# A conjecture on the lengths of filling pairs

## TL;DR

This paper proves a conjecture that the total length of any filling pair of simple closed geodesics on a hyperbolic surface is bounded below by a specific value related to the surface's genus, using a new isoperimetric inequality.

## Contribution

It introduces a generalized isoperimetric inequality for disconnected regions and confirms the Aougab-Huang conjecture on filling pair lengths.

## Key findings

- Proved the Aougab-Huang conjecture.
- Established a new isoperimetric inequality for disconnected regions.
- Linked filling pair lengths to the perimeter of a hyperbolic polygon.

## Abstract

A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In \cite{Aou}, Aougab-Huang conjectured that the length of any filling pair on $M$ is at least $\frac{m_{g}}{2}$, where $m_{g}$ is the perimeter of the regular right-angled hyperbolic $\left(8g-4\right)$-gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab-Huang conjecture as a corollary.