Journal d’Analyse Mathématique

, Volume 85, Issue 1, pp 371–396

# Earthquakes and circle packings

• G. Brock Williams
Article

## Abstract

We prove that earthquakes on hyperbolic surfaces can be approximated by discrete earthquakes constructed using circle packings. Consequently, we obtain a combinatorial version of Thurston’s Earthquake Theorem. Any surface can be approximated by combinatorial earthquakes of a packable surface. This provides a controlled combinatorial method for deforming hyperbolic surfaces.

## Keywords

Boundary Vertex Hyperbolic Surface Interior Vertex Circle Packing Geodesic Lamination
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. [1]
E. M. Andreev,Convex polyhedra in Lobacevskii space, Mat. Sb. (N.S.)10 (1970), 413–440 (English).
2. [2]
E. M. Andreev,Convex polyhedra of finite volume in Lobacevskii space, Math. USSR Sbornik12 (1970), 255–259 (English).
3. [3]
R. W. Barnard and G. B. Williams,Combinatorial excursions in moduli space, Pacific J. Math., to appear.Google Scholar
4. [4]
A. F. Beardon and K. Stephenson,The uniformization theorem for circle packings, Indian Univ. Math. J.39 (1990), 1383–1425.
5. [5]
F. Bonahon,Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Amer. Math. Soc.330 (1992), 69–95.
6. [6]
P. L. Bowers and K. Stephenson,Uniformizing dessins and Belyi maps via circle packing, preprint.Google Scholar
7. [7]
P. L. Bowers and K. Stephenson,The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense, Math. Proc. Camb. Phil. Soc.111 (1992), 487–513.
8. [8]
P. L. Bowers and K. Stephenson,Circle packings in surfaces of finite type: An in situ approach with application to moduli, Topology32 (1993), 157–183.
9. [9]
P. L. Bowers and K. Stephenson,A regular pentagonal tiling of the plane, Conform. Geom. Dynam.1 (1997), 58–68.
10. [10]
R. Brooks,Circle packings and co-compact extensions of Kleinian groups, Invent. Math.86 (1986), 461–469.
11. [11]
R. Brooks,Some relations between graph theory and Riemann surfaces, inProceedings of the Ashkelon Workshop on Complex Function Theory (L. Zalcman, ed.), Israels Math. Conf. Proc.11 (1997), 61–73.Google Scholar
12. [12]
P. Buser,Geometry and Spectra of Compact Riemann Surfaces, Birkhäusers, Boston, 1992.
13. [13]
A. J. Casson and S. A. Bleiler,Automorphisms of Surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, Vol. 9, Cambridge University Press, Cambridge, 1988.
14. [14]
C. Collins and K. Stephenson,A circle packing algorithm, preprint.Google Scholar
15. [15]
T. Dubejko and K. Stephenson,Circle packing: Experiments in discretes analytic function theory, Experiment. Math.4 (1995), 307–348.
16. [16]
A. Fathi et al.,Travaux de Thurston sur les surfaces, Astériques (Orsay Séminaire), 1979.Google Scholar
17. [17]
F. P. Gardiner and N. Lakic,Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, Vol. 76, American Mathematical Society, Providence, RI, 2000.
18. [18]
Zheng-Xu He and O. Rodin,Convergence of circle packing of finite valence to Riemann mappings, Comm. Anal. Geom.1 (1993), 31–41.
19. [19]
Zheng-Xu He and O. Schramm,On the convergence of circle packings to the Riemann map, Invent. Math.125 (1996), 285–305.
20. [20]
Y. Imayoshi and M. Taniguchi,An Introduction to Teichmüller Spaces, Springer-Verlag, Berlin, 1992.
21. [21]
S. P. Kerckhoff,The Nielsen realization problem, Ann. of Math.117 (1983), 235–265.
22. [22]
P. Koebe,Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig. Math.-Phys. Kl.88 (1936), 141–164.Google Scholar
23. [23]
S. G. Krantz,Conformal mappings, American Scientist87 (1999), 436–445.Google Scholar
24. [24]
O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, second edn., Springer-Verlag, Berlin-Heidelberg-New York, 1973.
25. [25]
D. Minda and B. Rodin,Circle packing and Riemann surfaces, J. Analyse Math.57 (1991), 221–249.
26. [26]
A. Papadopoulos,On Thurston’s boundary of Teichmüller space and the extension of earthquakes, Topology Appl.41 (1991), 147–177.
27. [27]
B. Rodin and D. Sullivan,The convergence of circle packings to the Riemann mapping, J. Differential Geom.26 (1987), 349–360.
28. [28]
K. Stephenson,Circle packing and discretes analytic function theory, preprint.Google Scholar
29. [29]
K. Stephenson,A probabilistic proof of Thurston’s conjecture on circle packings, Rend. Sem. Mat. Fis. Milano66 (1996), 201–291.
30. [30]
K. Stephenson,The approximation of conformal structures via circle packing, inComputational Methods and Function Theory 1997 (Nicosia) (N. Papamichael, S. Ruscheweyh and E. B. Saff, eds.), Ser. Approx. Decompos., World Scientific, River Edge, NJ, 1997, pp. 551–582.Google Scholar
31. [31]
W. Thurston,The Geometry and Topology of 3-Manifolds, Princeton University Notes, preprint.Google Scholar
32. [32]
W. Thurston,The finite Riemann mapping theorem, 1985, Invited talk, An International Symposium at Purdue University on the occasion of the proof of the Bieberbachs conjecture, March 1985.Google Scholar
33. [33]
W. Thurston,Earthquakes in two-dimensional hyperbolic geometry, inLow-dimensional Topology and Kleinian Groups (Conventry/Durham, 1984), London Math. Soc. Lecture Note Ser., Vol. 112, Cambridge University Press, 1986, pp. 90–112.Google Scholar
34. [34]
W. Thurston,On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.)19 (1988), 417–431.
35. [35]
G. B. Williams,Discrete conformal welding, Ph.D. thesis, University of Tennessee, Knoxville, May 1999.Google Scholar
36. [36]
G. B. Williams,Approximating quasisymmetries using circle packings, Discrete Comput. Geom.25 (2001), 103–124.