圆模式的Andreev-Thurston定理的能量函数方法证明

A Proof of Andreev-Thurston Theorems for Circle Patterns via the Energy Function Approach

对于一个给定边界标号的加权三角剖分(T,θ)(θ∈[0,π/2]),通过构造(T,θ)的内部顶点上的能量函数,推出对应于内部顶点的标号向量是由它的锥向量唯一决定的.导出一个向量是锥向量当且仅当它满足锥向量不等式.通过证明所要求的圆模式决定的相关(T,θ)所有内部顶点的角总和向量满足锥向量不等式,得到在复平面上实现该加权三角剖分(T,θ)的平面单叶圆模式和有分枝圆模式的存在性和唯一性.这为圆模式的存在唯一性定理提供了一种新的证明方法.

Given a weighted triangulation （T,θ）（θ∈[0,π/2]） with prescribed boundary labels, the construction of energy functions on its interior vertices results in that label vectors associated with the interior vertices are uniquely determined by their cone ones. It is derived that a vector is a cone one if and only if it satisfies a cone vector inequality. It is proved that the angle sum vector at all interior vertices of （T, θ） determined by desired circle patterns satisfies the cone vector inequality, which implies the existence and uniqueness of univalent and branched circle patterns realizing （T, θ） in the complex plane. This provides a new proof of the existence and uniqueness theorems for circle patterns.