The indefinite affine Cauchy problem

Abstract

We solve the geometric Cauchy problem for the class of affine maximal surfaces, with indefinite affine metric, in the affine space R3, that is, we find all the surfaces of this class which contain a given regular curve of R3 with prescribed affine normal and affine conormal along it. We prove the problem is well-posed when the initial data are non-characteristic and show that uniqueness of the solution can fail at characteristic (asymptotic) directions. As application we obtain some results about geodesics and symmetries of indefinite affine maximal surfaces.