×

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.

MSC:

53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid