The Cauchy problem for indefinite improper affine spheres and their Hessian equation

TL;DR

This paper provides a conformal representation for indefinite improper affine spheres solving the Cauchy problem for their Hessian equation, classifies helicoidal examples, and explores singularities and symmetries.

Contribution

It introduces a new conformal representation, classifies helicoidal indefinite improper affine spheres, and constructs examples with singularities and complete metrics.

Findings

Characterization of geodesics and symmetries of indefinite improper affine spheres.

Classification of helicoidal indefinite improper affine spheres.

Construction of examples with singularities and complete non-flat affine metrics.

Abstract

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with complete non flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.