Length minimization of filling pairs on hyperbolic surfaces

TL;DR

This paper classifies all minimal filling pairs on genus two hyperbolic surfaces, determining their properties and the shortest length among them, advancing understanding of surface decompositions.

Contribution

It provides a complete classification of minimal filling pairs on genus two surfaces and identifies the shortest such pairs.

Findings

All minimal filling pairs on genus two surfaces are classified.

The length of the shortest minimal filling pair is determined.

Minimal filling pairs are characterized up to the mapping class group action.

Abstract

A filling pair $(\alpha, \beta)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $\alpha\cup\beta$ 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.