Hyperbolic polygons of minimal perimeter in punctured discs

TL;DR

This paper proves that in punctured discs, polygons with fixed angles have minimal perimeter when inscribed in a horocycle centered at the puncture, and extends this to more complex surfaces, linking to moduli space geometry.

Contribution

It establishes a minimal perimeter characterization for polygons in punctured surfaces and generalizes it to various geometric settings, connecting to moduli space analysis.

Findings

Perimeter minimized by polygons inscribed in a horocycle at the puncture

Generalization to discs with cone points and annuli with geodesic boundaries

Application to the minimum spine systole in moduli space

Abstract

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to describe the minimum of the spine systole on the moduli space of punctured surfaces.