By Terrence Napier, Mohan Ramachandran

This textbook offers a unified method of compact and noncompact Riemann surfaces from the perspective of the so-called L2 $\bar{\delta}$-method. this technique is a strong procedure from the idea of numerous complicated variables, and offers for a different method of the essentially assorted features of compact and noncompact Riemann surfaces.

The inclusion of continuous workouts operating during the booklet, which result in generalizations of the most theorems, in addition to the routines incorporated in every one bankruptcy make this article perfect for a one- or two-semester graduate path. the necessities are a operating wisdom of ordinary subject matters in graduate point actual and intricate research, and a few familiarity of manifolds and differential forms.

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**An Introduction to Riemann Surfaces**

This textbook offers a unified method of compact and noncompact Riemann surfaces from the viewpoint of the so-called L2 $\bar{\delta}$-method. this system is a robust strategy from the speculation of a number of complicated variables, and gives for a different method of the essentially assorted features of compact and noncompact Riemann surfaces.

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

**Example text**

C) For each open set ⊂ X, E r,s ( ) denotes the vector space of C ∞ differential forms of type (r, s). The vector space of C ∞ (r, s)-forms with compact support in is denoted by Dr,s ( ). (d) Let α be a differential form of degree q on a set S ⊂ X. , α = a dz ∧ d z¯ ) ⎪ ⎪ ⎪ dz ∧ d z¯ ⎪ ⎩ 0 if q = 0, if q = 1, if q = 2, if q > 2. Remarks 1. For each point p ∈ X, we have α ∧ β = α¯ ∧ β¯ = 0 for all α, β ∈ 1,0 T ∗ X. Observe also that ξ ∧ ζ is of type (r + t, s + u) if ξ ∈ r,s T ∗ X and p p ζ ∈ t,u Tp∗ X.

It follows that each of the functions f + g, f g, and f/g is holomorphic on a neighborhood of p or has a removable singularity or a pole at p. Consequently, any sum, product, or quotient (provided the denominator does not vanish identically on any open set) of meromorphic functions is a meromorphic function, provided we holomorphically extend the resulting function over any removable singularities. In other words, for any connected open subset of a complex 1-manifold, M( ) is a field. 2. 6. Then f has a pole of order m ∈ Z>0 at p if and only if cn = 0 for all n < −m and c−m = 0.

2. A holomorphic structure on X determines a unique underlying real 2-dimensional C ∞ (in fact, real analytic) structure, with C ∞ atlas consisting of local C ∞ coordinates given by (x, y) = (Re z, Im z) for any local holomorphic coordinate z (see Chap. 9). A map from (to) an open subset of X from (respectively, to) an open subset of a manifold or complex 1-manifold is said to be of class C k if the map is of class C k with respect to the underlying C k structures. 3. In many treatments, manifolds are assumed to be second countable (often without comment).