Rigidity of the Delaunay triangulations of the plane

We prove a rigidity result for Delaunay triangulations of the plane under Luo's notion of discrete conformality, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We follow Zhengxu He's analytical approach in his work on the rigidity of disk patterns, and develop a discrete Schwarz lemma and a discrete Liouville theorem. As a key ingredient to prove the discrete Schwarz lemma, we establish a correspondence between the Euclidean discrete conformality and the hyperbolic discrete conformality, for geodesic embeddings of triangulations. Other major tools include conformal modulus, discrete extremal length, and maximum principles in discrete conformal geometry.