Rigidity of Infinite Hexagonal Triangulation of the Plane

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles are in $[\delta, \pi/2 -\delta]$ for any constant $\delta > 0$, then it is the regular hexagonal triangulation. This partially solves a conjecture of Luo [4]. The proof uses the concept of \emph{quasi-harmonic} functions to unfold the properties of the mesh.