论文标题
圆形包装类型的一些标准和组合高斯定理
Some criteria for circle packing types and combinatorial Gauss-Bonnet Theorem
论文作者
论文摘要
我们研究了圆圈包装的标准（CP）磁盘三角剖分图的类型，这些图嵌入了$ \ Mathbb {C} $中的简单连接域中。特别是，通过研究涉及边界转弯的组合曲率和组合式高斯定理，我们表明磁盘三角剖分图是CP抛物线的，如果\ [ \ sum_ {n = 1}^\ infty \ frac {1} {\ sum_ {j = 0}^{n-1}（k_j +6）} = \ infty， \] $ k_n $是\ [k_n = \ sum_ {v \ in b_n}（\ mbox {deg} \，v -6）\]的\ [k_n = \ sum_ {v \ in b_n}（\ mbox {deg} \ 6）\]，用于radius $ n $的组合球$ b_n $，并以固定的顶点为中心。还表明，如果\ [\ [ \ sum_ {n = 1}^\ infty \ frac {1} {\ sum_ {j = 0}^{n-1}（k_j +6） +\ sum_ {j = 0}^{n}^{n}（k_j +6）} = \ infty。 \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998.
We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in B_n} (\mbox{deg}\, v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)+\sum_{j=0}^{n} (k_j +6)} = \infty. \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998.
