaces in the $3$-sphere ${\mathbb{S}}^{3}$ with $0, as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in ${\mathbb{S}}^{3}$ 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 $K\ne 0$, 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.

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