Journal of Differential Geometry

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

David Brander and Martin Svensson

Full-text: Open access

Abstract

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and suffcient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.

Article information

Source
J. Differential Geom. Volume 93, Number 1 (2013), 37-66.

Dates
First available in Project Euclid: 2 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1357141506

Digital Object Identifier
doi:10.4310/jdg/1357141506

Mathematical Reviews number (MathSciNet)
MR3019511

Zentralblatt MATH identifier
1275.53050

Citation

Brander, David; Svensson, Martin. The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps. J. Differential Geom. 93 (2013), no. 1, 37--66. doi:10.4310/jdg/1357141506. https://projecteuclid.org/euclid.jdg/1357141506.


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