A conjecture on the lengths of filling pairs

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 $\alpha \cup \beta$ on $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In Aougab and Huang (Algebr Geom Topol 15:903–932, 2015), Aougab–Huang conjectured that the length of any filling pair on $M_{g}$ 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.