×

zbMATH — the first resource for mathematics

The moduli of compact continuations of an open Riemann surface of genus one. (English) Zbl 0626.30046
Let R be an open Riemann surface of genus one with a fixed canonical homology basis \(\{\) A,B\(\}\) modulo dividing cycles. We call the pair (R,\(\{\) A,B\(\})\) a marked open Riemann surface. By a marked realization of (R,\(\{\) A,B\(\})\) we mean a triple \((T,\{A_ T,B_ T\},i)\), where \((T,\{A_ T,B_ T\})\) is a marked torus and i is a conformal embedding of R into T with i(A) and i(B) homotopic respectively to \(A_ T\) and \(B_ T\). \((T,\{A_ T,B_ T\},i)\) and \((T',\{A_{T'},B_{T'}\},i')\) are said to be equivalent if \(i'\circ i^{-1}\) extends to a conformal mapping of T onto T’. The equivalence classes are called compact continuations of (R,\(\{\) A,B\(\})\) and the set of moduli of compact continuations of (R,\(\{\) A,B\(\})\) is denoted by \(M=M(R,\{A,B\})\). For any t, \(-1<t\leq 1\), there exists a unique holomorphic differential \(\phi_ t\) on R such that \(Im[e^{-\pi it/2}\phi_ t]\) is distinguished and \(\int_{A}\phi_ t=1\). Furthermore, there exists a marked realization \((T_ t,\{A_ t,B_ t\},i_ t)\) of (R,\(\{\) A,B\(\})\) such that \(\phi_ t\circ i_ t^{-1}\) extends to the normal holomorphic differential \(\phi^{T_ t}\) on \(T_ t\) w.r.t. the basis \(\{A_ t,B_ t\}\). The compact continuation represented by \((T_ t,\{A_ t,B_ t\},i_ t)\) is called the hydrodynamic continuation of (R,\(\{\) A,B\(\})\) w.r.t. \(\phi_ t.\)
The author proves the following results: (i) M(R,\(\{\) A,B\(\})\) is a closed disk in the Teichmüller space of genus one (the upper half plane); (ii) To each boundary point of M(R,\(\{\) A,B\(\})\) there corresponds a unique hydrodynamic continuation of (R,\(\{\) A,B\(\})\), and vice versa; (iii) To an interior point of M(R,\(\{\) A,B\(\})\) there correspond in general more than one compact continuation of (R,\(\{\) A,B\(\})\); (iv) The radius \(\rho\) (R) of M(R,\(\{\) A,B\(\})\) vanishes if and only if \(R\in O_{AD}\). The quantity \(\rho\) (R) gives a generalization of Schiffer’s span for planar regions. In the proof of (i)-(iv), the results of the author’s preceding papers [M. Shiba, Hiroshima Math. J. 14, 371-399 (1984; Zbl 0567.30033); M. Shiba and K. Shibata, Complex Variables 8, 205-211 (1987; Zbl 0586.30041)] are used.
Reviewer: H.Mizumoto

