No abstract

No abstract

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 (resp. CP hyperbolic) if there is a locally finite disk packing P in the plane (resp. the unit disk) with contacts graph G. Several criteria for deciding whether G 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 on G is recurrent, then G is CP parabolic. Conversely, if G has bounded valence and the random walk on G is transient, then G is CP hyperbolic. We also give a new proof that G 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 if G is CP hyperbolic and D is any simply connected proper subdomain of the plane, then there is a disk packing P with contacts graph G such that P is contained and locally finite in D.

If the domain of definition of 1 is a finite, disjoint union of oriented Jordan curves in C (and 1 has no fixed points), then the index of 1 is defined as the sum of the indices of the restrictions of 1 to the individual curves.

We prove the following results: A unique smooth solution exists for a short time for the heat equation associated with the Möbius energy of loops in a euclidean space, starting with any simple smooth loop. 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 the author, we show that any local minimizer of the energy must be smooth. Circles are the only two‐dimensional critical loops with cube‐integrable curvature. The technique also applies to a family of other knot energies. Similar problems are open for energies of surfaces or, more generally, for embedded submanifolds in a fixed Riemannian manifold. © 2000 John Wiley & Sons, Inc.