Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Existence and uniqueness of rectilinear slit maps
HTML articles powered by AMS MathViewer

by Carl H. FitzGerald and Frederick Weening PDF
Trans. Amer. Math. Soc. 352 (2000), 485-513 Request permission


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.
  • 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 440018, DOI 10.1112/blms/9.2.129
  • Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
  • 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
  • H. Grötzsch, Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche, Ber. Verh. sächs Akad. Wiss. Leipzig, Math.-phys. Kl. 84 (1932), 15–36.
  • 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.
  • A. N. Harrington, Conformal Mappings on domains with arbitrarily specified boundary shapes, Journal D’analyse Mathématique 41 (1982), 39–53.
  • 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, DOI 10.2307/2946541
  • James A. Jenkins, Univalent functions and conformal mapping, Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806
  • P. Koebe, Abhandlungen zur Theorie der konformen Abbildung: V. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche, Math. Z. 2 (1919), 198–236.
  • Fumio Maitani and David Minda, Rectilinear slit conformal mappings, J. Math. Kyoto Univ. 36 (1996), no. 4, 659–668. MR 1443742, DOI 10.1215/kjm/1250518446
  • 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 203003, DOI 10.1007/BF02392209
  • 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
  • R. de Possel, Zum Parallelschlitzentheorem unendlich-vielfach zusammenhängender Gebeite, Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. (1931), 192–202.
  • Edgar Reich and S. E. Warschawski, On canonical conformal maps of regions of arbitrary connectivity, Pacific J. Math. 10 (1960), 965–985. MR 117339
  • Burton Rodin and Leo Sario, Principal functions, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. In collaboration with Mitsuru Nakai. MR 0229812
  • Oded Schramm, Conformal uniformization and packings, Israel J. Math. 93 (1996), 399–428. MR 1380655, DOI 10.1007/BF02761115
  • Oded Schramm, Transboundary extremal length, J. Anal. Math. 66 (1995), 307–329. MR 1370355, DOI 10.1007/BF02788827
  • Masakazu Shiba, On the Riemann-Roch theorem on open Riemann surfaces, J. Math. Kyoto Univ. 11 (1971), 495–525. MR 291445, DOI 10.1215/kjm/1250523617
  • 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:
  • 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:
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 485-513
  • MSC (1991): Primary 30C35; Secondary 30C20, 31A15
  • DOI:
  • MathSciNet review: 1694289