A conjecture on the lengths of filling pairs (Q2039276)

Let \(M_g\) be the closed oriented hyperbolic surface of genus \(g \geq 2\). A pair \((\alpha, \beta)\) of simple closed curves on \(M_g\) is called a \textit{filling} if \(M_g \setminus (\alpha \cup \beta)\) is a disjoint union of \(k \geq 1\) disks. A filling pair is called \textit{minimal} when \(k=1\). Let \(\ell_{M_g}(\alpha)\) denote the length of the unique geodesic representative in the free homotopy class of \(\alpha\), and let \[\mathcal{F}_g(M_g) = \min\{\ell_{M_g}(\alpha) + \ell_{M_g}(\beta) : (\alpha,\beta) \text{ is a minimal filling pair of } M_g\}.\] When \((\alpha,\beta)\) is a minimal filling pair, \(M_g \setminus (\alpha \cup \beta)\) is a \((8g-4)\)-gon of area \(4\pi(g-1)\) such that \(\ell_{M_g}(\alpha) + \ell_{M_g}(\beta)\) is half the perimeter of the polygon. Let \(\mathcal{M}_g\) denote the moduli space of genus \(g\). Considering that the regular \(n\)-gon has the least perimeter among all hyperbolic \(n\)-gons of fixed area, it was shown in [\textit{K. Bezdek}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 27, 107--112 (1984; Zbl 0578.52008)] that \[\mathcal{F}_g(M_g) \geq m_g, \text{ for all } M_g \in \mathcal{M}_g,\] where \(m_g\) is the perimeter of a hyperbolic regular right-angled \((8g-4)\)-gon. Furthermore, by defining the filling pair systole function \(\mathcal{Y}_g : \mathcal{M}_g \to \mathbb{R}\) by \[\mathcal{Y}_g(M_g) = \min\{\ell_{M_g}(\alpha) + \ell_{M_g}(\beta) : (\alpha,\beta) \text{ is a filling pair of } M_g\},\] it was conjectured by \textit{T. Aougab} and \textit{S. Huang} [Algebr. Geom. Topol. 15, No. 2, 903--932 (2015; Zbl 1334.57013)] that \(\mathcal{Y}_g(M_g) \geq m_g/2\), for all \(M_g \in \mathcal{M}_g\). In fact, this conjecture is shown in [loc. cit.] to hold true when \(M_g \setminus (\alpha \cup \beta)\) has exactly two components. Let the area and the perimeter of a polygon \(P\) be denoted by \(\mathrm{area}(P)\) and \(\mathrm{Perim}(P)\), respectively. In this paper, the authors settle the conjecture of Aougab-Huang in the affirmative by proving the following more general result. Theorem. Suppose that for \(1 \leq i \leq k\), \(P_i\) is a regular hyperbolic \(2m_i\)-gon with \(m_i \geq 2\). Let \(R\) be a regular \(N\)-gon such that \(N = 4(1- k)+2\sum_{i=1}^km_i\) and \(\mathrm{area}(R) = \sum_{i=1}^k \mathrm{area}(P_i)\). If \(R\) is not acute, then \[\sum_{i=1}^k \mathrm{Perim}(P_i) \geq \mathrm{Perim}(R).\] A key ingredient in the proof of the main theorem is the establishment of the following proposition. Proposition. Let \(P\) be a regular hyperbolic \(2n\)-gon with interior angle \(\geq \pi/2\). Suppose that for \(i=1,2\), \(P_i\) is a regular hyperbolic \(m_i\)-gon with \(m_i \geq 2\) such that \(m_1 +m_2 = n+2\) and \(\mathrm{area} (P_1) +\mathrm{area} (P_2) = \mathrm{area}(P)\). Then \[\mathrm{Perim}(P) \leq \mathrm{Perim}(P_1) + \mathrm{Perim}(P_2).\] For \(n \geq 3\) and \(0 < x < (n-2)\pi\), let \(P_n(x)\) be a regular hyperbolic \(n\)-gon with area \(x\). A crucial step in the proof of the proposition above is showing that for \(n \geq 4\), the function \( \mathrm{Perim}(P_n(x)) - \mathrm{Perim}(P_{n+1}(x))\) is monotonically increasing in \(x\). The proof also draws inspiration from some of the ideas in [\textit{J. Gaster}, Geom. Dedicata 213, No. 1, 339--343 (2021; Zbl 1467.51014)].