Rigidity of inversive distance circle packings revisited

Abstract

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34]. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [28] proved the global rigidity. In this paper, based on an observation of Zhou [37], we prove this conjecture for inversive distance in (−1, +∞) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in (−1, +∞).

On the global rigidity of sphere packings on $3$-dimensional manifolds

“…Bobenko, Pinkall and Springborn [2] introduced a method to extend a locally convex function on a nonconvex domain to a convex function and solved affirmably a conjecture of Luo [23] on the global rigidity of piecewise linear metrics on surfaces. Using the method of extension, Luo [24] proved the global rigidity of inversive distance circle packing metrics for nonnegative inversive distance and the author [31] proved the global rigidity of inversive distance circle packing metrics when the inversive distance is in (−1, +∞). The method of extension was further used to study the deformation of combinatorial curvatures on surfaces in [8,9,10,11,16].…”

“…For α ∈ R, the local rigidity of Euclidean sphere packing metrics with constant combinatorial α-curvature on 3-dimensional triangulated manifolds was proven in [14]. Results similar to Theorem 1.5 were proven for Thurston's circle packing metrics on surfaces in [13,15] and for inversive distance circle packing metrics on surfaces in [9,10,11,16,31].…”

On a combinatorial curvature for surfaces with inversive distance circle packing metrics

“…Motivated by an observation of Zhou [33], the second author [29] recently proved the global rigidity of inversive distance circle packings in both Euclidean and hyperbolic background geometry when the inversive distance I > −1 and satisfies…”

A new class of discrete conformal structures on surfaces with boundary

“…Guo's vertex scaling on surfaces with boundary in Definition 3 is an analogue of Luo's vertex scaling on closed surfaces. Comparing Lemma 3.1 with Lemma 3.6 in [26], one can see that the discrete conformal structure on surfaces with boundary in Definition 1 is an analogue of the circle packings on closed surfaces.…”

“…The structure condition (1.2) has been previously used in the study of discrete conformal structures on closed surfaces. See, for instance, [26][27][28]33] and others.…”