CONTENTS

1. Introduction 1

2. Riemann mappings, Green's functions, and extremal

disks 5

3. Uniqueness of Riemann mappings, and Riemann

mappings onto circled domains 11

4. Riemann mappings and the Kobayashi indicatrix 16

5. Existence of Riemann mappings whose image is

a given smooth, strongly convex domain 18

6. Riemann mappings and HCMA, part 1 20

7. Riemann mappings and HCMA, part 2 26

8. Riemann mappings and liftings to C 34

9. Spaces of Riemann mappings, spaces of domains 41

10. Spaces of Riemann mappings as complex varieties 43

11. Homogeneous mappings, completely circled

domains, and the Kobayashi indicatrix

48

12. A natural action on it 55

13. The action of Ti on domains in C" 60

14. Riemannian geometry on 2°°; preliminary

discussion 62

15. Some basic facts and definitions concerning the

metric on Vf0 64

16. The metric on X~, circled domains, and the

Kobayashi indicatrix 69

17.

The Riemannian metric and the action of Ji 72

18. The first variation of the energy of a curve in V™ 75

19. Geometry on 1Z°° 78

20. Another approach to Riemannian geometry

on ft00 87

21.

A few remarks about the Hermitian geometry

on ft00 92

Some notations and conventions 95

References 97

V