使用牛顿法构造双曲多面体
Constructing hyperbolic polyhedra using Newton's Method
AI总结：
本文利用牛顿法和同伦方法实现了Andreev定理的存在性证明，从而构造具有非钝二面角的紧致三维双曲多面体，并应用于构造反射群生成的双曲3-orbifold并研究其体积和闭测地线谱。
摘要：
我们演示了如何使用牛顿法构造三维紧致双曲多面体。在二面角非钝的限制下，Andreev定理给出了实现给定组合结构$C$的紧致双曲多面体的二面角必须满足的五类线性不等式作为充要条件。Andreev定理还表明，所得多面体在双曲等距意义下是唯一的。我们的构造使用牛顿法和同伦方法显式地遵循Andreev给出的存在性证明，既非常清晰地说明了Andreev定理的证明，又提供了一种方便的方法来构造具有非钝二面角的三维紧致双曲多面体。作为应用，我们构造了二面角为$π$的（真）整数子倍数的紧致双曲多面体，使得由面反射生成的群$Γ$是双曲空间的离散等距群。因此商空间$\mathbb{H}^3/Γ$是一个紧致双曲3-orbifold，我们使用SnapPea研究其双曲体积和闭测地线长度谱。一个结果是Mednykh和Vesnin（见参考文献）考虑的“超椭圆”流形的体积估计。
英文摘要：
We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure $C$. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of proof of Andreev's Theorem as well as a convenient way to construct three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer sub-multiples of $π$, so that the group $Γ$ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient $\mathbb{H}^3/Γ$ is hence a compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and spectrum of closed geodesic lengths using SnapPea. One consequence is a volume estimate for a ``hyperelliptic'' manifold considered by Mednykh and Vesnin (see references).
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