[Paper Review] Rigidity of inversive distance circle packings revisited

Xu Xu|arXiv (Cornell University)|May 8, 2017

Geometric Analysis and Curvature Flows18 references3 citations

TL;DR

This paper proves the global rigidity of inversive distance circle packings for inversive distances in $(-1, +∞)$ using variational principles and convex energy functions, extending prior results on nonnegative inversive distances. It establishes that a given combinatorial curvature or $α$-curvature uniquely determines the circle packing metric in hyperbolic background geometry.

ABSTRACT

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity and then Luo \cite{L3} proved the global rigidity. In this paper, based on an observation of Zhou \cite{Z}, we prove this conjecture for inversive distance in $(-1, +\infty)$ by variational principles. We also study the global rigidity of a combinatorial curvature introduced in \cite{GJ4,GX4,GX6} with respect to the inversive distance circle packing metrics where the inversive distance is in $(-1, +\infty)$.

Motivation & Objective

• To resolve Bowers and Stephenson's conjecture on the rigidity of inversive distance circle packings for inversive distances in $(-1, +\infty)$.
• To extend the global rigidity result from nonnegative inversive distances to the full range $(-1, +\infty)$.
• To investigate the global rigidity of a generalized combinatorial curvature (including $\alpha$-curvature) under hyperbolic inversive distance circle packing metrics.
• To establish the uniqueness of circle packing metrics corresponding to prescribed combinatorial curvature or $\alpha$-curvature in hyperbolic geometry.

Proposed method

• Utilizes the variational principle established by Guo [22] for inversive distance circle packings.
• Applies the extension of locally convex functions by Bobenko, Pinkall, and Springborn, further developed by Luo [28].
• Defines a modified energy function $\widetilde{\mathcal{E}}(u)$ as the sum of concave terms $-\widetilde{\mathcal{E}}_{ijk}(u)$ and a convex integral term involving the curvature deficit.
• Proves that the extended energy $\widetilde{\mathcal{E}}(u)$ is $C^1$-smooth and convex on $\mathbb{R}^N$, ensuring uniqueness of critical points.
• Employs a convexity argument on the path between two potential metrics: if both yield the same curvature, their energy derivative vanishes identically, implying identical radii.
• Uses the positive definiteness of the Hessian matrix $\Lambda^H$ to conclude that the difference between two metrics must be zero, proving uniqueness.

Experimental results

Research questions

• RQ1 Is the inversive distance circle packing globally rigid for inversive distances in $(-1, +\infty)$?
• RQ2 Can the global rigidity result for nonnegative inversive distances be extended to negative values down to $-1$?
• RQ3 Does the generalized combinatorial $\alpha$-curvature uniquely determine the hyperbolic inversive distance circle packing metric?
• RQ4 What role does the convexity of the extended energy function play in proving uniqueness of circle packing metrics?
• RQ5 How does the variational principle extend to the full range $(-1, +\infty)$ compared to previous results restricted to $[0, +\infty)$?

Key findings

• The paper proves that for inversive distances in $(-1, +\infty)$, there exists at most one hyperbolic inversive distance circle packing metric with a given combinatorial curvature $\overline{K}$.
• The global rigidity of the standard combinatorial curvature $\overline{K}$ is established for $I > -1$ via the convexity of the extended energy function $\widetilde{\mathcal{E}}(u)$.
• The Hessian matrix $\Lambda^H$ of the energy function is shown to be positive definite, which implies that any two metrics with the same curvature must be identical.
• The proof technique extends to $\alpha$-curvature, showing that for $\alpha\overline{R} \leq 0$, there exists at most one hyperbolic inversive distance circle packing metric with a given $\alpha$-curvature $\overline{R}$.
• The result generalizes Luo’s earlier work on nonnegative inversive distances to the full interval $(-1, +\infty)$, resolving Bowers and Stephenson’s long-standing conjecture in this range.
• The energy function $\widetilde{\mathcal{E}}(u)$ is $C^1$-smooth and convex, ensuring that critical points correspond to unique solutions, which is key to proving rigidity.