Laminations, Balloon Animals & Sewing Up Holes - Conformal Embedding of Trees

Given a tree on \$\mathbb{C}\$, there exists a Riemann map from the complement of the closed unit disk onto the complement of the tree. By continuous extension, this map induces an equivalence relation on \$\mathbb{S}^1\$ (a lamination). In this talk, we will be investigating the inverse procedure, namely, given a lamination, can we find a Riemann map to “sew up” the unit disk with it and make a tree? In particular, we will introduce a gluing condition that implies the existence of Hölder continuous gluing map. We will also go through some interesting examples of trees and their associated laminations.