Square integrable differentials on Riemann surfaces and quasiconformal mappings

Minda, Carl David

doi:10.1090/S0002-9947-1974-0340589-3

Square integrable differentials on Riemann surfaces and quasiconformal mappings
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Trans. Amer. Math. Soc. 195 (1974), 365-381 Request permission

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 $K(f)^{1/2}$ 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

Robert D. M. Accola, The bilinear relation on open Riemann surfaces, Trans. Amer. Math. Soc. 96 (1960), 143–161. MR 125216, DOI 10.1090/S0002-9947-1960-0125216-2
Robert D. M. Accola, Differentials and extremal length on Riemann surfaces, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 540–543. MR 118829, DOI 10.1073/pnas.46.4.540
Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
Irwin Kra, Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0357775
O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin-New York, 1965 (German). MR 0188434
Albert Marden, The weakly reproducing differentials on open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I No. 359 (1965), 32. MR 0188430
Mitsuru Nakai, On a ring isomorphism induced by quasiconformal mappings, Nagoya Math. J. 14 (1959), 201–221. MR 102589
Mitsuru Nakai, A function algebra on Riemann surfaces, Nagoya Math. J. 15 (1959), 1–7. MR 106997
Mitsuru Nakai, Purely algebraic characterization of quasiconformality, Proc. Japan Acad. 35 (1959), 440–443. MR 112968

On a problem of Roy den on quasiconformal equivalence of Riemann surfaces

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Mitsuru Nakai, Algebraic criterion on quasiconformal equivalence of Riemann surfaces, Nagoya Math. J. 16 (1960), 157–184. MR 110801
Albert Pfluger, Theorie der Riemannschen Flächen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0084031
Richard Rochberg, Almost isometries of Banach spaces and moduli of planar domains, Pacific J. Math. 49 (1973), 455–466. MR 379866, DOI 10.2140/pjm.1973.49.445
Richard Rochberg, Almost isometries of Banach spaces and moduli of Riemann surfaces, Duke Math. J. 40 (1973), 41–52. MR 310234
Burton Rodin and Leo Sario, Principal functions, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. In collaboration with Mitsuru Nakai. MR 0229812
H. L. Royden, On a class of null-bounded Riemann surfaces, Comment. Math. Helv. 34 (1960), 52–66. MR 110795, DOI 10.1007/BF02565926
L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. MR 0264064
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009