A conjecture on the lengths of filling pairs

Bidyut Sanki; Arya Vadnere

doi:10.1007/s10711-020-00586-8

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arXiv · 1907.07096

A conjecture on the lengths of filling pairs

Bidyut Sanki·Arya Vadnere

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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.

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Bidyut Sanki, Arya Vadnere. 2019-07-16. A conjecture on the lengths of filling pairs. https://doi.org/10.1007/s10711-020-00586-8

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