Ebook: An Introduction to Riemann Surfaces

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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This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the L? -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 Sch?nflies 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.

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This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the L? -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 Sch?nflies 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.

Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
Complex Analysis in ?....Pages 3-23
Riemann Surfaces and the L 2 $bar{partial}$ -Method for Scalar-Valued Forms....Pages 25-99
The L 2 $bar{partial}$ -Method in a Holomorphic Line Bundle....Pages 101-154
Front Matter....Pages 155-155
Compact Riemann Surfaces....Pages 157-189
Uniformization and Embedding of Riemann Surfaces....Pages 191-309
Holomorphic Structures on Topological Surfaces....Pages 311-371
Front Matter....Pages 373-373
Background Material on Analysis in ? n and Hilbert Space Theory....Pages 375-405
Background Material on Linear Algebra....Pages 407-414
Background Material on Manifolds....Pages 415-476
Background Material on Fundamental Groups, Covering Spaces, and (Co)homology....Pages 477-530
Background Material on Sobolev Spaces and Regularity....Pages 531-543
Back Matter....Pages 545-560

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This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the L? -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 Sch?nflies 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.

Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
Complex Analysis in ?....Pages 3-23
Riemann Surfaces and the L 2 $bar{partial}$ -Method for Scalar-Valued Forms....Pages 25-99
The L 2 $bar{partial}$ -Method in a Holomorphic Line Bundle....Pages 101-154
Front Matter....Pages 155-155
Compact Riemann Surfaces....Pages 157-189
Uniformization and Embedding of Riemann Surfaces....Pages 191-309
Holomorphic Structures on Topological Surfaces....Pages 311-371
Front Matter....Pages 373-373
Background Material on Analysis in ? n and Hilbert Space Theory....Pages 375-405
Background Material on Linear Algebra....Pages 407-414
Background Material on Manifolds....Pages 415-476
Background Material on Fundamental Groups, Covering Spaces, and (Co)homology....Pages 477-530
Background Material on Sobolev Spaces and Regularity....Pages 531-543
Back Matter....Pages 545-560
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