发表机构
École nationale des ponts et chaussées, Univ. Gustave Eiffel; Univ. Gustave Eiffel; Universidad de Granada（巴黎路桥大学、古斯塔夫埃菲尔大学; 古斯塔夫埃菲尔大学; 格拉纳达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对张拉结构中由缆索围成的极小曲面问题，利用Plateau共轭方法构造了两族嵌入极小曲面（张拉悬链面和张拉k-noid），并证明其存在性与对称性，为建筑网格壳设计提供数学基础。
AI 中文摘要
由缆索或线束围成的极小曲面在张拉结构中自然出现：均匀张力下的薄膜呈现极小曲面形状，其柔性不可伸长边界沿恒定测地曲率的渐近线分布。自20世纪60年代以来，斯图加特轻型结构研究所对此类构型进行了实验研究，近期更在沿渐近线和测地线网络建造的网格壳中得以实现。受此建筑背景启发，我们利用Plateau共轭方法，应用于极小圆盘的部分自由边界问题（该圆盘沿自由边界分量与单位球面正交相交）的解，构造了两族由有限条常曲率渐近弧围成的嵌入极小曲面。对每个整数$m\geq 3$，我们证明存在一族单参数嵌入极小环面，称为张拉悬链面，其每个边界分量由$m$条在尖点处相交的常曲率渐近弧组成。随着参数变化，该族从平面构型退化到$m$个极小圆盘的并集。对每个整数$k\geq 3$，我们证明存在一个嵌入极小曲面，其拓扑为球面去掉$k$个圆盘，称为张拉$k$-noid，其每个$k$边界分量由四条在尖点处连接的渐近弧组成。两族曲面均关于水平面对称，并具有多个垂直对称平面。嵌入性通过证明共轭得到的基本片是包含在对称平面界定区域内的图来确立。
英文摘要
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 $m\geq 3$ , 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 $k\geq 3$, 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.
Comments32 pages, 34 figures
