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.
Similar content being viewed by others
References
E. M. Andreev,Convex polyhedra in Lobacevskii space, Mat. Sb. (N.S.)10 (1970), 413–440 (English).
E. M. Andreev,Convex polyhedra of finite volume in Lobacevskii space, Math. USSR Sbornik12 (1970), 255–259 (English).
R. W. Barnard and G. B. Williams,Combinatorial excursions in moduli space, Pacific J. Math., to appear.
A. F. Beardon and K. Stephenson,The uniformization theorem for circle packings, Indian Univ. Math. J.39 (1990), 1383–1425.
F. Bonahon,Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Amer. Math. Soc.330 (1992), 69–95.
P. L. Bowers and K. Stephenson,Uniformizing dessins and Belyi maps via circle packing, preprint.
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.
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.
P. L. Bowers and K. Stephenson,A regular pentagonal tiling of the plane, Conform. Geom. Dynam.1 (1997), 58–68.
R. Brooks,Circle packings and co-compact extensions of Kleinian groups, Invent. Math.86 (1986), 461–469.
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.
P. Buser,Geometry and Spectra of Compact Riemann Surfaces, Birkhäusers, Boston, 1992.
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.
C. Collins and K. Stephenson,A circle packing algorithm, preprint.
T. Dubejko and K. Stephenson,Circle packing: Experiments in discretes analytic function theory, Experiment. Math.4 (1995), 307–348.
A. Fathi et al.,Travaux de Thurston sur les surfaces, Astériques (Orsay Séminaire), 1979.
F. P. Gardiner and N. Lakic,Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, Vol. 76, American Mathematical Society, Providence, RI, 2000.
Zheng-Xu He and O. Rodin,Convergence of circle packing of finite valence to Riemann mappings, Comm. Anal. Geom.1 (1993), 31–41.
Zheng-Xu He and O. Schramm,On the convergence of circle packings to the Riemann map, Invent. Math.125 (1996), 285–305.
Y. Imayoshi and M. Taniguchi,An Introduction to Teichmüller Spaces, Springer-Verlag, Berlin, 1992.
S. P. Kerckhoff,The Nielsen realization problem, Ann. of Math.117 (1983), 235–265.
P. Koebe,Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig. Math.-Phys. Kl.88 (1936), 141–164.
S. G. Krantz,Conformal mappings, American Scientist87 (1999), 436–445.
O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, second edn., Springer-Verlag, Berlin-Heidelberg-New York, 1973.
D. Minda and B. Rodin,Circle packing and Riemann surfaces, J. Analyse Math.57 (1991), 221–249.
A. Papadopoulos,On Thurston’s boundary of Teichmüller space and the extension of earthquakes, Topology Appl.41 (1991), 147–177.
B. Rodin and D. Sullivan,The convergence of circle packings to the Riemann mapping, J. Differential Geom.26 (1987), 349–360.
K. Stephenson,Circle packing and discretes analytic function theory, preprint.
K. Stephenson,A probabilistic proof of Thurston’s conjecture on circle packings, Rend. Sem. Mat. Fis. Milano66 (1996), 201–291.
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.
W. Thurston,The Geometry and Topology of 3-Manifolds, Princeton University Notes, preprint.
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.
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.
W. Thurston,On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.)19 (1988), 417–431.
G. B. Williams,Discrete conformal welding, Ph.D. thesis, University of Tennessee, Knoxville, May 1999.
G. B. Williams,Approximating quasisymmetries using circle packings, Discrete Comput. Geom.25 (2001), 103–124.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Williams, G.B. Earthquakes and circle packings. J. Anal. Math. 85, 371–396 (2001). https://doi.org/10.1007/BF02788088
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02788088