Gluing doubly periodic Scherk surfaces into minimal surfaces

2025/07/17 by Hao Chen, Yecun Wu, Chen, Hao +1
Materials Science · Mathematics · #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2507.12859

openalex publication_date 2025/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct minimal surfaces by stacking doubly periodic Scherk surfaces one above another and gluing them along their ends. It is previously known that the Karcher--Meeks--Rosenberg (KMR) doubly periodic minimal surfaces and Meeks' family of triply periodic minimal surfaces can both be obtained by gluing two Scherk surfaces. There have been hope and failed attempts to glue more Scherk surfaces. But our analysis shows that: Except for the special case where the doubly periodic Scherk surfaces all have triangular horizontal lattice, a glue construction can only produce the trivial Scherk surface itself, the KMR examples, or Meeks' surfaces.

Citations

• A chiral family of triply-periodic minimal surfaces derived from the quartz network
• Opening infinitely many nodes
• The classification of doubly periodic minimal tori with parallel ends

Related

• Higher genus doubly periodic minimal surfaces
• Handle Addition for doubly-periodic Scherk Surfaces
• The doubly periodic Scherk-Costa surfaces
• Regarding a uniqueness property of singly-periodic Scherk surfaces
• An end-to-end-construction for singly periodic minimal surfaces