Preprint number: CP3-Origins-2010-40
The geometric Cauchy problem for a class of surfaces in a 3-dimensional pseudo-Riemannian manifold is to find the surface which contains a given curve with a prescribed normal vector field along the curve. We consider this problem for constant negative Gauss surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike non-minimal constant 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 sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a method of constructing all solutions using loop group techniques. The method used is the infinite dimensional D’Alembert type representation for surfaces associated to Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990’s. Explicit formulae, in terms of the prescribed data, are given here for the potentials used to construct the solutions, and some applications considered.