Tensile minimal surfaces and thread boundary problems
Abstract
Minimal surfaces bounded by cables or threads arise naturally in tensile architecture: a membrane under uniform tension takes the shape of a minimal surface, and its flexible, inextensible boundary lies along an asymptotic line of constant geodesic curvature. Such configurations have been studied experimentally since the 1960s at the Institute for Lightweight Structures in Stuttgart and more recently realized in gridshells built along networks of asymptotic and geodesic curves. Motivated by this architectural context, we construct two new families of embedded minimal surfaces bounded by finitely many asymptotic arcs of constant curvature, using the Plateau-conjugate method applied to the solution of a partially-free boundary problem for minimal disks that meet the unit sphere orthogonally along the free boundary component. For every integer m≥ 3 , we prove the existence of a one-parameter family of embedded minimal annuli, called tensile catenoids, whose boundary components each consist of m asymptotic arcs of constant curvature that meet at cusps. As the parameter varies, the family degenerates from a planar configuration to a union of m minimal disks. For every integer k≥ 3, we prove the existence of an embedded minimal surface with the topology of a sphere minus k disks, called a tensile k-noid, each of whose k boundary components consists of four asymptotic arcs joined by cusps. Both families are symmetric with respect to a horizontal plane and possess several vertical planes of symmetry. Embeddedness follows from showing that the fundamental piece obtained by conjugation is a graph contained in the region delimited by the planes of symmetry.
Related
- Calibrations for Minimal Surfaces with Free Boundary and Cheeger-Type Problems
- Free boundary minimal surfaces with connected boundary and arbitrary genus
- Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
- Inequivalent complexity criteria for free boundary minimal surfaces
- Free boundary minimal surfaces of genus zero
BibTeX
@misc{boutillier2026tensile,
title = {{Tensile minimal surfaces and thread boundary problems}},
author = {Romane Boutillier and Laurent Hauswirth and Magdalena Rodríguez},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36717},
eprint = {2609.36717},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05772041v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tensile minimal surfaces and thread boundary problems
AU - Romane Boutillier
AU - Laurent Hauswirth
AU - Magdalena Rodríguez
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.36717
UR - https://hal.science/hal-05772041v1
AB - Minimal surfaces bounded by cables or threads arise naturally in tensile architecture: a membrane under uniform tension takes the shape of a minimal surface, and its flexible, inextensible boundary lies along an asymptotic line of constant geodesic curvature. Such configurations have been studied experimentally since the 1960s at the Institute for Lightweight Structures in Stuttgart and more recently realized in gridshells built along networks of asymptotic and geodesic curves. Motivated by this architectural context, we construct two new families of embedded minimal surfaces bounded by finitely many asymptotic arcs of constant curvature, using the Plateau-conjugate method applied to the solution of a partially-free boundary problem for minimal disks that meet the unit sphere orthogonally along the free boundary component. For every integer m≥ 3 , we prove the existence of a one-parameter family of embedded minimal annuli, called tensile catenoids, whose boundary components each consist of m asymptotic arcs of constant curvature that meet at cusps. As the parameter varies, the family degenerates from a planar configuration to a union of m minimal disks. For every integer k≥ 3, we prove the existence of an embedded minimal surface with the topology of a sphere minus k disks, called a tensile k-noid, each of whose k boundary components consists of four asymptotic arcs joined by cusps. Both families are symmetric with respect to a horizontal plane and possess several vertical planes of symmetry. Embeddedness follows from showing that the fundamental piece obtained by conjugation is a graph contained in the region delimited by the planes of symmetry.
ER -