Week 3
1 Week 3
This week, I worked to understand and prove the Discrete Schwartz Lemma which is a key result in theory of circle packings. The lemma is a extension of the classical Schwartz lemma from complex analysis to the discrete setting of circle packings. The statement of the lemma is as follows:
Lemma 1.1 (Discrete Schwartz Lemma) Let be a closed connected simplicially-triangulated surface, where . Let such that and in the case Euclidean background, . Suppose and are two circle packing radius assignments on with either Euclidean or hyperbolic background such that and their curvatures and satisfy . Then and for any two vertices and
Furthermore, I worked to learn more about circle packings through reading Stephanson's book on circle packings.