### Abstract

Original language | English |
---|---|

Journal | Journal of Differential Geometry |

Volume | 93 |

Issue number | 1 |

Pages (from-to) | 37-66 |

ISSN | 0022-040X |

DOIs | |

Publication status | Published - 2013 |

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### Cite this

*Journal of Differential Geometry*,

*93*(1), 37-66. https://doi.org/10.4310/jdg/1357141506

}

*Journal of Differential Geometry*, vol. 93, no. 1, pp. 37-66. https://doi.org/10.4310/jdg/1357141506

**The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps.** / Svensson, Martin; Brander, David.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

AU - Svensson, Martin

AU - Brander, David

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

U2 - 10.4310/jdg/1357141506

DO - 10.4310/jdg/1357141506

M3 - Journal article

VL - 93

SP - 37

EP - 66

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -