Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Existence and uniqueness of rectilinear slit maps


Authors: Carl H. FitzGerald and Frederick Weening
Journal: Trans. Amer. Math. Soc. 352 (2000), 485-513
MSC (1991): Primary 30C35; Secondary 30C20, 31A15
Published electronically: October 5, 1999
MathSciNet review: 1694289
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' ${\theta}$ module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.


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

  • [ABB] J. M. Anderson, K. F. Barth, and D. A. Brannan, Research problems in complex analysis, Bull. London Math. Soc. 9 (1977), no. 2, 129–162. MR 0440018
  • [AS] Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
  • [Go] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • [Gr1] H. Grötzsch, Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche, Ber. Verh. sächs Akad. Wiss. Leipzig, Math.-phys. Kl. 84 (1932), 15-36.
  • [Gr2] H. Grötzsch, Zum Parallelschlitztheorem der konformen Abbildung schlichter unendlich-vielfach zussamenhängender Bereiche, Ber. Verh. sächs Akad. Wiss. Leipzig, Math.-phys. Kl. 83 (1931), 185-200.
  • [Ha] A. N. Harrington, Conformal Mappings on domains with arbitrarily specified boundary shapes, Journal D'analyse Mathématique 41 (1982), 39-53.
  • [HS] Zheng-Xu He and Oded Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2) 137 (1993), no. 2, 369–406. MR 1207210, 10.2307/2946541
  • [J] James A. Jenkins, Univalent functions and conformal mapping, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806
  • [K] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung: V. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche, Math. Z. 2 (1919), 198-236.
  • [MM] Fumio Maitani and David Minda, Rectilinear slit conformal mappings, J. Math. Kyoto Univ. 36 (1996), no. 4, 659–668. MR 1443742
  • [MR] A. Marden and B. Rodin, Extremal and conjugate extremal distance on open Riemann surfaces with applications to circular-radial slit mappings, Acta Math. 115 (1966), 237–269. MR 0203003
  • [NS] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. MR 0264064
  • [dP] R. de Possel, Zum Parallelschlitzentheorem unendlich-vielfach zusammenhängender Gebeite, Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. (1931), 192-202.
  • [RW] Edgar Reich and S. E. Warschawski, On canonical conformal maps of regions of arbitrary connectivity, Pacific J. Math. 10 (1960), 965–985. MR 0117339
  • [RS] Burton Rodin and Leo Sario, Principal functions, In collaboration with Mitsuru Nakai, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0229812
  • [Sc1] Oded Schramm, Conformal uniformization and packings, Israel J. Math. 93 (1996), 399–428. MR 1380655, 10.1007/BF02761115
  • [Sc2] Oded Schramm, Transboundary extremal length, J. Anal. Math. 66 (1995), 307–329. MR 1370355, 10.1007/BF02788827
  • [Sh] Masakazu Shiba, On the Riemann-Roch theorem on open Riemann surfaces, J. Math. Kyoto Univ. 11 (1971), 495–525. MR 0291445
  • [W] F. Weening, Existence and Uniqueness of Non-parallel Slit Maps, Ph. D. dissertation, 1994.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 30C35, 30C20, 31A15

Retrieve articles in all journals with MSC (1991): 30C35, 30C20, 31A15


Additional Information

Carl H. FitzGerald
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email: cfitzgerald@ucsd.edu

Frederick Weening
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Address at time of publication: Department of Mathematics and Computer Science, Doucette Hall, Edinboro University of Pennsylvania, Edinboro, Pennsylvania 16444
Email: fweening@edinboro.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02538-6
Keywords: Conformal uniformization, slit maps, extremal length
Received by editor(s): July 3, 1995
Received by editor(s) in revised form: October 24, 1996, and February 19, 1999
Published electronically: October 5, 1999
Article copyright: © Copyright 1999 American Mathematical Society