Abstract: The Schwarz lemma plays a pivotal role in proving the uniqueness of the complete hyperbolic metric in the uniformization theorem. The discrete Schwarz lemma for circle packings on surfaces was established by Beardon-Stephenson and was utilized to establish the uniqueness of infinite circle packing on the open disk. In this work, we prove a discrete Schwarz lemma for hyperbolic polyhedral surfaces in discrete conformal geometry. As a consequence, we establish the uniqueness of the discrete uniformization metric in the open disk.

The relationship between the uniqueness result, Cauchy rigidity, and the Weyl problem on convex surfaces will be discussed. This work is a collaboration with Yanwen Luo.