Dressing Actions on Proper Definite Affine Spheres

TL;DR

This paper explores the loop group formulations of definite affine spheres in R^3, classifies rational elements, and computes dressing actions to generate new examples of these surfaces.

Contribution

It introduces a classification of rational elements with specific poles in a real twisted loop group and applies dressing actions to produce new affine sphere examples.

Findings

Classification of rational elements with 3 or 6 poles

Explicit computation of dressing actions on affine spheres

Generation of new affine sphere examples with visualizations

Abstract

We will first clarify the loop group formulations for both hyperbolic and elliptic definite affine spheres in R^3. Then we classify the rational elements with 3 poles or 6 poles in a real twisted loop group, and compute dressing actions of them on such surfaces. Some new examples with pictures will be produced at last.