# A generalization of Riemann mappings and geometric structures on a space of domains in ${\bf C}^n$

### About this Title

**Stephen Semmes**

Publication: Memoirs of the American Mathematical Society

Publication Year
1992: Volume 98, Number 472

ISBNs: 978-0-8218-2532-7 (print); 978-1-4704-0898-5 (online)

DOI: http://dx.doi.org/10.1090/memo/0472

MathSciNet review: 1113614

MSC (1991): Primary 32H99; Secondary 32G99

### 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 $