The classification of complete improper affine spheres with singularities of low total curvature and new examples

Abstract

We provide a classification of complete improper affine spheres with singularities (say \emph{improper affine fronts}) in unimodular affine three-space $\boldsymbol{R}^3$ whose total curvature is greater than or equal to $-6\pi$, and a partial classification in the case of total curvature $-8\pi$. For the case of total curvature $-8\pi$, we give a complete classification for genus $0$ case and show the existence of an example and a one parameter family with genus $1$. We also study the asymptotic behavior of embedded ends of complete improper affine fronts. Moreover, we give new examples for this class of surfaces, including one which satisfies the equality condition of an Osserman-type inequality and is of positive genus.