Abstract
This paper is devoted to the development of Beckenbach’s theory of meromorphic minimal surfaces. We get an estimate of the sum of Petrenko’s deviations of the meromorphic minimal surface of finite lower order in term of Valiron’s defect \(\Delta ({\textbf {0}}, S_u)\). We also give an example showing that the estimate is sharp.
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References
Baernstein, A., II.: Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139–169 (1974)
Beckenbach, E.F., Radó, T.: Subharmonic functions and minimal surfaces. Trans. Am. Math. Soc. 35, 648–661 (1933)
Beckenbach, E.F., Hutchison, G.A.: Meromorphic minimal surfaces. Pac. J. Math. 28, 17–47 (1969)
Beckenbach, E.F., Cootz, T.: The second fundamental theorem for meromorphic minimal surfaces. Bull. Am. Math. Soc. 76, 711–716 (1970)
Beckenbach, E.F., Eng, F.H., Tafel, R.E.: Global properties of rational and logarithmico-rational minimal surfaces. Pac. J. Math. 50, 355–381 (1974)
Blaschke, W.: Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Russian transl, ONTI, Moscow (1935)
Essen, M., Shea, D.F.: Applications of Denjoy integral inequalities and differential inequalities to growth problems for subharmonic and meromorphic functions. Proc. Roy. Irish Acad. Sect. A 82, 201–216 (1982)
Fuchs, W. H. J.: Topics in Nevanlinna theory, Proc. NRL Conf. on Classical Function Theory, Naval Res. Lab., Washington, D.C., 1–32 (1970)
Fujimoto, H.: Value distribution theory of the gauss map of minimal surfaces. Aspects Math., E21 (1993)
Gariepy, R., Lewis, J.L.: Space analogues of some theorems for subharmonic and meromorphic functions. Ark. Mat. 13, 91–105 (1975)
Goldberg, A. A., Ostrovskii, I. V.: Distribution of values of meromophic functions, Nauka, Moscow, 1970 (Russian). Engl. Transl. AMS Transl., 236, Providence, (2008)
Hayman, W.K.: Multivalent Functions. Cambridge Univ. Press, Cambridge (1958)
Hayman, W.K., Kennedy, P.B.: Subharmonic Functions, vol. 1. Academic Press, New York (1976)
Kowalski, A., Marchenko, I.I.: On deviations and spreads of meromorphic minimal surfaces. Osaka J. Math. 57(1), 85–101 (2020)
Kowalski, A., Marchenko, I.I.: On Shea estimate for deviation of minimal surfaces. Houston J. Math. 48, 725–740 (2022)
Laine, I.: Nevanlinna Theory and Complex Differential Equations. De Gruyter, Berlin, New York (1993)
Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin, Heidelberg (1972)
Marchenko, I.I.: The growth of meromorphic minimal surfaces. Teor. Funkts. Funkts. Anal. Prilozh. 34, 95–98 (1979)
Marchenko, I.I.: An analogue of the second main theorem for uniform metric. Mat. Fiz. Anal. Geom. 5(3/4), 212–227 (1998)
Marchenko, I.I.: On deviations and defects of meromorphic functions of finite lower order, Ukr. Mat. Zhurn. 51(6), 796–803 (1999) (Russian; English transl. : Ukr. Math. J. 51(6) , 889–898 (1999))
Marchenko, I.I., Shcherba, A.I.: On magnitudes of deviations of meromorphic functions, Mat. Sb. 181, 3–24 (1990) (Russian; English transl.: Math. USSR Sb. 69(1), 1–24 (1991))
Markushevich, A.I.: Theory of Functions of Complex Variable. Prentice-Halls, USA (1965)
Petrenko, V.P.: Growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR Ser. Mat. 33(2), 414–454 (1969) (Russian; English transl.: Mathematics of the USSR-Izvestiya 33(2), 391-432 (1969))
Petrenko, V.P.: Growth of Meromorphic Functions. Vyshcha Shkola, Kharkov (1978). (Russian)
Petrenko, V.P.: Growth and distribution of values of minimal surfaces. Dokl. Akad. Nauk SSSR 256, 40–43 (1981). (Russian)
Petrenko, V.P.: The Entire Curves. Vyshcha Shkola, Kharkov (1984). (Russian)
Privalov, I.I.: Generalization of Jensen’s formula I. Izv. Akad. Nauk SSSR Ser. Mat. 6–7, 837–847 (1935). (Russian)
Rhoads, G., Weitsman, A.: The logarithmic derivative for minimal surfaces in \(\mathbb{R} ^3\). Comput. Methods Funct. Theory 4(1), 59–75 (2004)
Ronkin, L.I.: Introduction to the Theory of Entire Functions of Several Variables. Nauka, Moscow (1971). (Russian)
Ru, M.: Minimal Surfaces Through Nevanlinna Theory. De Gruyter, Berlin, Boston (2023)
Tafel, R.E.: Further Results in the Theory of Meromorphic Minimal Surfaces. In PhD. Thesis, University of California, Los Angeles, (1970)
Weitsman, A.: A theorem on Nevanlinna deficiencies. Acta Math. 128, 41–52 (1972)
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Communicated by Patrick Tuen Wai Ng.
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Kowalski, A., Marchenko, I.I. On Deviations of Meromorphic Minimal Surfaces of Finite Lower Order. Comput. Methods Funct. Theory (2024). https://doi.org/10.1007/s40315-024-00522-x
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DOI: https://doi.org/10.1007/s40315-024-00522-x
Keywords
- Minimal surfaces
- Deviations
- Subharmonic functions
- Baernstein’s \(T^{\star }-\)function
- Nevanlinna theory