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An Introduction to Riemann Surfaces de Terrence Napier

An Introduction to Riemann Surfaces: Cornerstones

Autor Terrence Napier, Mohan Ramachandran
en Limba Engleză Hardback – 28 sep 2011
This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact Riemann surfaces.
The inclusion of continuing exercises running throughout the book, which lead to generalizations of the main theorems, as well as the exercises included in each chapter make this text ideal for a one- or two-semester graduate course. The prerequisites are a working knowledge of standard topics in graduate level real and complex analysis, and some familiarity of manifolds and differential forms.
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Specificații

ISBN-13: 9780817646929
ISBN-10: 0817646922
Pagini: 560
Ilustrații: XVII, 560 p. 42 illus.
Dimensiuni: 155 x 235 x 32 mm
Greutate: 0.98 kg
Ediția:2012
Editura: Birkhäuser Boston
Colecția Birkhäuser
Seria Cornerstones

Locul publicării:Boston, MA, United States

Public țintă

Graduate

Cuprins

Preface.- Introduction.- Complex analysis in C.- Riemann Surfaces and the L2 \delta-Method for Scalar-Valued Forms.- The L2 \delta-Method in a Holomorphic Line Bundle.- Compact Riemann Surfaces.- Uniformization and Embedding of Riemann Surfaces.-Holomorphic Structures on Topological Surfaces.- Background Material on Analysis in Rn and Hilbert Space Theory.- Background Material on Linear Algebra.- Background Material on Manifolds.- Background Material on Fundamental Groups, Covering Spaces, and (Co)homology.- Background Material on Sobolev Spaces and Regularity.- References.- Notation Index.- Subject Index.

Recenzii

From the reviews:
“The present book gives a solid introduction to the theory of both compact and non-compact Riemann surfaces. While modern introductions often take the view point of algebraic geometry, the present book tries to also cover the analytical aspects. … The book is well written and constitutes a nice contribution to the existing literature on this topic.” (G. Teschl, Internationale Mathematische Nachrichten, Issue 225, 2014)
“This book takes the point of view that Riemann surface theory lies at the root of much of modern analysis, and … illustrate some of the interactions of analysis with geometry and topology. … While much of the book is intended for students at the second-year graduate level, Chapters 1 and 2 and Section 5.2 (along with the required background material) could serve as the basis for the complex analytic analysis component of a year-long first-year graduate-level course on real and complex analysis.” (V. V. Chueshev, Zentralblatt MATH, Vol. 1237, 2012)

Textul de pe ultima copertă

This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the L² -method, a powerful technique used in the theory of several complex variables.  The work features a simple construction of a strictly subharmonic exhaustion function and a related construction of a positive-curvature Hermitian metric in a holomorphic line bundle, topics which serve as starting points for proofs of standard results such as the Mittag-Leffler, Weierstrass, and Runge theorems; the Riemann−Roch theorem; the Serre duality and Hodge decomposition theorems; and the uniformization theorem. The book also contains treatments of other facts concerning the holomorphic, smooth, and topological structure of a Riemann surface, such as the biholomorphic classification of Riemann surfaces, the embedding theorems, the integrability of almost complex structures, the Schönflies theorem (and the Jordan curve theorem), and the existence of smooth structures on second countable surfaces. 
Although some previous experience with complex analysis, Hilbert space theory, and analysis on manifolds would be helpful, the only prerequisite for this book is a working knowledge of point-set topology and elementary measure theory. The work includes numerous exercises—many of which lead to further development of the theory—and  presents (with proofs) streamlined treatments of background topics from analysis and topology on manifolds in easily-accessible reference chapters, making it ideal for a one- or two-semester graduate course.

Caracteristici

Presents a unified and competitive approach to compact and noncompact Riemann surfaces Includes continuing exercises that run throughout the book and lead to generalizations of the main theorems Will help expand and reinforce a student’s knowledge of analysis, geometry, and topology Includes supplementary material: sn.pub/extras