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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Riemann surfaces and bounded holomorphic functions


Author: Walter Pranger
Journal: Trans. Amer. Math. Soc. 259 (1980), 393-400
MSC: Primary 30F99; Secondary 14F05, 32L05
MathSciNet review: 567086
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Abstract: The principal result of this article asserts the equivalence of the following four conditions on a hyperbolic Riemann surface X:

(a) the following set $ z\vert\,\vert f(z)\vert\, \leqslant \,{\text{sup}}\,\vert f\vert$ on K for every bounded holomorphic section f of $ \xi $ is compact for every unitary vector bundle $ \xi $ and every compact set K;

(b) every unitary line bundle has nontrivial bounded holomorphic sections and the condition in (a) holds for $ \xi \, = \,{i_d}$;

(c) every unitary line bundle has nontrivial bounded holomorphic sections and X is regular for potential theory;

(d) every unitary line bundle has nontrivial bounded holomorphic sections and X is its own B-envelope of holomorphy.

If X is a subset of C, these are also equivalent to the following:

(e) for every unitary line bundle $ \xi $ the bounded holomorphic sections are dense in the holomorphic sections.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0567086-7
PII: S 0002-9947(1980)0567086-7
Article copyright: © Copyright 1980 American Mathematical Society