Analytic formulas for complete hyperbolic affine spheres

TL;DR

This paper classifies certain convex cones in three dimensions and provides explicit analytic formulas for the hyperbolic affine spheres asymptotic to their boundaries, involving elliptic integrals.

Contribution

It offers a complete classification of three-dimensional convex cones with specific automorphism groups and derives explicit formulas for associated affine spheres.

Findings

Classification of convex cones with automorphism groups of dimension ≥ 2

Explicit formulas for hyperbolic affine spheres asymptotic to cone boundaries

Representation of affine spheres via elliptic integrals

Abstract

We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions into three-dimensional real space. The generic member of the family of immersions is given by elliptic integrals.