具有凸多面体边界的双曲尖点的Gauss像
Gauss images of hyperbolic cusps with convex polyhedral boundary
摘要：
证明了具有凸多面体边界的三维双曲尖点由其高斯象唯一确定。此外，在具有负曲率的锥奇点和所有长度大于$2π$的闭可缩测地线的环面上，任何球面度量都是凸多面体尖点的高斯像的度量.这个结果是Rivin-Hodgson定理的一个类比，该定理用高斯像刻画了紧致凸双曲多面体。证明使用了变分方法。即，具有给定高斯像的尖点被识别为沿着一族半直线具有锥型奇点的尖点空间上的泛函的临界点。的功能被证明是凹的，并达到最大值的内点的域。作为一个副产品，我们证明刚性声明的高斯图像尖锥型奇点或没有。在一个特殊的情况下，我们的定理是等价的存在一个圆形模式的环面，规定的组合和相交角。这是瑟斯顿定理的一个亏格.实际上，我们的定理推广了Thurston定理，就像Rivin-Hodgson定理推广了Andreev定理一样，推广了具有非钝角二面角的紧致凸多面体。证明中使用的泛函是体积项和曲率项之和。我们证明了在Thurston定理的情形下，它是Chow和Luo所考虑的组合里奇流的位势。我们的定理代表了关于紧致曲面的等距浸入的一般陈述的最后一个特例。
英文摘要：
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a functional on the space of cusps with cone-type singularities along a family of half-lines. The functional is shown to be concave and to attain maximum at an interior point of its domain. As a byproduct, we prove rigidity statements with respect to the Gauss image for cusps with or without cone-type singularities. In a special case, our theorem is equivalent to existence of a circle pattern on the torus, with prescribed combinatorics and intersection angles. This is the genus one case of a theorem by Thurston. In fact, our theorem extends Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's theorem on compact convex polyhedra with non-obtuse dihedral angles. The functional used in the proof is the sum of a volume term and curvature term. We show that, in the situation of Thurston's theorem, it is the potential for the combinatorial Ricci flow considered by Chow and Luo. Our theorem represents the last special case of a general statement about isometric immersions of compact surfaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
