Abstract

The radius of a packing of metric discs embedded in a compact hyperbolic surface is bounded by an extremal value dependent upon the topology of the surface and the number of discs in the packing. In this paper we discuss the possibility of finding multiple extremal disc-packings within a given surface, determining the combinatorial-arithmetic conditions on the topology of the surface and the number of discs of the packing that allow such a phenomenon to happen. Moreover, we provide explicit examples of surfaces containing multiple extremal packings for each type of packing and each topological type of surface possible. Our construction relies in computer experimentation in two ways: first, by performing numerical computations that suggest certain surfaces as good candidates to contain more than one extremal packing, and second by checking with computer algebra software some lengthy necessary algebraic conditions in certain number fields that prove that the surfaces numerically constructed do indeed contain multiple extremal disc-packings.

摘要

嵌入致密双曲曲面中的公制圆盘填料的半径受极值的限制，该极值取决于表面的拓扑结构和填料中圆盘的数量。在本文中，我们讨论了在给定表面内找到多个极值盘填料的可能性，确定表面拓扑结构的组合算术条件以及允许这种现象发生的填料盘的数量。此外，我们为每种类型的包装和可能的每种拓扑类型的表面提供了包含多个极值包装的表面的明确示例。我们的构造以两种方式依赖于计算机实验：首先，通过执行数值计算，建议某些表面作为包含多个极值包装的良好候选者，