Riemann surfaces and bounded holomorphic functions

Pranger, Walter

doi:10.1090/S0002-9947-1980-0567086-7

Riemann surfaces and bounded holomorphic functions
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Trans. Amer. Math. Soc. 259 (1980), 393-400 Request permission

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 $z| |f(z)| \leqslant {\text {sup}} |f|$ 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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