The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps. (English) Zbl 1275.53050

Authors’ 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 time-like 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 time-like CMC) it is necessary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means for 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 1990s. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.


53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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