Improper affine spheres and the Hessian one equation

TL;DR

This paper reviews twenty years of research on improper affine spheres and their connection to the Hessian one equation, highlighting key developments and geometric methods used in this area.

Contribution

It provides a comprehensive overview of advances in the study of improper affine spheres and their relation to the Hessian one equation over the past two decades.

Findings

Significant progress in understanding the geometry of improper affine spheres.

Development of new methods for analyzing the Hessian one equation.

Enhanced classification and solution techniques for related geometric problems.

Abstract

Improper affine spheres have played an important role in the development of geometric methods for the study of the Hessian one equation. Here, we review most of the advances we have made in this direction during the last twenty years.