Rigidity of Acute Triangulations of the Plane

We show that a uniformly acute triangulation of the plane is rigid under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We followed He's analytical approach in his work on the rigidity of disk patterns. The main tools include maximum principles, a discrete Liouville theorem, smooth and discrete extremal lengths on networks. The key step is relating the Euclidean discrete conformality to the hyperbolic discrete conformality, to obtain an L-infinity bound on the discrete conformal factor.