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Existence and uniqueness of rectilinear slit maps. (English) Zbl 0941.30005

This extensive (29 pp.) paper contains a few very interesting and brand new results. The authors present these results in an extremely synthetic way as highlighted below. 1. “Normalized rectilinear slit maps are shown to be uniquely determined on domains of countable connectivity (see Theorem 2.3). 2. An example of a domain of countable connectivity and a (discontinuous) angle assignment for which no normalized rectilinear slit map achieving the assignment exists is demonstrated (see the end of section 2). 3. A new extremal length tool called the crossing – module is developed. The crossing-module extends the definition of the classical module to allow arcs to cross over specified boundary components of domains. Useful results regarding the cross – module of a domain whose outer boundary component is a rectangle are obtained (see Theorem 3.3 and 3.6). 4. The crossing – module is used to state an extremal length condition, called \((\text{NP}_\theta)\), generalizing Jenkins’ \((\text{P}_\theta)\) condition in the case of an angle assignment which is continuous and has finite range (see Theorem 4.4). 5. For an arbitrary domain it is shown that there exists a rectilinear slip map achieving a given continuous angle assignment of finite range which satisfies the \((\text{NP}_\theta)\) condition (see Theorem 4.4)”. All the definitions necessary to understand the paper are provided and the notation used is clearly explained.

MSC:

30C35 General theory of conformal mappings
30C20 Conformal mappings of special domains
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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[1] 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. · Zbl 0354.30002
[2] Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. · Zbl 0196.33801
[3] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. · Zbl 0183.07502
[4] H. Grötzsch, Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche, Ber. Verh. sächs Akad. Wiss. Leipzig, Math.-phys. Kl. 84 (1932), 15-36.
[5] 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.
[6] A. N. Harrington, Conformal Mappings on domains with arbitrarily specified boundary shapes, Journal D’analyse Mathématique 41 (1982), 39-53. · Zbl 0528.30006
[7] Zheng-Xu He and Oded Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2) 137 (1993), no. 2, 369 – 406. · Zbl 0777.30002
[8] 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. · Zbl 0083.29606
[9] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung: V. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche, Math. Z. 2 (1919), 198-236. · JFM 46.0546.01
[10] Fumio Maitani and David Minda, Rectilinear slit conformal mappings, J. Math. Kyoto Univ. 36 (1996), no. 4, 659 – 668. · Zbl 0948.30013
[11] 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. · Zbl 0142.33104
[12] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603
[13] R. de Possel, Zum Parallelschlitzentheorem unendlich-vielfach zusammenhängender Gebeite, Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. (1931), 192-202.
[14] Edgar Reich and S. E. Warschawski, On canonical conformal maps of regions of arbitrary connectivity, Pacific J. Math. 10 (1960), 965 – 985. · Zbl 0091.25503
[15] Burton Rodin and Leo Sario, Principal functions, In collaboration with Mitsuru Nakai, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. · Zbl 0159.10701
[16] Oded Schramm, Conformal uniformization and packings, Israel J. Math. 93 (1996), 399 – 428. · Zbl 0872.30005
[17] Oded Schramm, Transboundary extremal length, J. Anal. Math. 66 (1995), 307 – 329. · Zbl 0842.30006
[18] Masakazu Shiba, On the Riemann-Roch theorem on open Riemann surfaces, J. Math. Kyoto Univ. 11 (1971), 495 – 525. · Zbl 0227.30022
[19] F. Weening, Existence and Uniqueness of Non-parallel Slit Maps, Ph. D. dissertation, 1994.
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