Rigidity of Acute Triangulations of the Plane

TL;DR

This paper proves that uniformly acute triangulations of the plane are rigid under discrete conformal changes, extending known results and employing analytical tools like maximum principles and extremal lengths.

Contribution

It establishes a discrete analogue of conformal rigidity for the plane, extending previous results to acute triangulations and relating Euclidean and hyperbolic conformality.

Findings

Uniformly acute triangulations are rigid under discrete conformal change.

The proof involves relating Euclidean and hyperbolic conformality.

Key tools include maximum principles and extremal length techniques.

Abstract

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.