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arXiv · 1301.5999

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

David Brander·Jun-ichi Inoguchi·Shimpei Kobayashi
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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 $0<K<1$.

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David Brander, Jun-ichi Inoguchi, Shimpei Kobayashi. 2013-01-25. Constant Gaussian curvature surfaces in the 3-sphere via loop groups. https://doi.org/10.2140/pjm.2014.269.281

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