An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres

Abstract

where Ω is a planar domain and f is in the usual Holder space C2,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. Jorgens which asserts that any solution of (1) on Ω = R2 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 conveniently 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).