Week 8

This week was presentation week so most of the time was spent preparing for the presentation. But I was able to make some progress on the Discrete Schwarz Lemma for Circle Packings. I was able to prove that the discrete Schwarz lemma holds for circle packings with more general intersection angles.

Let be a weighted simplicially-triangulated surface. is a weight function on the edges of the triangulation that describes the intersection angles of the circles in the cirle packing.

For simplicity of notation, forall triangles let where . Then .

{ Let be a closed connected weighted simplicially-triangulated surface where for any topological triangle we have . 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 . }

This theorem encompasses the known case for circle packings with intersection angles less than or equal to 90 degrees, as well as a subset of the case of intersection angles greater than 90 degrees. The proof is similar to the proof of the discrete Schwarz lemma for circle packings with intersection angles less than or equal to 90 degrees, but it requires a more careful analysis of the intersection angles and the curvature conditions.