[Paper Review] Construction of Triply Periodic Minimal Surfaces

Rami Younes|arXiv (Cornell University)|Feb 25, 2010

Quasicrystal Structures and Properties18 references3 citations

TL;DR

This paper constructs embedded triply periodic minimal surfaces by desingularizing a periodic tiling $τ$ of the plane with straight edges, using a nodal gluing technique inspired by Traizet. By opening nodes at vertices and inserting simply periodic Karcher saddle towers whose wings align with tiling edges, the method produces a continuous family of minimal surfaces that resolve the singularities of $\mathcal{T} \times \mathbb{R}$, offering a new construction for triply periodic minimal surfaces from combinatorial tilings.

ABSTRACT

Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes $\mathcal{T} imes\mathbb{R}$. For this purpose, inspired by the work of Martin Traizet, we open the nodes of singular Riemann surfaces to glue together simply periodic Karcher saddle towers, each placed at a vertex of the tiling in such a way that its wings go along the corresponding edges of the tiling ending at that vertex.

Motivation & Objective

• To develop a systematic method for constructing embedded triply periodic minimal surfaces from planar tilings with translational symmetry.
• To resolve the singularities of the product space $\mathcal{T} \times \mathbb{R}$, where $\mathcal{T}$ is a tiling by polygons invariant under two independent translations.
• To generalize Traizet's nodal gluing technique to the triply periodic setting by inserting periodic saddle towers at tiling vertices.
• To ensure the resulting surfaces are minimal and embedded by aligning the wings of saddle towers with the edges of the tiling.

Proposed method

• Utilize a tiling $\mathcal{T}$ of the plane by straight-edge polygons that is invariant under two independent translations.
• Apply a nodal gluing procedure inspired by Martin Traizet to open singularities at vertices of the tiling.
• Insert simply periodic Karcher saddle towers at each vertex of $\mathcal{T}$, positioning them so their wings align with the incident edges of the tiling.
• Ensure the geometric compatibility of the gluing by matching the asymptotic behavior of the saddle towers to the local structure of the tiling.
• Construct a continuous 1-parameter family of minimal surfaces by deforming the nodal configuration.
• Verify the resulting surfaces are embedded and triply periodic through analysis of the gluing parameters and asymptotic data.

Experimental results

Research questions

• RQ1 Can a family of embedded triply periodic minimal surfaces be constructed from a given periodic tiling of the plane by polygons with straight edges?
• RQ2 How can the singularities of $\mathcal{T} \times \mathbb{R}$ be resolved via geometric gluing techniques to produce minimal surfaces?
• RQ3 What role do the asymptotic properties of Karcher saddle towers play in aligning with the tiling edges during the gluing process?
• RQ4 Under what conditions does the nodal gluing of periodic saddle towers yield embedded minimal surfaces?
• RQ5 How does the choice of tiling structure influence the topology and geometry of the resulting triply periodic minimal surface?

Key findings

• The construction yields a continuous 1-parameter family of embedded triply periodic minimal surfaces that desingularize $\mathcal{T} \times \mathbb{R}$.
• The resulting surfaces are minimal and embedded due to the precise alignment of saddle tower wings with the edges of the tiling.
• The method generalizes Traizet's nodal gluing technique to the triply periodic setting, enabling new constructions from combinatorial tilings.
• The gluing process preserves the translational symmetry of the original tiling, ensuring the minimal surfaces inherit the same periodicity.
• The construction is robust for any tiling $\mathcal{T}$ with straight edges and two independent translational symmetries.
• The asymptotic behavior of the Karcher saddle towers ensures smooth matching at the gluing locus, avoiding self-intersections.