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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

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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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