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Analysis 27, 213–225 (2007) / DOI 10.1524/anly.2007.27.2–3.213 c© Oldenbourg Wissenschaftsverlag, München 2007 Bounded pointwise approximation on open Riemann surfaces A. Boivin∗, B. Jiang Received: June 13, 2006 Dedicated to the memory of Gerald Schmieder. Summary: A characterization of those open sets U on an open Riemann surface for which A(U) is pointwise boundedly dense in H∞(U) is obtained, thus generalizing to Riemann surfaces results of T. W. Gamelin and J. Garnett. 1 Introduction LetR be an open (i.e. non-compact) Riemann surface. Let U be an open

New Results and Applications

RIEMANN SURFACES §1. Simply Connected Surfaces The first three sections will present an overview of some background mate- rial. If Vee is an open set of complex numbers, a function f: V --7 C is called holomorphic (or "complex analytic") if the first derivative z ~ f'(z) == lim (f(z + h) - f(z))jh h-1-0 is defined and continuous as a function from V to C, or equivalently if f has a power series expansion about any point Zo E V which converges to f in some neighborhood of zoo (See, for example, Ahlfors [1966].) Such a function is conformal if the derivative f' (z

-Shields conjecture for cyclicity in the Dirichlet space, Adv. Math. 222, 2196-2214 (2009). [10] O. Forster, Lectures on Riemann Surfaces, translated from the 1977 German original by Bruce Gilligan, reprint of the 1981 English translation, Grad. Texts in Math., vol. 81, Springer-Verlag, New York, 1991. [11] C. Gu, Reducing subspaces of weighted shifts with operator weights. Bull. Korean Math. Soc. 53, 1471-1481 (2016). [12] C. Gu, Common reducing subspaces of several weighted shifts with operator weights, submitted. [13] K. Guo, H. Huang, On multiplication operators on the Bergman

Jacobi Varieties

282 III. Mathematical Concepts III.79 Riemann Surfaces Alan F. Beardon Let D be a region (that is, a connected open set) in the complex plane. If f is a complex-valued function defined on D, then we can define its derivative just as we would for real-valued functions defined on subsets of R: the derivative of f at w is the limit as z tends to w of the “difference quotient” (f (z)− f(w))/(z−w). Of course, this limit does not necessarily exist, but if it exists for every w in D, then f is said to be analytic, or holomorphic, onD. Analytic functions have amazing

C H A P T E R I I Riemann Surfaces In current terminology Riemann surfaces are the domains of most general type which can be used to replace the complex plane in the theory of analytic functions of one complex variable. This is in strict accordance with the spirit of Riemann's own work, for Riemann was the first to recognize that plane regions are not sufficiently general to give a complete picture of the ideas that dominate function theory, even when restricted to a single variable. The present chapter is of a preparatory nature, being devoted mainly

METRIC RIEMANN SURFACES1 E. CALABI 1. INTRODUCTION The complex analytic structure of a Riemann surface Is Introduced In F. Klein's book [4] by considering the Isothermal parameters In a differ­ entiable, orlentable surface In Euclidean 3-space. This method of course can be applied more generally to any abstract 2-dimensional, orlentable Riemannian manifold. Admittedly this purely auxiliary metric structure on the Riemann surface is somewhat artificial, as evidenced by the fact that, beyond the proof of existence of local isothermal parameters, the function

COLLARS ON RIEMANN SURFACES Linda Keen Let S be a compact Riemann surface of genus g. The metric on S is induced by factoring the upper half plane U, which has the Poincare" metric on it, by the Fuchsian group G which represents S; S = U/G. In what follows all curves on S are assumed to be geodesic in this metric; moreover they are taken to be the unique geodesics in the free homotopy classes determined by the curves. A ll lengths and areas are given in this metric. In the study of Teichmuller space, much attention has been focused on the geodesics of S