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Links between the integrable systems of CMC surfaces, isothermic surfaces and constrained Willmore surfaces

Katrin Leschke

TL;DR

The paper addresses how constant mean curvature (CMC) surfaces in $\mathbb{R}^3$ sit at the intersection of three integrable systems: isothermic surfaces, constrained Willmore surfaces, and the harmonic Gauss map framework. It develops a unified view by organizing these systems around their associated families of flat connections ($d^N_\lambda$, $d_\lambda$, $d^S_\lambda$) and their parallel sections, and proves that parallel sections from one system algebraically determine those of the others. A key finding is a 2:1 correspondence between the CMC/Constrained Willmore spectral parameter $\mu$ and the isothermic parameter $\varrho$, via $\mu_\pm = 1-2\varrho \pm 2i\sqrt{\varrho(1-\varrho)}$, enabling translation between $\mu$-Darboux and $\varrho$-Darboux transforms and their simple factor dressings. The results yield a coherent framework for deriving and relating Darboux transforms, simple factor dressings, and associated families across all three integrable structures, with the CMC associated family emerging as a limit of the other two pictures.

Abstract

Since constant mean curvature surfaces in 3-space are special cases of isothermic and constrained Willmore surfaces, they give rise to three, apriori distinct, integrable systems. We provide a comprehensive and unified view of these integrable systems in terms of the associated families of flat connections and their parallel sections: in case of a CMC surface, parallel sections of all three associated families of flat connections are given algebraically by parallel sections of either one of the families. As a consequence, we provide a complete description of the links between the simple factor dressing given by the conformal Gauss map, the simple factor dressing given by isothermicity, the simple factor dressing given by the harmonic Gauss map, as well as the relationship to the classical, the $μ$- and the $\varrho$-Darboux transforms of a CMC surface. Moreover, we establish the associated family of the CMC surfaces as limits of the associated family of isothermic surfaces and constrained Willmore surfaces.

Links between the integrable systems of CMC surfaces, isothermic surfaces and constrained Willmore surfaces

TL;DR

The paper addresses how constant mean curvature (CMC) surfaces in sit at the intersection of three integrable systems: isothermic surfaces, constrained Willmore surfaces, and the harmonic Gauss map framework. It develops a unified view by organizing these systems around their associated families of flat connections (, , ) and their parallel sections, and proves that parallel sections from one system algebraically determine those of the others. A key finding is a 2:1 correspondence between the CMC/Constrained Willmore spectral parameter and the isothermic parameter , via , enabling translation between -Darboux and -Darboux transforms and their simple factor dressings. The results yield a coherent framework for deriving and relating Darboux transforms, simple factor dressings, and associated families across all three integrable structures, with the CMC associated family emerging as a limit of the other two pictures.

Abstract

Since constant mean curvature surfaces in 3-space are special cases of isothermic and constrained Willmore surfaces, they give rise to three, apriori distinct, integrable systems. We provide a comprehensive and unified view of these integrable systems in terms of the associated families of flat connections and their parallel sections: in case of a CMC surface, parallel sections of all three associated families of flat connections are given algebraically by parallel sections of either one of the families. As a consequence, we provide a complete description of the links between the simple factor dressing given by the conformal Gauss map, the simple factor dressing given by isothermicity, the simple factor dressing given by the harmonic Gauss map, as well as the relationship to the classical, the - and the -Darboux transforms of a CMC surface. Moreover, we establish the associated family of the CMC surfaces as limits of the associated family of isothermic surfaces and constrained Willmore surfaces.

Paper Structure

This paper contains 20 sections, 31 theorems, 218 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal immersions
  4. Isothermic surfaces
  5. The classical Darboux transformation
  6. The associated family of flat connections
  7. Holomorphic sections and (generalised) Darboux transformation
  8. Isothermic surfaces
  9. A $\mathbb{ C}$--family of flat connections
  10. The $\varrho$--Darboux transformation
  11. Simple factor dressing of isothermic surfaces
  12. CMC surfaces
  13. The associated family of flat connections of a harmonic map
  14. The $\mu$--Darboux transformation of CMC surfaces
  15. Simple factor dressing of CMC surfaces
  16. ...and 5 more sections

Key Result

Theorem 2.5

If $f^d$ is a fixed dual surface of the isothermic $f$, and $\hat{f} =f +T$ is a classical Darboux transform of $f$ with parameter $r\in\mathbb{ R}_*$ then $\hat{f}^d = f^d + \frac{1}{r} T^{-1}$ is a classical Darboux transform of $f^d$ with parameter $r$. In particular, a classical Darboux transfor

Figures (13)

  • Figure 2: Classical Darboux transforms of the surface of revolution $f(x,y)= i( -x + \frac{x^3}{3}) + j(1+x^2)e^{-iy}$ at spectral parameter $r=\frac{3}{4}$, with different initial conditions. We give a complete description of all Darboux transforms of a surface of revolution in Example \ref{['ex:rhodt of sor']}.
  • Figure 3: Elements of the associated family of the surface of revolution $f(x,y)= i( -x + \frac{x^3}{3}) + j(1+x^2)e^{-iy}$, for $r=-\frac{1}{5}$, $r=\frac{1}{4}$ and $r=\frac{3}{4}$.
  • Figure 4: The surface on the right is the common Darboux transform of the two Darboux transforms on the left (with spectral parameter $\frac{3}{4}$ and $2$ respectively) of the surface of revolution $f(x,y)= i( -x + \frac{x^3}{3}) + j(1+x^2)e^{-iy}$.
  • Figure 5: $\varrho$--Darboux transforms $f_\pm$ of the surface of revolution $f(x,y)= i( -x + \frac{x^3}{3}) + j(1+x^2)e^{-iy}$ for $\varrho=1+i$, both orthogonal projections to 3--space of rotation surfaces in 4--space.
  • Figure 6: Isothermic Bubbletons: classical Darboux transforms of the surface of revolution $f(x,y)= i( -x + \frac{x^3}{3}) + j(1+x^2)e^{-iy}$ at the resonance points $r_k=\frac{k^2-1}{4}$, $k=2,3$ with various initial conditions.
  • ...and 8 more figures

Theorems & Definitions (74)

  • Definition 2.1: christoffelhertrich-jeromin_supplement_1997
  • Example 2.2
  • Definition 2.3: darbouxdarboux_isothermic
  • Remark 2.4
  • Theorem 2.5: darbouxdarboux_isothermic
  • proof
  • Theorem 2.6: see e.g. udo_habil
  • Theorem 2.7: udo_habilburstall_conformal_2010burstall_isothermic_2011sym-darboux
  • Remark 2.8
  • Theorem 2.9: see hertrich_musso_nicolodifran_epos
  • ...and 64 more