Blowing up sequences of constant mean curvature tori in $\mathbb{R}^3$ to minimal surfaces

TL;DR

This paper investigates how sequences of constant mean curvature tori in three-dimensional space can be rescaled to converge to minimal surfaces, revealing a connection between different integrable systems through algebraic-geometric methods.

Contribution

It demonstrates a specific blow-up process for cmc tori of finite type, linking solutions of the sinh-Gordon system to Liouville's equation and minimal surfaces.

Findings

Blow-up sequences of cmc tori can converge to minimal surfaces.

The process involves solutions to sinh-Gordon and Liouville's equations.

Algebraic-geometric correspondence is key to understanding the transition.

Abstract

This paper is motivated by the question of whether a sequence of solutions of a given integrable system can be blown up to obtain a solution of a different integrable system in the limit. We study a specific example of this phenomenon. Namely, we describe a blow-up for immersed constant mean curvature (cmc) planes of finite type with unbounded principal curvatures and derive sufficient conditions under which this blow-up converges to a minimal surface immersion. Passing to the respective Gauss-Codazzi equations, we are blowing up a sequence of solutions to the sinh-Gordon integrable system to obtain a solution to Liouville's equation, whose integrable system will turn out to be closely related to the Korteweg-de Vries integrable system. Our most important tool for this investigation is the algebraic-geometric correspondence that was established by Pinkall/Sterling and by Hitchin for cmc planes of finite type, which include all cmc tori.