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 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Holomorphic motions and Teichmüller spaces

Author(s): C. J. Earle; I. Kra; S. L. KrushkalЬ
Journal: Trans. Amer. Math. Soc. 343 (1994), 927-948.
MSC: Primary 32G15; Secondary 30F60
MathSciNet review: 1214783
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Abstract: We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane $ \hat{\mathbb{C}}$ extends to a holomorphic motion of $ \widehat{\mathbb{C}}$. As a consequence we prove that every holomorphic map of the unit disc into Teichmüller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmüller metric.

References:

[1]
K. Astala and G.J. Martin, Holomorphic motions, (to appear). MR 1886611 (2003e:37055)

[2]
C.A. Berenstein and R. Gay, Complex variables, an introduction, Graduate Texts in Math., vol. 125, Springer-Verlag, 1991. MR 1107514 (92f:30001)

[3]
L. Bers, Extremal quasiconformal mappings, Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 27-52. MR 0285706 (44:2924)

[4]
-, Holomorphic families of isomorphisms of Möbius groups, J. Math. Kyoto Univ. 26 (1986), 73-76. MR 827159 (87j:32067)

[5]
L. Bers and H.L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), 243-257. MR 857675 (88i:30034)

[6]
S. Dineen, The Schwarz lemma, Oxford Univ. Press, Oxford, 1989. MR 1033739 (91f:46064)

[7]
C.J. Earle, On holomorphic cross-sections in Teichmüller spaces, Duke Math. J. 36 (1969), 409-416. MR 0254233 (40:7442)

[8]
-, On holomorphic cross-sections in Teichmüller spaces. II, Rice Univ. Studies 56 (1970), 109-113. MR 0294632 (45:3700)

[9]
-, On the moduli of closed Riemann surfaces with symmetries, Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 119-130. MR 0296282 (45:5343)

[10]
-, Teichmüller theory, Discrete Groups and Automorphic Functions, Academic Press, 1977, pp. 143-162.

[11]
C.J. Earle and J. Eells, Jr., On the differential geometry of Teichmüller spaces, J. Analyse Math. 19 (1967), 35-52. MR 0220923 (36:3975)

[12]
C.J. Earle and I. Kra, On sections of some holomorphic families of closed Riemann surfaces, Acta Math. 137 (1976), 49-79. MR 0425183 (54:13140)

[13]
H.M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Math., vol. 71, Springer-Verlag, New York, Heidelberg and Berlin, 1992. MR 1139765 (93a:30047)

[14]
F.P. Gardiner, Teichmüller theory and quadratic differentials, Wiley-Interscience, New York, 1987. MR 903027 (88m:32044)

[15]
R.S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399-406. MR 0245787 (39:7093)

[16]
L.A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in Holomorphy, North-Holland Math. Studies, vol. 34, North-Holland, Amsterdam, 1979, pp. 345-406. MR 520667 (80j:32043)

[17]
M. Hervé, Analyticity in infinite dimensional spaces, de Gruyter, Berlin, 1989. MR 986066 (90f:46074)

[18]
E. Hille and R. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, RI, 1957. MR 0089373 (19:664d)

[19]
I. Kra, On new kinds of Teichmüller spaces, Israel J. Math 6 (1973), 237-25. MR 0364632 (51:886)

[20]
-, On the Nielsen-Thurston-Bers type of some of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231-270. MR 611385 (82m:32019)

[21]
-, Families of univalent functions and Kleinian groups, Israel J. Math. 60 (1987), 89-127. MR 931871 (89d:30008)

[22]
S. Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 278 (1959), 1-35. MR 0148906 (26:6402)

[23]
S.L. Krushkal', Extremal problems for conformal and quasiconformal mappings of Riemann surfaces, Sibirsk Mat. Z. 14 (1973), 582-598 and 694. MR 0340591 (49:5343)

[24]
R. Mañé, P. Sad, and D. Sullivan, On dynamics of rational maps, Ann. Sci. École Norm. Sup. 16 (1983), 193-217. MR 732343 (85j:58089)

[25]
C. McMullen, Amenability. Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), 95-127. MR 999314 (90e:30048)

[26]
S. Nag, Non-geodesic discs embedded in Teichmüller spaces, Amer. J. Math. 104 (1982), 399-408. MR 654412 (83e:32027)

[27]
-, The complex analytic theory of Teichmüller space, Wiley-Interscience, New York, 1988.

[28]
B. O'Byrne, On Finsler geometry and applications to Teichmüller spaces, Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 317-328. MR 0286141 (44:3355)

[29]
E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to Analysis, Academic Press, San Diego, 1974, pp. 375-391. MR 0361065 (50:13511)

[30]
H.L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton, Univ. Press, Princeton, NJ, 1971, pp. 369-383. MR 0288254 (44:5452)

[31]
-, The Ahlfors-Schwarz lemma: the case of equality, J. Analyse Math. 46 (1986), 261-270. MR 861705 (87m:30045)

[32]
W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987. MR 924157 (88k:00002)

[33]
Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347-355. MR 1037218 (91f:58078)

[34]
-, Invariant extensions of holomorphic motions, Abstracts Amer. Math. Soc. 13 (1992), 259.

[35]
K. Strebel, On quasiconformal mappings of open Riemann surfaces, Comment. Math. Helv. 53 (1978), 301-321. MR 505549 (81i:30041)

[36]
D. Sullivan, Quasiconformal homeomorphisms and dynamics. I: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), 401-418. MR 819553 (87i:58103)

[37]
-, Quasiconformal homeomorphisms and dynamics. II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), 243-260. MR 806415 (87i:58104)

[38]
-, Quasiconformal homeomorphisms and dynamics. III: Topological conjugacy classes of analytic diffeomorphisms (to appear).

[39]
D. Sullivan and W.P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), 243-257. MR 857674 (88i:30033)

[40]
H. Tanigawa, Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces, Nagoya Math. J. 127 (1992), 117-128. MR 1183655 (93i:32027)

[41]
Li Zhong, Nonuniqueness of geodesics in infinite dimensional Teichmüller spaces, Complex Variables Theory Appl. 16 (1991), 261-272. MR 1105200 (92c:32024)

[42]
A. Douady, Prolongement de mouvements holomorphes [d'après Slodkowski et autres], Séminaire N. Bourbaki, 1993-1994, Exposé 775, pp. 1-12.

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Additional Information:

DOI: 10.1090/S0002-9947-1994-1214783-6
PII: S0002-9947-1994-1214783-6
Copyright of article: Copyright 1994, American Mathematical Society




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