MSC:
30F99 Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101 – 129. · Zbl 0041.20301 · doi:10.1007/BF02392634 · doi.org
[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] S. Bochner, Fortsetzung Riemannscher Flächen, Math. Ann. 98 (1928), no. 1, 406 – 421 (German). · JFM 53.0322.01 · doi:10.1007/BF01451602 · doi.org
[4] T. Carleman, Über ein Minimalproblem der mathematischen Physik, Math. Z. 1 (1918), no. 2-3, 208 – 212 (German). · JFM 46.0765.02 · doi:10.1007/BF01203612 · doi.org
[5] Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. · Zbl 0764.30001
[6] H. Grötzsch, Die Werte des Doppelverhältnisses bei schlichter konformer Abbildung, Sitzungsber. Preuss. Akad. Wiss. Berlin (1933), 501-515. · Zbl 0007.21404
[7] -, Über Flächensätze der konformen Abbildung, Jber. Deutsch. Math.-Verein. 44 (1934), 266-269. · Zbl 0011.12202
[8] -, Einige Bemerkungen zur schlichten konformen Abbildung, Jber. Deutsch. Math.-Verein. 44 (1934), 270-275. · Zbl 0010.30802
[9] Maurice Heins, A problem concerning the continuation of Riemann surfaces, Contributions to the theory of Riemann surfaces, Annals of Mathematics Studies, no. 30, Princeton University Press, Princeton, N. J., 1953, pp. 55 – 62. · Zbl 0052.30504
[10] Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl. Vierte vermehrte und verbesserte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). · JFM 51.0236.12
[11] M. S. Ioffe, Conformal and quasi-conformal imbedding of one finite Riemann surface into another, Soviet Math. Dokl. 13 (1972), 75-78.
[12] -, On a problem of the variational calculus in the large for conformal and quasi-conformal mappings of one finite Riemann surface in another, Soviet Math. Dokl. 14 (1973), 1576-1579. · Zbl 0314.30018
[13] 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
[14] Akira Mori, A remark on the prolongation of Riemann surfaces of finite genus, J. Math. Soc. Japan 4 (1952), 27 – 30. · Zbl 0048.05903 · doi:10.2969/jmsj/00410027 · doi.org
[15] Kôtaro Oikawa, On the prolongation of an open Riemann surface of finite genus, Kōdai Math. Sem. Rep. 9 (1957), 34 – 41. · Zbl 0078.06901
[16] Kôtaro Oikawa, On the uniqueness of the prolongation of an open Riemann surface of finite genus, Proc. Amer. Math. Soc. 11 (1960), 785 – 787. · Zbl 0097.06502
[17] 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
[18] E. Rengel, Existenzbeweise für schlichte Abbildungen mehrfach zusammenhängender Bereiche auf gewisse Normalbereiche, Jber. Deutsch. Math.-Verein. 44 (1934), 51-55. · JFM 60.0286.01
[19] H. Renggli, Structural instability and extensions of Riemann surfaces, Duke Math. J. 42 (1975), 211 – 224. · Zbl 0335.30013
[20] Ian Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259 – 269. · Zbl 0156.22203
[21] 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
[22] L. Sario and K. Oikawa, Capacity functions, Die Grundlehren der mathematischen Wissenschaften, Band 149, Springer-Verlag New York Inc., New York, 1969. · Zbl 0184.10503
[23] Akira Sakai, On minimal slit domains, Proc. Japan Acad. 35 (1959), 128 – 133. · Zbl 0122.08001
[24] Menahem Schiffer, The span of multiply connected domains, Duke Math. J. 10 (1943), 209 – 216. · Zbl 0060.23704
[25] Masakazu Shiba, On the Riemann-Roch theorem on open Riemann surfaces, J. Math. Kyoto Univ. 11 (1971), 495 – 525. · Zbl 0227.30022
[26] Masakazu Shiba, The Riemann-Hurwitz relation, parallel slit covering map, and continuation of an open Riemann surface of finite genus, Hiroshima Math. J. 14 (1984), no. 2, 371 – 399. · Zbl 0567.30033
[27] M. Shiba and K. Shibata, Hydrodynamic continuations of an open Riemann surface of finite genus, Complex Variables Theory Appl. 8 (1987), no. 3-4, 205 – 211. · Zbl 0586.30041
[28] C. L. Siegel, Topics in complex function theory. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Elliptic functions and uniformization theory; Translated from the German by A. Shenitzer and D. Solitar; With a preface by Wilhelm Magnus; Reprint of the 1969 edition; A Wiley-Interscience Publication. · Zbl 0635.30002
[29] George Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957. · Zbl 0078.06602
[30] Vo Dang Thao, Über einige Flächeninhaltsformeln bei schlicht-konformer Abbildung von Kreisbogenschlitzgebieten, Math. Nachr. 74 (1976), 253 – 261 (German). · Zbl 0355.30007 · doi:10.1002/mana.3210740119 · doi.org
[31] Steffan Timmann, Einbettungen endlicher Riemannscher Flächen, Math. Ann. 217 (1975), no. 1, 81 – 85 (German). · Zbl 0297.32014 · doi:10.1007/BF01363243 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.