Finite type minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

TL;DR

This paper explores finite type minimal annuli in the product space ^2 b7 R, relating them to harmonic maps and spectral data, and investigates their geometric and spectral properties.

Contribution

It introduces a new framework connecting finite type minimal annuli to harmonic maps and spectral curves, extending Pinkall-Sterling iteration and analyzing the isospectral set structure.

Findings

Characterization of finite type minimal annuli via spectral curves

Analysis of polynomial Killing fields with zeroes and singular spectral curves

Description of the differentiable structure on the isospectral set

Abstract

We study minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ of finite type by relating them to harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set. We consider polynomial Killing fields with zeroes and the corresponding singular spectral curves, bubbletons and simple factors. We investigate the differentiable structure on the isospectral set of any finite type minimal annulus. We apply the theory to a 2-parameter family of embedded minimal annuli foliated by horizontal circles.