By Terrence Napier, Mohan Ramachandran

ISBN-10: 0817646922

ISBN-13: 9780817646929

ISBN-10: 0817646930

ISBN-13: 9780817646936

This textbook provides 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 method from the speculation of numerous advanced variables, and gives for a distinct method of the essentially assorted features of compact and noncompact Riemann surfaces.

The inclusion of constant workouts operating during the booklet, which bring about generalizations of the most theorems, in addition to the workouts incorporated in every one bankruptcy make this article excellent for a one- or two-semester graduate direction. the necessities are a operating wisdom of normal themes in graduate point actual and complicated research, and a few familiarity of manifolds and differential forms.

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B) The fundamental theorem of algebra (Gauss). Every nonconstant complex polynomial has a zero. Hint. Given a nonvanishing complex polynomial function g, consider the holomorphic function 1/g. 3 Holomorphic Attachment One may produce infinitely many examples of Riemann surfaces by holomorphic attachment. In fact, one goal of this book is a proof (appearing in Chap. 5) that every Riemann surface may be obtained by holomorphic attachment of tubes to a domain 36 2 ¯ for Scalar-Valued Forms Riemann Surfaces and the L2 ∂-Method in the Riemann sphere P1 .

3, we will consider the analogue for forms with values in a holomorphic line bundle. The solution in line bundles is more efficient in some ways, and it also generalizes more readily to higher-dimensional complex manifolds. In Sects. 9, we consider the definition and basic properties of a Riemann surface, the L2 spaces of differential forms, and the fundamental theorem regarding the solution of the (inhomogeneous) Cauchy–Riemann equation for scalar-valued differential forms. In the remaining sections, we apply the above to obtain some important facts, namely, the existence of meromorphic 1-forms and functions, Radó’s theorem on second countability of Riemann surfaces, the Mittag-Leffler theorem, and the Runge approximation theorem (see [R] for a historical perspective).

It is customary and convenient to identify X and X with their (disjoint) images in X X . However, although it is customary to leave out any explicit mention of the inclusion maps ι and ι , we will often mention the inclusion maps when considering specific mappings of the above spaces. This will allow us to avoid any danger of confusion (for example, there is danger of confusion whenever X = X ). 2. The natural identification of X X with X X gives a natural identification of X ∪ X with X ∪ −1 X.

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