The moduli of compact continuations of an open Riemann surface of genus one
HTML articles powered by AMS MathViewer
- by M. Shiba PDF
- Trans. Amer. Math. Soc. 301 (1987), 299-311 Request permission
Abstract:
Let $(R, \{ A, B\} )$ be a marked open Riemann surface of genus one. Denote by $(T, \{ {A_T}, {B_T}\} ,i)$ a pair of a marked torus $(T, \{ {A_T}, {B_T}\} )$ and a conformal embedding $i$ of $R$ into $T$ with $i(A)$ and $i(B)$ homotopic respectively to ${A_T}$ and ${B_T}$. We say that $(T, \{ {A_T}, {B_T}\} ,i)$ and $(T’, \{ {A_T’}, {B_T’}\} ,i’)$ are equivalent if $i’ \circ {i^{ - 1}}$ extends to a conformal mapping of $T$ onto ${T^\prime }$. 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 $M = M(R, \{ A, B\} )$. Then $M$ is a closed disk in the upper half plane. The radius of $M$ represents the size of the ideal boundary of $R$ and gives a generalization of Schiffer’s span for planar domains; in particular, it vanishes if and only if $R$ belongs to the class ${O_{AD}}$. On the other hand, any holomorphic differential on $R$ with distinguished imaginary part produces in a canonical manner a compact continuation of $(R, \{ A, B\} )$. Such a compact continuation is referred to as a hydrodynamic continuation of $(R, \{ A, B\} )$. The boundary of $M$ parametrizes in a natural way the space of hydrodynamic continuations; i.e., the hydrodynamic continuations have extremal properties.References
- Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129. MR 36841, DOI 10.1007/BF02392634
- Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
- S. Bochner, Fortsetzung Riemannscher Flächen, Math. Ann. 98 (1928), no. 1, 406–421 (German). MR 1512413, DOI 10.1007/BF01451602
- T. Carleman, Über ein Minimalproblem der mathematischen Physik, Math. Z. 1 (1918), no. 2-3, 208–212 (German). MR 1544292, DOI 10.1007/BF01203612
- Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745 H. Grötzsch, Die Werte des Doppelverhältnisses bei schlichter konformer Abbildung, Sitzungsber. Preuss. Akad. Wiss. Berlin (1933), 501-515. —, Über Flächensätze der konformen Abbildung, Jber. Deutsch. Math.-Verein. 44 (1934), 266-269. —, Einige Bemerkungen zur schlichten konformen Abbildung, Jber. Deutsch. Math.-Verein. 44 (1934), 270-275.
- 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. MR 0056097
- Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Die Grundlehren der mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl; Vierte vermehrte und verbesserte Auflage. MR 0173749 M. S. Ioffe, Conformal and quasi-conformal imbedding of one finite Riemann surface into another, Soviet Math. Dokl. 13 (1972), 75-78. —, 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.
- James A. Jenkins, Univalent functions and conformal mapping, Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806
- Akira Mori, A remark on the prolongation of Riemann surfaces of finite genus, J. Math. Soc. Japan 4 (1952), 27–30. MR 50024, DOI 10.2969/jmsj/00410027
- Kôtaro Oikawa, On the prolongation of an open Riemann surface of finite genus, K\B{o}dai Math. Sem. Rep. 9 (1957), 34–41. MR 86872
- Kôtaro Oikawa, On the uniqueness of the prolongation of an open Riemann surface of finite genus, Proc. Amer. Math. Soc. 11 (1960), 785–787. MR 122982, DOI 10.1090/S0002-9939-1960-0122982-2
- Edgar Reich and S. E. Warschawski, On canonical conformal maps of regions of arbitrary connectivity, Pacific J. Math. 10 (1960), 965–985. MR 117339 E. Rengel, Existenzbeweise für schlichte Abbildungen mehrfach zusammenhängender Bereiche auf gewisse Normalbereiche, Jber. Deutsch. Math.-Verein. 44 (1934), 51-55.
- H. Renggli, Structural instability and extensions of Riemann surfaces, Duke Math. J. 42 (1975), 211–224. MR 372190
- Ian Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259–269. MR 143186, DOI 10.1090/S0002-9947-1963-0143186-0
- 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
- L. Sario and K. Oikawa, Capacity functions, Die Grundlehren der mathematischen Wissenschaften, Band 149, Springer-Verlag New York, Inc., New York, 1969. MR 0254232
- Akira Sakai, On minimal slit domains, Proc. Japan Acad. 35 (1959), 128–133. MR 107713
- Menahem Schiffer, The span of multiply connected domains, Duke Math. J. 10 (1943), 209–216. MR 8259
- 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
- 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. MR 764457
- 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. MR 898063, DOI 10.1080/17476938708814232
- 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. MR 1008930
- George Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957. MR 0092855
- Vo Dang Thao, Über einige Flächeninhaltsformeln bei schlicht-konformer Abbildung von Kreisbogenschlitzgebieten, Math. Nachr. 74 (1976), 253–261 (German). MR 492198, DOI 10.1002/mana.3210740119
- Steffan Timmann, Einbettungen endlicher Riemannscher Flächen, Math. Ann. 217 (1975), no. 1, 81–85 (German). MR 390207, DOI 10.1007/BF01363243
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 301 (1987), 299-311
- MSC: Primary 30F30; Secondary 14H15, 30F25, 32G15
- DOI: https://doi.org/10.1090/S0002-9947-1987-0879575-2
- MathSciNet review: 879575