DOI: 10.1007/s00209-004-0728-y Math. Z. 249, 755–766 (2005) Mathematische Zeitschrift Improper affine maps A. Mart´ ınez Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (e-mail: amartine@ugr.es) Received: 23 February 2004; in final form: 23 July 2004 / Published online: 15 October 2004 – © Springer-Verlag 2004 Abstract. We study a class of improper affine spheres with singularities called improper affine maps. New examples of genus 0 and 1 are described and the clas- sification of complete improper affine maps with at most two embedded ends will be given. Mathematics Subject Classification (2000): 53A15, 53A35 1. Introduction The equation det(∇ 2 u) = 1 arises in the context of an affine differential prob- lem such as the equation of improper affine spheres (i.e. locally strongly convex immersions with parallel affine normal lines) in the unimodular affine 3-space. In [4], Calabi proved that there is an interesting local correspondence between solutions of the above Monge-Ampère type equation and solutions of the equation of minimal surfaces in the Euclidean 3-dimensional space. From this, one expects improper affine spheres (with their canonical conformal structure) to be conve- niently described (at least locally) in terms of meromorphic data. Actually, in [11], the reader may find a global complex representation for improper affine spheres which is particularly useful to describe examples and to understand their behavior at infinity. In the regular case, global properties have already been studied and nowadays it is well known (see [3], [5], [14], [15], [18]) that any euclidean complete (or affine complete) improper affine sphere must be an elliptic paraboloid. However, by studying the Cauchy problem for an improper affine immersion, one learns that singularities can determine the immersion, (see [1]). Thus, it is natural to consider Research partially supported by MCYT-FEDER Grant No. BFM2001-3318 and Junta de Andaluc´ ıa CEC: FQM0804.