发表机构
Zhejiang Normal University（浙江师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对双曲平面及闭曲面上的测地线单调瓦片，证明了直径与面积的正下界，并给出显式常数，同时构造了边数可任意大但直径趋于零的示例，表明边数需附加条件限制。
AI 中文摘要
对于每个固定的 $n$，我们证明了双曲平面中紧致简单测地线 $n$ 边形单调瓦片的直径存在正下界。我们还证明了在闭双曲曲面的有限单面体铺砌中，此类瓦片的面积存在正下界，且该下界与拓扑和度量无关。对于顶点适当地铺砌（其中每个真正的瓦片顶点至少属于三个不同的瓦片），这两个下界均与 $n$ 无关。这包括凸多边形铺砌。允许反射角和非边对边相接。在定性证明之后，我们通过关于具有任意耦合非负整数约束的堆积多面体的尖锐间隙估计获得显式常数。精确的角平衡适用于闭曲面；在平面中，覆盖对偶性和球计数给出了依赖于瓦片直径的边界因子。遵循 Zare 的方法，对于每个整数 $q\ge2$，我们给出一个具有 $2q+3$ 条边且直径小于 $3/q$ 的测地线构造，表明在没有附加条件的情况下，边数不能不受限制。闭曲面的结果可推广到正则曲边。几何常数是有效的，但不声称是最优的。
英文摘要
For each fixed $n$, we prove a positive lower bound for the diameter of a compact simple geodesic $n$-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of $n$ for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer $q\ge2$, a geodesic construction with $2q+3$ sides and diameter less than $3/q$, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.
Comments27 pages, 2 figures
