Year of Graduation
2023
Level of Access
Open Access Thesis
Embargo Period
5-18-2023
Department or Program
Mathematics
First Advisor
Chandrika Sadanand
Abstract
This work builds up the theory surrounding a recent result of Erlandsson, Leininger, and Sadanand: the Current Support Theorem. This theorem states precisely when a hyperbolic cone metric on a surface is determined by the support of its Liouville current. To provide background for this theorem, we will cover hyperbolic geometry and hyperbolic surfaces more generally, cone surfaces, covering spaces of surfaces, the notion of an orbifold, and geodesic currents. A corollary to this theorem found in the original paper is discussed which asserts that a surface with more than $32(g-1)$ cone points must be rigid. We extend this result to the case that there are more than $3(g-1)$ cone points. An infinite family of cone surfaces which are not rigid and which have precisely $3(g-1)$ cone points is also provided, hence demonstrating tightness.