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A rigidity theorem for holomorphic disks in Teichmüller space


Author: Hideki Miyachi
Journal: Proc. Amer. Math. Soc. 143 (2015), 2949-2957
MSC (2010): Primary 32G15; Secondary 32F10, 32T15, 32E35
DOI: https://doi.org/10.1090/S0002-9939-2015-12488-4
Published electronically: February 16, 2015
MathSciNet review: 3336619
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Abstract: In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give an improvement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains with $ C^{2}$-boundaries, but the rigidity property does not hold for product manifolds.


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  • [1] William Abikoff, Two theorems on totally degenerate Kleinian groups, Amer. J. Math. 98 (1976), no. 1, 109-118. MR 0396937 (53 #797)
  • [2] Zoltán M. Balogh and Mario Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000), no. 3, 504-533. MR 1793800 (2001k:32046), https://doi.org/10.1007/s000140050138
  • [3] Lipman Bers, An inequality for Riemann surfaces, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 87-93. MR 780038 (86h:30076)
  • [4] Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570-600. MR 0297992 (45 #7044)
  • [5] Francis Bonahon, Bouts des variétés hyperboliques de dimension $ 3$, Ann. of Math. (2) 124 (1986), no. 1, 71-158 (French). MR 847953 (88c:57013), https://doi.org/10.2307/1971388
  • [6] Francis Bonahon, Geodesic laminations on surfaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 1-37. MR 1810534 (2001m:57023), https://doi.org/10.1090/conm/269/04327
  • [7] J. F. Brock, Continuity of Thurston's length function, Geom. Funct. Anal. 10 (2000), no. 4, 741-797. MR 1791139 (2001g:57028), https://doi.org/10.1007/PL00001637
  • [8] Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1-149. MR 2925381, https://doi.org/10.4007/annals.2012.176.1.1
  • [9] Frederick P. Gardiner and Howard Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991), no. 2-3, 209-237. MR 1099913 (92f:32034)
  • [10] Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $ C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219-240. MR 0372252 (51 #8468)
  • [11] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. MR 919829 (89e:20070), https://doi.org/10.1007/978-1-4613-9586-7_3
  • [12] Yôichi Imayoshi and Hiroshige Shiga, A finiteness theorem for holomorphic families of Riemann surfaces, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 207-219. MR 955842 (89i:32046), https://doi.org/10.1007/978-1-4613-9611-6_15
  • [13] Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23-41. MR 559474 (81f:32029), https://doi.org/10.1016/0040-9383(80)90029-4
  • [14] Shoshichi Kobayashi, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460-480. MR 0232411 (38 #736)
  • [15] N. Lusin and J. Priwaloff, Sur l'unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup. (3) 42 (1925), 143-191 (French). MR 1509265
  • [16] Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381-386. MR 802500 (87c:30062), https://doi.org/10.5186/aasfm.1985.1042
  • [17] Howard A. Masur and Michael Wolf, Teichmüller space is not Gromov hyperbolic, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 259-267. MR 1346811 (96f:30048)
  • [18] Yair N. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), no. 2, 559-626. MR 1689341 (2000f:30028), https://doi.org/10.2307/120976
  • [19] Hideki Miyachi, Unification of extremal length geometry on Teichmüller space via intersection number, Math. Z. 278 (2014), no. 3-4, 1065-1095. MR 3278905, https://doi.org/10.1007/s00209-014-1346-y
  • [20] Hideki Miyachi, Mappings which are conservative with the Gromov product at infinity, preprint, ArXiv.org http://arxiv.org/abs/1306.1424
  • [21] Ken'ichi Ohshika, Limits of geometrically tame Kleinian groups, Invent. Math. 99 (1990), no. 1, 185-203. MR 1029395 (91c:30087), https://doi.org/10.1007/BF01234417
  • [22] R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770 (94b:57018)
  • [23] H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J., 1971, pp. 369-383. MR 0288254 (44 #5452)
  • [24] Hiroshige Shiga, On analytic and geometric properties of Teichmüller spaces, J. Math. Kyoto Univ. 24 (1984), no. 3, 441-452. MR 766636 (86c:32024)
  • [25] Hiroshige Shiga, Remarks on holomorphic families of Riemann surfaces, Tohoku Math. J. (2) 38 (1986), no. 4, 539-549. MR 867060 (88b:32052), https://doi.org/10.2748/tmj/1178228406
  • [26] Harumi Tanigawa, Holomorphic mappings into Teichmüller spaces, Proc. Amer. Math. Soc. 117 (1993), no. 1, 71-78. MR 1113649 (93c:32029), https://doi.org/10.2307/2159700
  • [27] W. Thurston, The geometry and Topology of Three-Manifolds, http://www.msri.org/publications/books/gt3m/.
  • [28] M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894 (22 #5712)

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Additional Information

Hideki Miyachi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan

DOI: https://doi.org/10.1090/S0002-9939-2015-12488-4
Keywords: Teichm\"uller space, Teichm\"uller distance, Kobayashi distance, pseudoconvex domains
Received by editor(s): December 26, 2013
Received by editor(s) in revised form: January 3, 2014, and February 8, 2014
Published electronically: February 16, 2015
Additional Notes: The author was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 21540177.
Dedicated: This paper is dedicated to Professor Hiroshige Shiga on the occasion of his 60th birthday.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2015 American Mathematical Society

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