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Voss surfaces in sine-Gordon hierarchies

Michal Marvan

TL;DR

The paper unifies the classical Guichard construction of Voss surfaces with modern sine-Gordon integrable-systems techniques by identifying Guichard transformations with the Guthrie form of Olver's recursion operator acting on sine-Gordon symmetries. It develops a comprehensive framework: (i) linking Moutard solutions to Voss nets on pseudospherical surfaces, (ii) deriving a key lemma that ties the length of Guichard sequences to invariance properties of seed solutions, and (iii) introducing inverse and extended inverted operators to generate new Voss nets beyond Guichard's original sequences. The work yields explicit families of Voss nets from symmetry hierarchies (pmKdV and Khor'kova), analyzes degeneracy conditions, and constructs additional nets via inverse operators, including the arch and other non-Guichard nets. Overall, it expands the repertoire of Voss surfaces and deepens the connection between classical differential geometry of nets and modern integrable systems with potential applications in geometric modelling and architecture.

Abstract

We explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces by quadratures, starting from an arbitrarily chosen pseudospherical surface and a seed solution of the Moutard equation, by means of two simple transformations. In this paper we 1) identify the Guichard transformations with the well-known recursion operators for symmetries of the sine-Gordon equation; 2) prove a lemma which allows us to derive the length of Guichard's sequences from the invariance properties of the initial sine-Gordon solution; 3) introduce an extended class of inverted operators, increasing the number of Voss surfaces obtainable by quadratures. Several Voss nets are presented explicitly.

Voss surfaces in sine-Gordon hierarchies

TL;DR

The paper unifies the classical Guichard construction of Voss surfaces with modern sine-Gordon integrable-systems techniques by identifying Guichard transformations with the Guthrie form of Olver's recursion operator acting on sine-Gordon symmetries. It develops a comprehensive framework: (i) linking Moutard solutions to Voss nets on pseudospherical surfaces, (ii) deriving a key lemma that ties the length of Guichard sequences to invariance properties of seed solutions, and (iii) introducing inverse and extended inverted operators to generate new Voss nets beyond Guichard's original sequences. The work yields explicit families of Voss nets from symmetry hierarchies (pmKdV and Khor'kova), analyzes degeneracy conditions, and constructs additional nets via inverse operators, including the arch and other non-Guichard nets. Overall, it expands the repertoire of Voss surfaces and deepens the connection between classical differential geometry of nets and modern integrable systems with potential applications in geometric modelling and architecture.

Abstract

We explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces by quadratures, starting from an arbitrarily chosen pseudospherical surface and a seed solution of the Moutard equation, by means of two simple transformations. In this paper we 1) identify the Guichard transformations with the well-known recursion operators for symmetries of the sine-Gordon equation; 2) prove a lemma which allows us to derive the length of Guichard's sequences from the invariance properties of the initial sine-Gordon solution; 3) introduce an extended class of inverted operators, increasing the number of Voss surfaces obtainable by quadratures. Several Voss nets are presented explicitly.

Paper Structure

This paper contains 8 sections, 17 theorems, 102 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Essentials about Voss nets and Voss surfaces
  3. Voss nets from symmetries
  4. Recursion operators of the sine-Gordon equation
  5. Guichard sequences of Voss surfaces
  6. On Guichard's problem
  7. Another inverse operator
  8. Beyond the Guichard sequence

Key Result

Proposition 2.2

Let $\mathbf r(x,y)$ be a pseudospherical surface in Dini's parameterisation, let $\phi$ denote the Chebyshev angle. Let $\Phi$ be a solution of the Moutard equation $\Phi_{xy} = \Phi \cos\phi$. (i) The net if non-singular, is a Voss net, possessing the same normal vector $\mathbf n$ and satisfying $\mathbf q \cdot \mathbf n = \Phi$. Its fundamental forms are where The Gauss and mean curvature

Figures (9)

  • Figure 1: The Voss net on the right helicoid from the pseudosphere and the scaling symmetry
  • Figure 2: A helicoid from the Dini surface and translation (left); a part of its Voss net (right)
  • Figure 3: Singular and blow-up points of the koru surface ($\det'I = 0$ in blue, $\det'I = \infty$ in red)
  • Figure 4: Part of the koru surface next to the Hundertwasser's koru flag
  • Figure 5: The infinitely curled spinal curve of the koru surface in three magnifications
  • ...and 4 more figures

Theorems & Definitions (52)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Example 2.5
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3: Guichard CG-1890b
  • ...and 42 more