# A Generalization of Riemann Mappings and Geometric Structures on a Space of Domains in C\(^n\)

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*Stephen Semmes*

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.

#### Table of Contents

# Table of Contents

## A Generalization of Riemann Mappings and Geometric Structures on a Space of Domains in C$^{n}$

- Contents v6 free
- 1. Introduction 18 free
- 2. Riemann mappings, Green's functions, and extremal disks 512 free
- 3. Uniqueness of Riemann mappings, and Riemann mappings onto circled domains 1118
- 4. Riemann mappings and the Kobayashi indicatrix 1623
- 5. Existence of Riemann mappings whose image is a given smooth, strongly convex domain 1825
- 6. Riemann mappings and HCMA, part 1 2027
- 7. Riemann mappings and HCMA, part 2 2633
- 8. Riemann mappings and liftings to C 3441
- 9. Spaces of Riemann mappings, spaces of domains 4148
- 10. Spaces of Riemann mappings as complex varieties 4350
- 11. Homogeneous mappings, completely circled domains, and the Kobayashi indicatrix 4855
- 12. A natural action on R 5562
- 13. The action of H on domains in C[sup(n)] 6067
- 14. Riemannian geometry on D[sup(∞)]; preliminary discussion 6269
- 15. Some basic facts and definitions concerning the metric on D[sup(∞)][sub(co)] 6471
- 16. The metric on D[sup(∞)][sub(co)], circled domains, and the Kobayashi indicatrix 6976
- 17. The Riemannian metric and the action of H 7279
- 18. The first variation of the energy of a curve in D[sup(∞)][sub(co)] 7582
- 19. Geometry on R[sup(∞)] 7885
- 20. Another approach to Riemannian geometry on R[sup(∞)] 8794
- 21. A few remarks about the Hermitian geometry on R[sup(∞)] 9299
- Some notations and conventions 95102
- References 97104