The Cauchy problem for improper affine spheres and the Hessian one equation

Abstract

Abstract. 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 MongeAmpère equation.

An Extension of the Affine Bernstein Problem

“…So, when Σ is simply-connected, 1 2 N is locally the real part of a holomorphic curve Φ : Σ −→ C 3 determined by ψ up to a real translation which satisfies…”

“…(1.2) has recently received much attention [1][2][3][4]12,14,22] which has revealed an interesting global theory for this class of surfaces.…”

Ribaucour type transformations for the Hessian one equation

“…In particular, it is known that the only global C 2 solutions of this equation are the quadratic polynomials [4,7,18], and that the exterior Dirichlet problem has a solution [6,12,13]. The solutions defined on R 2 minus a finite number of points are classified in [14], and a local classification result for isolated singularities is obtained in [1]. Another important issue in the theory of geometric PDEs is the study of singularities.…”

“…Actually, improper affine maps are given locally as a pair (Ω, f ) of a solution of (1.1), and they can be recovered in terms of their singular set. Generically, the singularities are cuspidal edges and swallowtails, (see [1,17,24,25]). …”

“…Several methods as the method of perturbation, [28], integrable systems, [23] and Weierstrass' type representation, [1,11,22,24] have been used in the construction of solutions of (1.1). But the classical theory of surfaces shows that geometric transformations may also be used to construct new surfaces from a given one.…”