Barycentric discrete conformal maps between piecewise flat approximations of Riemannian surfaces converge, under fullness and local rigidity assumptions, to conformal maps, generalizing Rodin-Sullivan circle packing convergence.
Rigidity of infinite inversive distance circle packings in the plane
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abstract
In 2004, Bowers-Stephenson [2] introduced the inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured the rigidity of infinite inversive distance circle packings in the plane. Motivated by the recent work of Luo-Sun-Wu [22] on Luo's vertex scaling, we prove Bower-Stephenson's conjecture for inversive distance circle packings in the hexagonal triangulated plane. This generalizes Rodin-Sullivan's famous result [13] on the rigidity of infinite tangential circle packings in the hexagonal triangulated plane. The key tools include a maximal principle for generic weighted Delaunay inversive distance circle packings and a ring lemma for the inversive distance circle packings in the hexagonal triangulated plane.
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Convergence of discrete conformal mappings on surfaces
Barycentric discrete conformal maps between piecewise flat approximations of Riemannian surfaces converge, under fullness and local rigidity assumptions, to conformal maps, generalizing Rodin-Sullivan circle packing convergence.