两个最对称的平坦环面作为八顶点纸环面
The two most symmetric flat tori as eight-vertex paper tori
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中文总结 AI 辅助
本文构造了实现正方形和六边形这两个最对称平坦环面的八顶点纸环面，并证明其附近所有形状均可由八顶点纸环面实现。
中文摘要 AI 辅助
纸环面是嵌入在$\mathbb{R}^3$中的多面体环面，其内在度量是局部欧几里得的，即由一张平坦的纸折叠而成的环面。已知有许多构造，顶点数从几十个到数千个不等，因此自然要问最少需要多少个顶点。Schwartz证明了没有七顶点的纸环面，并构造了一个八顶点的。随后，Doyle和Schwartz构造了实现几乎所有平坦环面形状的八顶点纸环面，他们的构造在模空间中恰好遗漏了两条射线，即所有矩形环面，以及长宽比至少为$\sqrt3$的菱形环面。这些射线的端点正是两个最对称的平坦环面：正方形环面和六边形环面。我们构造了实现其中每一个的八顶点纸环面。我们还证明了八顶点纸环面实现了这两个附近的所有形状，因为这两者周围的平坦构型形成一个光滑的十七维流形，在该流形上模量是一个浸没。
英文摘要
A paper torus is a polyhedral torus embedded in $\mathbb{R}^3$ whose intrinsic metric is locally Euclidean, a torus folded from a flat sheet of paper. Many constructions are known, from dozens of vertices into the thousands, and it is natural to ask how few suffice. Schwartz proved that no paper torus has seven vertices, and built one with eight. Doyle and Schwartz then built paper tori realizing almost every shape of flat torus with eight vertices, their construction missing exactly two rays in moduli space, all the rectangular tori, and the rhombic tori of aspect ratio at least $\sqrt3$. These rays end at the two most symmetric flat tori, the square torus and the hexagonal torus. We build an eight-vertex paper torus realizing each of them. We also prove that eight-vertex paper tori realize every shape near these two, since the flat configurations around both form a smooth seventeen dimensional manifold on which the modulus is a submersion.
发表机构
- Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数理科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
