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 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 K0, 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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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
Authors
David Brander
Institut for Matematik og Computer Science
Matematiktorvet
Bygning 303B
Technical University of Denmark
DK-2800
Kongens Lyngby
Denmark
Jun-ichi Inoguchi
Department of Mathematical Sciences
Faculty of Science
Yamagata University
Yamagata 990-8560
Japan
Shimpei Kobayashi
Department of Mathematics
Hokkaido University
Sapporo 060-0810
Japan