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The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the 1-skeleton of a triangulation of an open disk. G is said to be CP parabolic respectively CP hyperbolic], if there is a locally nite disk packing P in the plane respectively, the unit disk] with contacts graph G. Several criteria… (More)

- Zheng-Xu He, Oded Schramm
- 1998

Let $ C be a simply connected domain. The Rodin-Sullivan Theorem states that a sequence of disk packings in the unit disk U converges, in a well deened sense, to a conformal map from to U. Moreover, it is known that the rst and second derivatives converge as well. Here, it is proven that for hexagonal disk packings the convergence is C 1. This is done by… (More)

- Zheng-Xu He, ZHENG-XU HE
- 1999

Let P be a locally finite disk pattern on the complex plane C whose com-binatorics is described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function Θ : E → [0, π/2] on the set of edges. Let P * be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function.… (More)

- Zheng-Xu He
- 1999

We prove the following results: (1) A unique smooth solution exists for short time for the heat equation associated with the MM obius energy of loops in a euclidean space, starting with any simple smooth loop. (2) A critical loop of the energy is smooth if it has cube-integrable curvature. Combining this with an earlier result of M. Freedman, Z. Wang and… (More)

- Zheng-Xu He
- 1998

A physically natural potential energy for simple closed curves in R 3 is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist… (More)

- Zheng-Xu He
- 2007

Let K be a hyperbolic knot, and let K 0 be a satellite of K of (homological) degree p; where p is an integer. We show that the crossing number of K 0 is at least area(E) length((m])(2?2 length((m])) p 2 , where area(E) is the area of the critical horo-torus of the hyperbolic structure on the knot complement and length((m]) is the length of the meridian in… (More)