Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Square integrable differentials on Riemann surfaces and quasiconformal mappings


Author: Carl David Minda
Journal: Trans. Amer. Math. Soc. 195 (1974), 365-381
MSC: Primary 30A52
MathSciNet review: 0340589
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ f:R \to R'$ is a quasiconformal homeomorphism of Riemann surfaces, then Marden showed that f naturally induces an isomorphism of the corresponding Hilbert spaces of square integrable first-order differential forms. It is demonstrated that this isomorphism preserves many important subspaces. Preliminary to this, various facts about the subspace of semiexact differentials are derived; especially, the orthogonal complement is identified. The norm of the isomorphism is $ K{(f)^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$ where $ K(f)$ is the maximal dilatation of f. In addition, f defines an isomorphism of the square integrable harmonic differentials and some important subspaces are preserved. It is shown that not all important subspaces are preserved. The relationship of this to other work is investigated; in particular, the connection with the work of Nakai on the isomorphism of Royden algebras induced by a quasiconformal mapping is explored. Finally, the induced isomorphisms are applied to the classification theory of Riemann surfaces to show that various types of degeneracy are quasiconformally invariant.


References [Enhancements On Off] (What's this?)

  • [1] R. D. M. Accola, The bilinear relation on open Riemann surfaces, Trans. Amer. Math. Soc. 96 (1960), 143-161. MR 23 #A2521. MR 0125216 (23:A2521)
  • [2] -, Differentials and extremal length on Riemann surfaces, Proc. Nat. Acad. Sci. U. S. A. 46 (1960), 540-543. MR 22 #9598. MR 0118829 (22:9598)
  • [3] L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Math. Series, no. 26, Princeton Univ. Press, Princeton, N. J., 1960. MR 22 #5729. MR 0114911 (22:5729)
  • [4] I. Kra, Automorphic forms and Kleinian groups, Benjamin, New York, 1972. MR 0357775 (50:10242)
  • [5] O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der math. Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin and New York, 1965. MR 32 #5872. MR 0188434 (32:5872)
  • [6] A. Marden, The weakly reproducing differentials on open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. AI No. 359 (1965), 32pp. MR 32 #5868. MR 0188430 (32:5868)
  • [7] M. Nakai, On a ring isomorphism induced by quasiconformal mappings, Nagoya Math. J. 14 (1959), 201-221. MR 21 #1379. MR 0102589 (21:1379)
  • [8] -, A function algebra on Riemann surfaces, Nagoya Math. J. 15 (1959), 1-7. MR 21 #5726. MR 0106997 (21:5726)
  • [9] -, Purely algebraic characterization of quasiconformality, Proc. Japan Acad. 35 (1959), 440-443. MR 22 #3813. MR 0112968 (22:3813)
  • [10] -, On a problem of Roy den on quasiconformal equivalence of Riemann surfaces, Proc. Japan Acad. 36 (1960), 33-37. MR 22 #2691.
  • [11] -, Algebraic criterion on quasiconformal equivalence of Riemann surfaces, Nagoya Math. J. 16 (1960), 157-184. MR 22 #1669. MR 0110801 (22:1669)
  • [12] A. Pfluger, Theorie der Riemannschen Flächen, Die Grundlehren der math. Wissenschaften, Band 89, Springer-Verlag, Berlin, 1957. MR 18, 796. MR 0084031 (18:796f)
  • [13] R. Rochberg, Almost isometries of Banach spaces and moduli of planar domains (to appear). MR 0379866 (52:771)
  • [14] -, Almost isometries of Banach spaces and moduli of Riemann surfaces, Duke Math. J. 40 (1973), 41-52. MR 0310234 (46:9336)
  • [15] B. Rodin and L. Sario, Principal functions, Van Nostrand, Princeton, N. J., 1968. MR 37 #5378. MR 0229812 (37:5378)
  • [16] H. L. Royden, On a class of null-bounded Riemann surfaces, Comment. Math. Helv. 34 (1960), 52-66. MR 22 # 1663. MR 0110795 (22:1663)
  • [17] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der math. Wissenschaften, Band 164, Springer-Verlag, New York and Berlin, 1970. MR 41 #8660. MR 0264064 (41:8660)
  • [18] J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., no. 229, Springer-Verlag, New York, 1971. MR 0454009 (56:12260)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A52

Retrieve articles in all journals with MSC: 30A52


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0340589-3
PII: S 0002-9947(1974)0340589-3
Keywords: Square integrable differential forms, semiexact differentials, classification theory, quasiconformal mappings, isomorphisms of differentials
Article copyright: © Copyright 1974 American Mathematical Society