The geometric Cauchy problem for a class of surfaces in a 3-dimensional (pseudo)-Riemannian manifold is to construct a solution which contains a given curve and with tangent bundle prescribed along the curve. This generalizes the classical Bjoerling problem for minimal surfaces, for which there is a unique solution due to the fact that the given data is enough to determine the holomorphic Weierstrass data along this curve. Other surfaces classes which have Weierstrass representations have an analogous solution. On the other hand, flat surfaces in the 3-sphere have a solution via a dAlembert representation.
In this talk we discuss how surfaces associated to integrable systems, which have infinite dimensional versions of the Weierstrass and dAlembert representations can also be solved. Examples include spacelike and timelike CMC surfaces in Lorentz 3-space and constant negative Gauss curvature surfaces in Euclidean 3-space.