The clasical Bers' theorem about pants decomposition says that any compact Riemann surface of genus $g \geq 2$ has a pants decomposition such that every cutting geodesic in this decomposition is of length $\leq \mathcal{B}_g$, where $\mathcal{B}_g$ is a constant (so-called Bers' constant) depending only on $g$. There are also estimations on Bers' constant.

My question is : What is known about Bers' constant on hyperbolic surfaces with boundary?

Thank you for your answers!


Balacheff, Parlier, and Sabourau proved that Bers's theorem also holds for arbitrary complete Riemannian metrics, as long as you rescale appropriately: there's a $C_{g,n}$ such that a complete surface of genus $g$ with $n$ ends and area $A$ has a pants decomposition where each curve has length at most $C_{g,n}\sqrt{A}$.

You can apply this to surfaces with boundary by adding a cylinder and a cusp to each boundary component -- i.e., to every boundary component of length $L$, attach a cylinder of height $L$ and circumference $L$, and to the end of that, attach a hyperbolic cusp. Take a pants decomposition. Any curve that goes into the cylinder or the cusp is homotopic to a shorter curve that stays in the original manifold, so a minimal pants decomposition stays in the original surface, and has all its curves of length at most $C_{g,n}\sqrt{A+\sum_i L_i^2+L_i}$.


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