Multiple Extremal Disc-Packings in Compact Hyperbolic Surfaces

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.