Leonor Ferrer

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

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).

Symmetry and uniqueness of parabolic affine spheres

The proof of a variety of problems about symmetry properties in partial differential equations and differential geometry is based on the Maximum Principle for elliptic linear equations (see [A1], [GNN] and [S]). In this paper we apply a similar technique to a locally strongly convex parabolic affine sphere (in short, PA-sphere) embedded in the unimodular affine real 3-space j~3. The study of PA-spheres is locally equivalent (see [C1], [C2]) to the study of convex solutions of the Monge-Amp~re equation

A UNIQUENESS THEOREM FOR THE SINGLY PERIODIC GENUS-ONE HELICOID ∗

The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Weis genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic genus-one helicoid provided the existence of one symmetry.

Singly-periodic improper affine spheres

Abstract In this paper we give a characterization of complete embedded ends of singly-periodic improper affine spheres in terms of their conformal representation.