We construct actions of fundamental groups of Riemann surfaces by automorphisms of the complex hyperbolic plane, which realize all possible values of Toledo's invariant. For integer values of these actions are discrete embeddings. The quotient complex hyperbolic surfaces are disc bundles over Riemann surfaces, whose topological type is determined in terms of .

The only connected minimal surfaces foliated by circles and lines are domains on one of the following surfaces: the helicoid, the catenoid, the plane, and the examples of Riemann ([Ri] p329-33, [En] p403-6, [Ni] p85-6). All these surfaces are complete and embedded. Topologically they are planar domains: the helicoid is simply-connected, the catenoid is an annulus (conformally a twice-punctured ...

We define and prove basic properties of Riemann surfaces, which we follow with a discussion of divisors and an elementary proof of the RiemannRoch theorem for compact Riemann surfaces. The Riemann-Roch theorem is used to prove the existence of a holomorphic embedding from any compact Riemann surface into n-dimensional complex projective space Pn. Using comparison principles such as Chow’s theor...

The level set of an elliptic function is a doubly periodic point set in C. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in C2 and its sections (“cuts”) by C. We give S a crystallographic isometry in C2 by defining a fundamental surface element as a conformal map of triangular domains and S as its extension by reflections in the triangle edg...

We compactify M(atrix) theory on Riemann surfaces Σ with genus g > 1. Following [1], we construct a projective unitary representation of π1(Σ) realized on L (H), with H the upper half–plane. As a first step we introduce a suitably gauged sl2(R) algebra. Then a uniquely determined gauge connection provides the central extension which is a 2–cocycle of the 2nd Hochschild cohomology group. Our con...

We present a new approach to the diierential geometry of surfaces in R 3 and R 4 that treats this theory as a \quaternioniied" version of the complex analysis and algebraic geometry of Riemann surfaces.

The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program Cars ((3]) that can be used to deene Riemann surfaces for computations. Cars allows one also to perform the Fenchel{Nielsen twist and other deformations on Riemann ...

We study holography for asymptotically AdS spaces with an arbitrary genus compact Riemann surface as the conformal boundary. Such spaces can be constructed from the Euclidean AdS3 by discrete identifications; the discrete groups one uses are the so-called classical Schottky groups. As we show, the spaces so constructed have an appealing interpretation of “analytic continuations” of the known Lo...

The uniformization theorem states that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere. We present three aproaches to the uniformization of Riemann surfaces. We first prove the uniformization theorem via the construction of a harmonic function by the Dirichlet principle. We then give an alternate proof by triangula...

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