The Cauchy problem for indefinite improper affine spheres and their Hessian equation (Q2445365)

Let \(\psi: \rightarrow \mathbb{R}^3\) be an improper affine sphere, denote by \(\xi\) its constant affine normal. Modulo a unimodular transformation \(\xi\) is given by \(\xi=(0,0,1)\), and \(\psi\) is the graph of a solution \(f\) of the PDE \[ f_{xx}f_{yy} - (f_{xy})^2 = - 1. \] The author gives a conformal representation for indefinite, improper affine spheres that solve the Cauchy problem for the foregoing Hessian equation. As application, he characterizes the geodesics of the surface, he also classifies indefinite helicoidal improper affine spheres.