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Transactions of the American Mathematical Society

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A general conformal geometric reflection principle


Author: Oliver Roth
Journal: Trans. Amer. Math. Soc. 359 (2007), 2501-2529
MSC (2000): Primary 30A99; Secondary 30F45
DOI: https://doi.org/10.1090/S0002-9947-07-03942-6
Published electronically: January 4, 2007
MathSciNet review: 2286042
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a generalization of the Schwarz-Carathéodory reflec- tion principle for analytic maps $ f$ from the unit disk into arbitrary Riemann surfaces equipped with a complete real analytic conformal Riemannian metric $ \lambda(w) \, \vert dw\vert$. This yields a necessary and sufficient condition for $ f$ to have an analytic continuation in terms of the pullback of the metric $ \lambda(w) \, \vert dw\vert$ under the map $ f$.


References [Enhancements On Off] (What's this?)

  • 1. L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton University Press, Princeton, 1960. MR 0114911 (22:5729)
  • 2. F. G. Avkhadiev, Conformal mappings that satisfy the boundary condition of equality of metrics, Doklady Akad. Nauk. (1996), 347 no. 3, 295-297. English transl.: Doklady Mathematics. (1996), 53 no. 2, 194-196. MR 1393054 (97d:30007)
  • 3. A. Beurling, An extension of the Riemann mapping theorem, Acta Math. (1953), 90, 117-130. MR 0060027 (15:614e)
  • 4. C. Carathéodory, Zum Schwarzschen Spiegelungsprinzip (Die Randwerte von meromorphen Funktionen), Comment. Math. Helv. (1946), 46, 263-278. MR 0020144 (8:508m)
  • 5. E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
  • 6. P. L. Duren, Univalent Functions, Springer-Verlag, Berlin, New York, 1983. MR 0708494 (85j:30034)
  • 7. R. Fournier and St. Ruscheweyh, Free boundary value problems for analytic functions in the closed unit disk, Proc. Amer. Math. Soc. (1999), 127 no. 11, 3287-3294. MR 1618666 (2000b:30059)
  • 8. R. Fournier and St. Ruscheweyh, A generalization of the Schwarz-Carathéodory reflection principle and spaces of pseudo-metrics, Math. Proc. Cambridge Phil. Soc. (2001), 130, 353-364. MR 1806784 (2003a:30022)
  • 9. F. W. Gehring, Characteristic properties of quasidisks, Les Presses de l'Université de Montréal, Montreal, Que., 1982. MR 0674294 (84a:30036)
  • 10. M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Birkhäuser, Boston, 1999. MR 1699320 (2000d:53065)
  • 11. J. Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, Springer-Verlag, Berlin, New York, 1991. MR 1632873 (2000k:32011)
  • 12. D. Kraus, Riccati-Differentialgleichungen im Komplexen, Diploma Thesis, Universität Würzburg, 2000.
  • 13. R. Kühnau, Längentreue Randverzerrung bei analytischer Abbildung in hyperbolischer und sphärischer Geometrie, Mitt. Math. Sem. Giessen (1997), 229, 45-53. MR 1439207 (98g:30011)
  • 14. I. Laine, Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin, New York, 1993. MR 1207139 (94d:34008)
  • 15. J. Liouville, Sur l'équation aux différences partielles $ \frac{d^2 \log \lambda}{du dv}\pm \frac{\lambda}{2 a^2}=0$, J. de Math. (1853), 16, 71-72.
  • 16. D. Minda, Conformal Metrics (unpublished).
  • 17. D. Minda, Bloch constants, J. Anal. Math. (1982), 41, 54-84. MR 0687945 (85e:30013)
  • 18. Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin, New York, 1992. MR 1217706 (95b:30008)
  • 19. O. Roth, An extension of the Schwarz-Carathéodory reflection principle, Habilitationsschrift, Universität Würzburg, 2003.
  • 20. H. A. Schwarz, Über einige Abbildungsaufgaben, J. Reine Angew. Math. (1869), 70, 105-120.
  • 21. G. Warnecke, Über einige Probleme bei einer nichtlinearen Differentialgleichung zweiter Ordnung im Komplexen, J. Reine Angew. Math. (1969), 239-240, 353-362. MR 0255955 (41:615)
  • 22. H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. (1936), 36, 63-89. MR 1501735

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Additional Information

Oliver Roth
Affiliation: Mathematisches Institut, Universität Würzburg, D–97074 Würzburg, Germany
Email: roth@mathematik.uni-wuerzburg.de

DOI: https://doi.org/10.1090/S0002-9947-07-03942-6
Received by editor(s): January 20, 2005
Published electronically: January 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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