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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
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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.

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Keywords: Square integrable differential forms, semiexact differentials, classification theory, quasiconformal mappings, isomorphisms of differentials
Article copyright: © Copyright 1974 American Mathematical Society