The affine Cauchy problem

The aim of this paper is to solve the Cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional. These surfaces are called affine maximal surfaces and here, we give a new complex representation which let us describe the solution to the corresponding Cauchy problem. As applications, we obtain a generalized symmetry principle, characterize when a curve in R³ can be a geodesic or pre-geodesic of a such surface and study the helicoidal affine maximal surfaces. Finally, we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set.