The associated families of semi-homogeneous complete hyperbolic affine spheres

TL;DR

This paper computes isothermal parametrizations and constructs associated families of semi-homogeneous hyperbolic affine spheres in , revealing rich geometric structures and connections to classical special functions.

Contribution

It introduces explicit isothermal parametrizations and constructs entire associated families for Hildebrand's semi-homogeneous cones, expanding understanding of hyperbolic affine spheres.

Findings

Explicit isothermal parametrizations for new examples.

Construction of associated families parametrized by S^1.

Representation of generic members using Weierstrass functions.

Abstract

Hildebrand classified all semi-homogeneous cones in $\mathbb{R}^3$ and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass $\mathscr{P}, ~\zeta ~\text{and} ~\sigma$ functions. In general any regular convex cone in $\mathbb{R}^3$ has a natural associated $S^1$-family of such cones, which deserve further studies.