A generalization of Riemann mappings and geometric structures on a space of domains in Cn / Stephen Semmes.

Format:

Book

Author/Creator:

Semmes, Stephen, 1962- author.

Series:

Memoirs of the American Mathematical Society ; Volume 98, Number 472.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 98, Number 472

Language:

English

Subjects (All):

Riemann surfaces.

Conformal mapping.

Physical Description:

1 online resource (111 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1992.

Language Note:

English

Summary:

Similar in philosophy to the study of moduli spaces in algebraic geometry, the central theme of this book is that spaces of (pseudoconvex) domains should admit geometrical structures that reflect the complex geometry of the underlying domains in a natural way. Semmes makes two main points in the book. The first is that there is a reasonable analogue of the universal Teichmüller space for domains in Cn, which has a great deal of interesting geometrical structure, some of which is surprisingly analogous to the classical situation in one complex variable. Second, there is a very natural notion of a Riemann mapping in several complex variables which is a modification of Lempert's, but which is defined in terms of first-order differential equations. In particular, the space of these Riemann mappings has a natural complex structure, which induces interesting geometry on the corresponding space of domains. With its unusual geometric perspective of some topics in several complex variables, this book appeals to those who view much of mathematics in broadly geometrical terms.

Contents:

""Contents""; ""1. Introduction""; ""2. Riemann mappings, Green's functions, and extremal disks""; ""3. Uniqueness of Riemann mappings, and Riemann mappings onto circled domains""; ""4. Riemann mappings and the Kobayashi indicatrix""; ""5. Existence of Riemann mappings whose image is a given smooth, strongly convex domain""; ""6. Riemann mappings and HCMA, part 1""; ""7. Riemann mappings and HCMA, part 2""; ""8. Riemann mappings and liftings to C""; ""9. Spaces of Riemann mappings, spaces of domains""; ""10. Spaces of Riemann mappings as complex varieties""

Notes:

July 1992, Volume 98, Number 472 (third of 4 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0898-8