The geometric cauchy problem for surfaces with lorentzian harmonic gauss maps

David Brander Technical University of Denmark Martin Svensson University of Southern Denmark

Differential Geometry mathscidoc:1609.10293

Journal of Differential Geometry, 93, (1), 37-66, 2013
The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to nd 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 noncharacteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and sucient for similar data to be prescribed along an additional characteristic curve that intersects the rst. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the in nite 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.
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@inproceedings{david2013the,
  title={THE GEOMETRIC CAUCHY PROBLEM FOR SURFACES WITH LORENTZIAN HARMONIC GAUSS MAPS},
  author={David Brander, and Martin Svensson},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914073025879520955},
  booktitle={Journal of Differential Geometry},
  volume={93},
  number={1},
  pages={37-66},
  year={2013},
}
David Brander, and Martin Svensson. THE GEOMETRIC CAUCHY PROBLEM FOR SURFACES WITH LORENTZIAN HARMONIC GAUSS MAPS. 2013. Vol. 93. In Journal of Differential Geometry. pp.37-66. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914073025879520955.
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