By Terrence Napier, Mohan Ramachandran

This textbook offers a unified method of compact and noncompact Riemann surfaces from the viewpoint of the so-called L2 $\bar{\delta}$-method. this technique is a robust process from the speculation of numerous complicated variables, and offers for a distinct method of the essentially diversified features of compact and noncompact Riemann surfaces.

The inclusion of continuous workouts working through the publication, which result in generalizations of the most theorems, in addition to the workouts incorporated in each one bankruptcy make this article perfect for a one- or two-semester graduate direction. the necessities are a operating wisdom of ordinary issues in graduate point actual and complicated research, and a few familiarity of manifolds and differential forms.

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**Additional info for An Introduction to Riemann Surfaces**

**Example text**

An annulus) T (equivalently, the boundaries of the unit disks are glued together). We call X the complex 1-manifold obtained by holomorphic attachment of the tube T to Y at the coordinate disks {(Dν , ν , (0; Rν ))}ν∈{0,1} (or simply at {Dν }ν∈{0,1} ). This is a slight abuse of language, since actually we first −1 removed the set −1 0 ( (0; 1)) ∪ 1 ( (0; 1)) before performing the attachment. 38 2 ¯ for Scalar-Valued Forms Riemann Surfaces and the L2 ∂-Method Observe that if Y is connected, then X is connected; and if Y is compact, then X is compact.

7 We identify any meromorphic function f on an open subset of a complex 1-manifold X with the associated holomorphic mapping → P1 . If p ∈ f −1 (∞) is a pole of f at which this mapping has multiplicity m, then we say that f has a pole of order m at p. We say that f has a zero of order m at q ∈ if q ∈ f −1 (0) and the holomorphic function f \f −1 (∞) has a zero of order m at q (note that this is consistent with the above identification). For any point p ∈ , the order of f at p is given by ⎧ 0 if p ∈ / f −1 ({0, ∞}), ⎪ ⎪ ⎪ ⎨m if f has a zero of order m at p, ordp f ≡ ⎪ −m if f has a pole of order m at p, ⎪ ⎪ ⎩ ∞ if f ≡ 0 on a neighborhood of p.

If such a biholomorphism exists, then we say that X and Y are biholomorphically equivalent (or simply biholomorphic). For Y = X, we also call an automorphism of X. (d) A holomorphic mapping : X → Y is a local biholomorphism (or a locally biholomorphic mapping) if maps a neighborhood of each point in X biholomorphically onto an open subset of Y . Remarks 1. A holomorphic function on an open subset of a complex 1-manifold X is precisely a holomorphic mapping of into C. 2. A function f on an open subset of a complex 1-manifold X is holomorphic if and only if for each point p ∈ , there exists a local holomorphic chart (U, , U ) in X such that p ∈ U and f ◦ −1 ∈ O( ( ∩ U )).