eBook ISBN:  9781470408985 
Product Code:  MEMO/98/472.E 
List Price:  $30.00 
MAA Member Price:  $27.00 
AMS Member Price:  $18.00 
eBook ISBN:  9781470408985 
Product Code:  MEMO/98/472.E 
List Price:  $30.00 
MAA Member Price:  $27.00 
AMS Member Price:  $18.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 98; 1992; 98 ppMSC: Primary 32
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 firstorder 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.
ReadershipMathematicians with a background in several complex variables and differential geometry.

Table of Contents

Chapters

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 $\mathcal {C}$

9. Spaces of Riemann mappings, spaces of domains

10. Spaces of Riemann mappings as complex varieties

11. Homogeneous mappings, completely circled domains, and the Kobayashi indicatrix

12. A natural action on $\hat {\mathcal {R}}$

13. The action of $\mathcal {H}$ on domains in $\mathbf {C}^n$

14. Riemannian geometry on $\mathcal {D}^\infty $; preliminary discussion

15. Some basic facts and definitions concerning the metric on $\mathcal {D}^\infty _{co}$

16. The metric on $\mathcal {D}^\infty _{co}$, circled domains, and the Kobayashi indicatrix

17. The Riemannian metric and the action of $\mathcal {H}$

18. The first variation of the energy of a curve in $\mathcal {D}^\infty _{co}$

19. Geometry on $\mathcal {R}^\infty $

20. Another approach to Riemannian geometry on $\mathcal {R}^\infty $

21. A few remarks about the Hermitian geometry on $\hat {\mathcal {R}}^\infty $


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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 firstorder 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.

Chapters

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 $\mathcal {C}$

9. Spaces of Riemann mappings, spaces of domains

10. Spaces of Riemann mappings as complex varieties

11. Homogeneous mappings, completely circled domains, and the Kobayashi indicatrix

12. A natural action on $\hat {\mathcal {R}}$

13. The action of $\mathcal {H}$ on domains in $\mathbf {C}^n$

14. Riemannian geometry on $\mathcal {D}^\infty $; preliminary discussion

15. Some basic facts and definitions concerning the metric on $\mathcal {D}^\infty _{co}$

16. The metric on $\mathcal {D}^\infty _{co}$, circled domains, and the Kobayashi indicatrix

17. The Riemannian metric and the action of $\mathcal {H}$

18. The first variation of the energy of a curve in $\mathcal {D}^\infty _{co}$

19. Geometry on $\mathcal {R}^\infty $

20. Another approach to Riemannian geometry on $\mathcal {R}^\infty $

21. A few remarks about the Hermitian geometry on $\hat {\mathcal {R}}^\infty $