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Gluing Karcher-Scherk saddle towers I: Triply periodic minimal surfaces

Hao Chen, Martin Traizet

Abstract

We construct minimal surfaces by gluing simply periodic Karcher--Scherk saddle towers along their wings. Such constructions were previously implemented assuming a horizontal reflection plane. We break this symmetry by prescribing phase differences between the saddle towers. It turns out that, in addition to the previously known horizontal balancing condition, the saddle towers must also be balanced under a subtle vertical interaction. This interaction vanishes in the presence of a horizontal reflection plane, hence was not perceived in previous works. Our construction will be presented in a series of papers. In this first paper of the series, we will explain the background of the project and establish the graph theoretical setup that will be useful for all papers in the series. The main task of the current paper is to glue saddle towers into triply periodic minimal surfaces (TPMSs). Our construction expands many previously known TPMSs into new 5-parameter families, therefore significantly advances our knowledge on the space of TPMSs.

Gluing Karcher-Scherk saddle towers I: Triply periodic minimal surfaces

Abstract

We construct minimal surfaces by gluing simply periodic Karcher--Scherk saddle towers along their wings. Such constructions were previously implemented assuming a horizontal reflection plane. We break this symmetry by prescribing phase differences between the saddle towers. It turns out that, in addition to the previously known horizontal balancing condition, the saddle towers must also be balanced under a subtle vertical interaction. This interaction vanishes in the presence of a horizontal reflection plane, hence was not perceived in previous works. Our construction will be presented in a series of papers. In this first paper of the series, we will explain the background of the project and establish the graph theoretical setup that will be useful for all papers in the series. The main task of the current paper is to glue saddle towers into triply periodic minimal surfaces (TPMSs). Our construction expands many previously known TPMSs into new 5-parameter families, therefore significantly advances our knowledge on the space of TPMSs.

Paper Structure

This paper contains 46 sections, 27 theorems, 195 equations, 10 figures.

Table of Contents

  1. Background
  2. First look on Karcher--Scherk saddle towers
  3. Graphs
  4. Intuitive picture
  5. Formal definition
  6. The multigraph
  7. Topological embedding
  8. Geometric representation
  9. Orientation
  10. Vector spaces on graphs
  11. Cycles and cuts
  12. Functions on half-edges
  13. Cut space and cycle space
  14. Discrete differential operators
  15. A divergence over shortest edges
  16. ...and 31 more sections

Key Result

Theorem 1.1

Let $\mathsf{G}$ be a balanced, rigid and orientable graph. Then $\mathsf{G}\times\mathbb{R}$ can be desingularized into a family of minimal surfaces $\mathscr{M}_\varepsilon$ that have vertical period $2\pi\varepsilon^2$ and are symmetric with respect to a horizontal plane.

Figures (10)

  • Figure 1: A rhombohedral (left) and a tetragonal (right) deformation of the Gyroid near the saddle tower limits. They do not have any horizontal symmetry plane.
  • Figure 2: A gallery of known examples of saddle tower limits of triply periodic minimal surfaces (Source: Matthias Weber). Generalizations of them will be constructed in this paper.
  • Figure 3: A four-wing saddle tower seen from a distance (left), then scaled at the "axis" (middle). The shape of a wing is illustrated on the right, featuring the undulation.
  • Figure 4: Two saddle towers with six and ten wings; see Section \ref{['sec:exampletower']}. The one on the right has five pairs of parallel wings. (Source: Matthias Weber)
  • Figure 5: Left: two saddle towers in phase. Right: two saddle towers in opposite phase. The solid and dotted lines represent the level lines $x_3=0$ and $x_3=\pi$, respectively.
  • ...and 5 more figures

Theorems & Definitions (78)

  • Theorem 1.1: Informal statement
  • Theorem 1.2: Informal statement
  • Definition 2.1: Phase
  • Remark 2.2
  • Remark 3.1
  • Remark 3.4
  • Definition 3.5
  • Remark 3.7
  • Proposition 3.8
  • proof
  • ...and 68 more