标题： 顶点最小的双曲折纸2环面 显示英文标题

标题： Vertex-minimal hyperbolic origami 2-torus

摘要： 在本文中，我们研究双曲三维空间$\mathbb{H}^3$中三角双曲2环面$\Sigma_2$的几何化。我们证明，对于所有$n \geq 10$，存在一个具有$n$个顶点的$\Sigma_2$的三角剖分以及一个嵌入$S: \Sigma_2 \hookrightarrow \mathbb{H}^3$，该嵌入在三角形上限制为等距映射。我们称这些嵌入为双曲折纸2环面。由于下界$n \geq 10$在组合上是严格的，本文解决了获得双曲嵌入所需的最少顶点数的问题。 显示英文摘要

摘要： In this paper, we explore geometrizations of triangulated hyperbolic 2-tori $\Sigma_2$ in the hyperbolic 3-space $\mathbb{H}^3$. We show that for all $n \geq 10$, there exists a triangulation of $\Sigma_2$ with $n$ vertices and an embedding $S: \Sigma_2 \hookrightarrow \mathbb{H}^3$ that restricts to isometries on the triangles. We call these embeddings hyperbolic origami 2-tori. Since the lower bound $n \geq 10$ is combinatorially sharp, this paper settles the question of minimum number of vertices required to obtain a hyperbolic embedding.