An isometry theorem for quadratic differentials on Riemann surfaces of finite genus
Author:
Nikola Lakic
Journal:
Trans. Amer. Math. Soc. 349 (1997), 29512967
MSC (1991):
Primary 32G15; Secondary 30C62, 30C75
MathSciNet review:
1390043
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Abstract: Assume both and are Riemann surfaces which are subsets of compact Riemann surfaces and respectively, and that the set has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on and are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of onto the Teichmüller space of is induced by some quasiconformal map of onto . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.
 [A]
Lars
V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math.
86 (1964), 413–429. MR 0167618
(29 #4890)
Lars
Ahlfors, Correction to “Finitely generated Kleinian
groups”, Amer. J. Math. 87 (1965), 759. MR 0180675
(31 #4906)
 [B]
Lipman
Bers, An approximation theorem, J. Analyse Math.
14 (1965), 1–4. MR 0178287
(31 #2545)
 [EG]
Armen
Edigarian, A remark on the Lempert theorem, Univ. Iagel. Acta
Math. 32 (1995), 83–88. MR 1345124
(96h:32034)
 [EK]
Clifford
J. Earle and Irwin
Kra, On holomorphic mappings between Teichmüller spaces,
Contributions to analysis (a collection of papers dedicated to Lipman
Bers), Academic Press, New York, 1974, pp. 107–124. MR 0430319
(55 #3324)
 [F]
Otto
Forster, Lectures on Riemann surfaces, Graduate Texts in
Mathematics, vol. 81, SpringerVerlag, New York, 1981. Translated from
the German by Bruce Gilligan. MR 648106
(83d:30046)
 [FK]
H.
M. Farkas and I.
Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics,
vol. 71, SpringerVerlag, New York, 1992. MR 1139765
(93a:30047)
 [G1]
Frederick
P. Gardiner, Approximation of infinitedimensional
Teichmüller spaces, Trans. Amer. Math.
Soc. 282 (1984), no. 1, 367–383. MR 728718
(85f:30082), http://dx.doi.org/10.1090/S00029947198407287187
 [G2]
Frederick
P. Gardiner, Teichmüller theory and quadratic
differentials, Pure and Applied Mathematics (New York), John Wiley
& Sons Inc., New York, 1987. A WileyInterscience Publication. MR 903027
(88m:32044)
 [H]
John
Hamal Hubbard, Sur les sections analytiques de la courbe
universelle de Teichmüller, Mem. Amer. Math. Soc.
4 (1976), no. 166, ix+137. MR 0430321
(55 #3326)
 [K]
Irwin
Kra, Automorphic forms and Kleinian groups, W. A. Benjamin,
Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. MR 0357775
(50 #10242)
 [R]
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)
 [A]
 L. V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413429; 87(1965), 759. MR 29:4890; MR 31:4906
 [B]
 L. Bers, An approximation theorem, J. Anal. Math. 14 (1965), 14. MR 31:2545
 [EG]
 C. J. Earle and F.P. Gardiner, Geometric isomorphisms between infinite dimensional
Teichmüller spaces, Trans. Am. Math. Soc. 348 (1996), 11631190. MR 96h:32034
 [EK]
 C. J. Earle and I. Kra, On holomorphic mappings between Teichmüller spaces, Contributions to Analysis, Academic Press, New York (1974), 107124. MR 55:3324
 [F]
 O. Forster, Lectures on Riemann Surfaces, SpringerVerlag, New York, Heidelberg, Berlin, 1981. MR 83d:30046
 [FK]
 H. M. Farkas and I.Kra, Riemann Surfaces, 2nd ed., SpringerVerlag, New York, Heidelberg, Berlin, 1992. MR 93a:30047
 [G1]
 F. P. Gardiner, Approximation of infinite dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 282 (1984), 367383. MR 85f:30082
 [G2]
 F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, WileyInterscience, New York, 1987. MR 88m:32044
 [H]
 J. Hubbard, Sur les sections analytiques de la courbe universelle de Teichmüller, Mem. Am. Math. Soc. No. 166 (1976), p. 116. MR 55:3326
 [K]
 I. Kra, Automorphic Forms and Kleinian Groups, Benjamin, Reading, Massachusetts, 1972. MR 50:10242
 [R]
 H. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces, Ann. Math. Stud. 66, Princeton University Press, 1971, 369384. MR 44:5452
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Additional Information
Nikola Lakic
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853
Email:
Nikola@math.cornell.edu
DOI:
http://dx.doi.org/10.1090/S0002994797017716
PII:
S 00029947(97)017716
Received by editor(s):
December 1, 1994
Received by editor(s) in revised form:
February 26, 1996
Article copyright:
© Copyright 1997 American Mathematical Society
