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!


1 Answer 1


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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