Rigidity of the Delaunay triangulations of the plane

In this talk, we will show the rigidity of the Delaunay triangulated plane under Luo’s discrete conformality (also called the vertex scaling). More precisely, let T=(V,E,F) be a topological triangulation of the plane. Let l,l’ be two PL-metrics of T such that the induced distance structures are isometric to the plane. Suppose l and l’ are discrete conformal in the sense that there exists a function u on V such that l’_{ij}=e^{u_i+u_j}l_{ij}. We further assume l satisfies the Delaunay condition, l’ satisfies the uniformly Delaunay condition and both l and l’ satisfy the uniformly nondegenerate condition. Then l and l’ differ by a constant factor. This a joint work with Tianqi Wu.