Length minimization of filling pairs on hyperbolic surfaces
Ni An, Bhola Nath Saha, Bidyut Sanki
TL;DR
This work classifies minimal filling pairs on genus-two surfaces up to mapping class group actions, showing there are exactly two types, $\{4,12\}$ and $\{8,8\}$. It then derives sharp lower bounds on the length of such pairs: $\ge 8\,\mathrm{cosh}^{-1}(\sqrt{2}+1)$ for the $\{8,8\}$ type and $\ge L_0$ with $L_0=6\,\mathrm{cosh}^{-1}(7/2)$ for the $\{4,12\}$ type, with the global minimum realized in the latter. The $\{4,12\}$ bound is obtained via a detailed quadrilateral construction and constrained optimization (Lagrange multipliers), supported by numerical verification and symmetry arguments; the $\{8,8\}$ bound leverages Gauss–Bonnet and isoperimetric inequalities for hyperbolic polygons. Collectively, these results establish a concrete, global lower bound on the length of any filling pair in genus two, contributing to extremal problems in Teichmüller theory and hyperbolic geometry.
Abstract
A filling pair $(α, β)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $α\cupβ$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is minimal among all filling pairs of $S_g$. For surfaces of genus $g \geq 3$, minimal filling pairs are well understood, whereas in genus two, such a pair divides the surface into exactly two disks. In this paper, we classify all minimal filling pairs up to the action of the mapping class group in genus two and determine the length of the shortest minimal filling pair.