Zheng-Xu He

Fixed points, Koebe uniformization and circle packings

A domain in the Riemann sphere \(\hat{\mathbb{C}}\) is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following conjecture, known as Koebe’s Kreisnormierungsproblem: Any plane domain is conformally homeomorphic to a circle domain in \(\hat{\mathbb{C}}\). When the domain is simply connected, this is the content of the Riemann mapping theorem.

Hyperbolic and parabolic packings

The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. LetG be the 1-skeleton of a triangulation of an open disk.G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk packingP in the plane (resp. the unit disk) with contacts graphG. Several criteria for deciding whetherG is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial criterion. A criterion in terms of the random walk says that if the random walk onG is recurrent, theG is CP parabolic. Conversely, ifG has bounded valence and the random walk onG is transient, thenG is CP hyperbolic.We also give a new proof thatG is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that ifG is CP hyperbolic andD is any simply connected proper subdomain of the plane, then there is a disk packingP with contacts graphG such thatP is contained and locally finite inD.

Rigidity of infinite disk patterns

Let P be a locally flnite disk pattern on the complex plane C whose combinatorics are described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function £ : E! [0;…=2] on the set of edges. Let P ⁄ be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function. We show that P and P ⁄ difier only by a euclidean similarity. In particular, when the dihedral angle function £ is identically zero, this yields the rigidity theorems of B. Rodin and D. Sullivan, and of O. Schramm, whose arguments rely essentially on the pairwise disjointness of the interiors of the disks. The approach here is analytical, and uses the maximum principle, the concept of vertex extremal length, and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern. A similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane, where the proof follows by a simple use of the maximum principle. Also, we have a uniformization result for disk patterns. In a future paper, the techniques of this paper will be extended to the case when 0 • £ <… . In particular, we will show a rigidity property for a class of inflnite convex polyhedra in the 3-dimensional hyperbolic space.

Möbius invariance of knot energy

A physically natural potential energy for simple closed curves in R 3 is shown to be invariant under Mobius 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 and are regular. Finally, the number of knot types with energy less than any constant M is estimated

Second derivatives of circle packings and conformal mappings

AbstractWilliam Thurston conjectured that the Riemann mapping functionf from a simply connected region Ω onto the unit disk % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFdcpraaa!41AC!

Rigidity of circle domains whose boundary hasσ-finite linear measure

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On the distortion of relative circle domain isomorphisms

can be approximated as follows. Almost fill Ω with circles of radius ɛ packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFdcpraaa!41AC!