Geodesically Complete Hyperbolic Structures

TL;DR

This paper investigates the geometry of infinite hyperbolic surfaces, establishing conditions for geodesic completeness, and explores the structure of their Teichmüller spaces, including cases of incompleteness in the length spectrum metric.

Contribution

It characterizes geodesically complete hyperbolic surfaces and demonstrates how to achieve completeness through specific twist choices during gluing, also analyzing the Teichmüller space structure.

Findings

Complete hyperbolic surfaces are formed by convex core with funnels and half-planes.

Existence of twist parameters ensures completeness despite cuff growth.

Teichmüller space is incomplete in the length spectrum metric for certain flute surfaces.

Abstract

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. Such a surface is not always complete; for example, if the cuffs grow fast enough and the twists are small. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichm\"uller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichm\"uller space.