The Cauchy Problem for Improper Affine Spheres and the Hessian One Equation.
2007 - J. A. Aledo, R.M.B. Chaves, J. A. Gálvez
Transactions of the American Mathematical Society 359, 4183-4208 (2007).
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.