Skip to main content
SpringerLink
Log in

Existence and uniqueness of packings with specified combinatorics

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Generalizations of the Andreev-Thurston circle packing theorem are proved. One such result is the following.

Let G=G(V) be a planar graph, and for each vertex v ∈ V, let ℱ v be a proper 3-manifold of smooth topological disks in S 2,with the property that the pattern of intersection of any two sets A, B ∈ ℱ v is topologically the pattern of intersection of two circles (i.e., there is a homeomorphism h:S 2→S 2 taking A and B to circles). Then there is a packing P=(P v :v ∈V)whose nerve is G, and which satisfies P v ∈ ℱ ν for v ∈ V. (‘The nerve is G’ means that two sets, P v ,P u ,touch, if, and only if, u ↔ v is an edge in G.)

In the case whereG is the 1-skeleton of a triangulation, we also give a precise uniqueness statement. Various examples and applications are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. M. Andreev,On convex polyhedra in Lobačevskii spaces, Mat. Sb. (N.S.)81(123) (1970), 445–478; English transl. in Math. USSR Sb.10 (1970), 413–440.

    MathSciNet  Google Scholar 

  2. E. M. Andreev,On convex polyhedra of finite volume in Lobačevskii space, Mat. Sb. (N.S.)83(125) (1970), 256–260; English transl. in Math. USSR Sb.12 (1970), 255–259.

    MathSciNet  Google Scholar 

  3. Zheng-Xu He,Solving Beltrami equations by circle packing, Trans. Am. Math. Soc., to appear.

  4. B. Rodin and D. Sullivan,The convergence of circle packings to the Riemann mapping, J. Differ. Geom.26 (1987), 349–360.

    MATH  MathSciNet  Google Scholar 

  5. O. Schramm,Packing two-dimensional bodies with prescribed combinatorics and applications to the construction of conformal and quasiconformal mappings, Ph.D. thesis, Princeton, 1990.

  6. O. Schramm,Rigidity of infinite (circle) packings, J. Am. Math. Soc., to appear.

  7. O. Schramm,How to cage an egg, preprint.

  8. E. Schulte,Analogues of Steinitz’s theorem about non-inscribable polytopes, inIntuitive Geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai, 48 (K. Böröczky and G. F. Tóth, eds.), North-Holland, Amsterdam, 1987, pp. 503–516.

    Google Scholar 

  9. W. P. Thurston,The Geometry and Topology of 3-Manifolds, Princeton University Notes.

  10. W. P. Thurston,The finite Riemann mapping theorem, invited talk in the International Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, March 1985.

Download references

Author information

Authors and Affiliations

  1. Mathematics Department C-012, University of California-San Diego, 92093, La Jolla, CA, USA

    Oded Schramm

Authors
  1. Oded Schramm
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schramm, O. Existence and uniqueness of packings with specified combinatorics. Israel J. Math. 73, 321–341 (1991). https://doi.org/10.1007/BF02773845

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773845

Keywords

Search

Navigation