TY  - BOOK
AU  - 
AU  - 
ED  - SpringerLink (Online service)
TI  - An Introduction to Riemann Surfaces
T2  - Cornerstones
SN  - 9780817646936
AV  - QA331.7 
U1  - 515.94 23
PY  - 2012///
CY  - Boston, MA
PB  - Birkhuser Boston, Imprint: Birkhuser
KW  - Mathematics
KW  - Global analysis (Mathematics)
KW  - Global analysis
KW  - Differential equations, partial
KW  - Several Complex Variables and Analytic Spaces
KW  - Global Analysis and Analysis on Manifolds
KW  - Analysis
N1  - 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 ofRiemann Surfaces.-Holomorphic Structures on Topological Surfaces -- Background Material on Analysis in Rnand 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; ZDB-2-SMA
N2  - This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the LÙ -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 RiemannRoch 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 Schȵnflies 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 exercisesmany of which lead to further development of the theoryand 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
UR  - http://dx.doi.org/10.1007/978-0-8176-4693-6
ER  -