Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Earle, Clifford; Gardiner, Frederick

doi:10.1090/S0002-9947-96-01490-0

Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces
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by Clifford J. Earle and Frederick P. Gardiner PDF

Trans. Amer. Math. Soc. 348 (1996), 1163-1190 Request permission

Abstract:

Let $X$ and $Y$ be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by a quasiconformal homeomorphism of $X$ onto $Y$. These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.

References

Richard Arens, The closed maximal ideals of algebras of functions holomorphic on a Riemann surface, Rend. Circ. Mat. Palermo (2) 7 (1958), 245–260. MR 105501, DOI 10.1007/BF02849323
Lipman Bers, Automorphic forms and Poincaré series for infinitely generated Fuchsian groups, Amer. J. Math. 87 (1965), 196–214. MR 174737, DOI 10.2307/2373231
Seán Dineen, The Schwarz lemma, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR 1033739
Clifford J. Earle, On holomorphic cross-sections in Teichmüller spaces, Duke Math. J. 36 (1969), 409–415. MR 254233
Clifford J. Earle, The integrable automorphic forms as a dual space, Modular functions in analysis and number theory, Lecture Notes Math. Statist., vol. 5, Univ. Pittsburgh, Pittsburgh, PA, 1983, pp. 30–40. MR 732960
Clifford J. Earle, The integrable automorphic forms as a dual space. II, Complex Variables Theory Appl. 12 (1989), no. 1-4, 153–158. MR 1040916, DOI 10.1080/17476938908814361
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
Clifford J. Earle and Irwin Kra, On isometries between Teichmüller spaces, Duke Math. J. 41 (1974), 583–591. MR 348098
C. J. Earle, I. Kra, and S. L. Krushkal′, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927–948. MR 1214783, DOI 10.1090/S0002-9947-1994-1214783-6
C. J. Earle and A. Marden, Projections to automorphic functions, Proc. Amer. Math. Soc. 19 (1968), 274–278. MR 224813, DOI 10.1090/S0002-9939-1968-0224813-6
H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
Frederick P. Gardiner, Approximation of infinite-dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 282 (1984), no. 1, 367–383. MR 728718, DOI 10.1090/S0002-9947-1984-0728718-7
Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
A. Harrington and M. Ortel, The dilatation of an extremal quasi-conformal mapping, Duke Math. J. 43 (1976), no. 3, 533–544. MR 425117, DOI 10.1215/S0012-7094-76-04343-X
Lawrence A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) North-Holland Math. Stud., vol. 34, North-Holland, Amsterdam-New York, 1979, pp. 345–406. MR 520667
Marvin I. Knopp, A corona theorem for automorphic forms and related results, Amer. J. Math. 91 (1969), 599–618. MR 251219, DOI 10.2307/2373341
Clinton J. Kolaski, Isometries of Bergman spaces over bounded Runge domains, Canadian J. Math. 33 (1981), no. 5, 1157–1164. MR 638372, DOI 10.4153/CJM-1981-087-1
Irwin Kra, Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0357775
N. Lakic, An isometry theorem for quadraric differentials on Riemann surfaces of finite genus, Trans. Amer. Math. Soc., to appear.
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Albert Marden and Howard Masur, A foliation of Teichmüller space by twist invariant disks, Math. Scand. 36 (1975), no. 2, 211–228. MR 393584, DOI 10.7146/math.scand.a-11572
Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
Kung Fu Ng, On a theorem of Dixmier, Math. Scand. 29 (1971), 279–280 (1972). MR 338740, DOI 10.7146/math.scand.a-11054
T. Ohsawa, On the analytic structure of certain infinite dimensional Teichmüller spaces.
H. L. Royden, Automorphisms and isometries of Teichmüller space, Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings (Braşov, 1969) Publ. House of the Acad. of the Socialist Republic of Romania, Bucharest, 1971, pp. 273–286. (errata insert). MR 0302894
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
—, Functional Analysis, Second Edition, McGraw-Hill, New York, 1992.
A. L. Shields and D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302. MR 283559, DOI 10.1090/S0002-9947-1971-0283559-3
Kurt Strebel, On quadratic differentials and extremal quasi-conformal mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 223–227. MR 0507848
Kurt Strebel, On the existence of extremal Teichmueller mappings, J. Analyse Math. 30 (1976), 464–480. MR 440031, DOI 10.1007/BF02786734
Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
Harumi Tanigawa, Holomorphic families of geodesic discs in infinite-dimensional Teichmüller spaces, Nagoya Math. J. 127 (1992), 117–128. MR 1183655, DOI 10.1017/S0027763000004128