  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 sufficient 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.
