Affine spheres with prescribed Blaschke metric

Abstract

It is proved that the equality $\Delta\ln|\kappa-\lambda|=6\kappa$, where $\kappa$ is the Gaussian curvature of a metric tensor $g$ on a 2-dimensional manifold is a sufficient and necessary condition for local realizability of the metric as the Blaschke metric of some affine sphere. Consequently, the set of improper local affine spheres with non-vanishing Pick invariant can be parametrized by harmonic functions.