Complete Flat Cone Metrics on Punctured Surfaces

TL;DR

This paper establishes that complete flat cone metrics on punctured surfaces can be triangulated with finitely many triangle types, derives a Gauss-Bonnet formula for these metrics, and proves the existence of geodesic representatives for free homotopy classes.

Contribution

It introduces a finite triangulation method for complete flat cone metrics on punctured surfaces and extends classical geometric formulas to this setting.

Findings

Triangulation with finitely many triangle types for flat cone metrics

Gauss-Bonnet formula adaptation for cone metrics

Existence of geodesic representatives in free homotopy classes

Abstract

We prove that each complete flat cone metric on a surface, perhaps with boundary and punctures, can be triangulated with finitely many types of triangles. We derive Gauss-Bonnet formula for this kind of cone metrics. In addition, we prove that each free homotopy class of paths has a geodesic representative.