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A Generalization of Riemann Mappings and Geometric Structures on a Space of Domains in C$$^n$$
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Memoirs of the American Mathematical Society
1992; 98 pp; softcover
Volume: 98
ISBN-10: 0-8218-2532-1
ISBN-13: 978-0-8218-2532-7
List Price: US$30 Individual Members: US$18
Institutional Members: US\$24
Order Code: MEMO/98/472

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 $${\mathbf C}^n$$, 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.

Mathematicians with a background in several complex variables and differential geometry.

• Riemann mappings, Green's functions, and extremal disks
• Uniqueness of Riemann mappings, and Riemann mappings onto circled domains
• Riemann mappings and the Kobayashi indicatrix
• Existence of Riemann mappings whose image is a given smooth, strongly convex domain
• Riemann mappings and HCMA, part $$1$$
• Riemann mappings and HCMA, part $$2$$
• Riemann mappings and liftings to $$\mathcal C$$
• Spaces of Riemann mappings, spaces of domains
• Spaces of Riemann mappings as complex varieties
• Homogeneous mappings, completely circled domains, and the Kobayashi indicatrix
• A natural action on $$\hat \mathcal R$$
• The action of $$\mathcal H$$ on domains in $$\mathbf C^n$$
• Riemannian geometry on $$\mathcal D^\infty$$ ; preliminary discussion
• Some basic facts and definitions concerning the metric on $$\mathcal D^\infty _co$$
• The metric on $$\mathcal D^\infty _co$$, circled domains, and the Kobayashi indicatrix
• The Riemannian metric and the action of $$\mathcal H$$
• The first variation of the energy of a curve in $$\mathcal D^\infty _co$$
• Geometry on $$\mathcal R^\infty$$
• Another approach to Riemannian geometry on $$\mathcal R^\infty$$
• A few remarks about the Hermitian geometry on $$\hat \mathcal R^\infty$$