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

DOI: 10.2140/pjm.2014.269.281

Abstract

In this paper we study constant positive Gauss curvature $K$ surfaces in the $3$-sphere ${\mathbb{S}}^3$ with $0<K<1$, as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in ${\mathbb{S}}^3$ 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 $0<K<1$.

Keywords

constant curvature, 3-sphere, generalized Weierstrass representation, nonlinear d'Alembert formula

Mathematical Subject Classification 2010

Primary: 53A10

Secondary: 53C42, 53C43

Milestones

Received: 25 January 2013

Accepted: 4 November 2013

Published: 26 July 2014