Math. Z. 230, 471–486 (1999) c  Springer-Verlag 1999 An extension of a theorem by K. J¨ orgens and a maximum principle at infinity for parabolic affine spheres ⋆ L. Ferrer, A. Mart´ ınez, F. Mil´ an Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Granada, E-18071 Granada, Spain (e-mail: lferrer@goliat.ugr.es; amartine@goliat.ugr.es; milan@goliat.ugr.es) Received March 7, 1997; in final form September 5, 1997 1. Introduction The aim of this paper is to study the following unimodular Hessian equation, det  ∂ 2 f ∂x i ∂x j  =1 in Ω, (1) where Ω is a planar domain and f is in the usual H¨ older space C 2,α ( Ω). Without loss of generality we shall consider only locally convex solutions of (1). This equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere (in short PA-sphere) in the unimodular affine real 3-space (see [C1], [C2], [CY] and [LSZ]). Contrary to the case of smooth bounded convex domains, little is known about solutions of (1) when the domain is unbounded. Here, we recall a famous result by K. J ¨ orgens which asserts that any solution of (1) on Ω = R 2 is a quadratic polynomial (see [J]) and we also mention a previous paper (see [FMM]) where the authors study solutions of (1) on the exterior of a planar domain that are regular at infinity. Since the underlying almost-complex structure of (1) is integrable, one expects PA-spheres (with their canonical conformal structure) to be conve- niently described in terms of meromorphic functions. The reader will find in Sect. 2 a complex representation of PA-spheres and, particularly, a complex description for the solutions of (1). ⋆ Research partially supported by DGICYT Grant No. PB94-0796 and the GADGET III program of the EU. Mathematics Subject Classification (1991): 53A15