We give a conformal representation for improper affine spheres which is used to solve the Cauchy problem for the Hessian one equation. With this representation, we characterize the geodesics of an improper affine sphere, study its symmetries and classify the helicoidal ones. Finally, we obtain the complete classification of the isolated singularities of the Hessian one Monge-Ampère equation.
30 September 1999 in Journal of Geometry and Physics