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

Emma Carberry, Sebastian Klein, Martin Ulrich Schmidt

TL;DR

This work investigates whether blow-ups of solutions to one integrable system can yield solutions to a different integrable system, focusing on sequences of cmc tori in $\mathbb{R}^3$ with unbounded principal curvatures. By blowing up both the domain and ambient space, the sinh-Gordon-based data are transformed into Liouville’s equation in the limit, producing a minimal-surface immersion, with the algebraic-geometric Pinkall–Sterling/Hitchin correspondence carried through to the limit. A main result provides explicit conditions under which the blown-up spectral data converge to a Liouville-type (and hence minimal) limit, with helicoid as a canonical example; the analysis relies on Symes’ method, Iwasawa/Birkhoff decompositions, and a gauge transformation linking Liouville to KdV structures. Overall, the paper establishes a rigorous bridge between cmc geometry and minimal-surface theory via spectral curves and polynomial Killing fields, illuminating how KdV/Liouville integrable systems arise in blow-up limits.

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.

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

TL;DR

This work investigates whether blow-ups of solutions to one integrable system can yield solutions to a different integrable system, focusing on sequences of cmc tori in with unbounded principal curvatures. By blowing up both the domain and ambient space, the sinh-Gordon-based data are transformed into Liouville’s equation in the limit, producing a minimal-surface immersion, with the algebraic-geometric Pinkall–Sterling/Hitchin correspondence carried through to the limit. A main result provides explicit conditions under which the blown-up spectral data converge to a Liouville-type (and hence minimal) limit, with helicoid as a canonical example; the analysis relies on Symes’ method, Iwasawa/Birkhoff decompositions, and a gauge transformation linking Liouville to KdV structures. Overall, the paper establishes a rigorous bridge between cmc geometry and minimal-surface theory via spectral curves and polynomial Killing fields, illuminating how KdV/Liouville integrable systems arise in blow-up limits.

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.

Paper Structure

This paper contains 4 sections, 16 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. The integrable system for constant mean curvature surfaces
  3. Polynomial Killing fields and Symes' method for minimal surface immersions
  4. Blowing up cmc tori to minimal surfaces

Key Result

Proposition 2.1

Let a real-valued, smooth function $\omega$ and constants $H \in \mathbb{R}^+$ , $Q \in \mathbb{C}^{\times}$ be given so that these data satisfy the sinh-Gordon equation eq:cmc:gauss, and let $F_\lambda$ be the corresponding extended frame. Further choose a Sym point$\lambda_s \in S^1$ . Then is an immersion $X \to \mathfrak{su}(2) \cong \mathbb{R}^3$ with induced metric $e^\omega\,\math

Theorems & Definitions (37)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 27 more