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Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

Published online by Cambridge University Press:  20 November 2018

Yu Kawakami
Yu Kawakami*
Affiliation:
Graduate School of Natural Science and Technology, Kanazawa university, Kanazawa, 920-1192, Japan. e-mail: [email protected]
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Abstract

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We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Keywords

53C4230D3530F4553A1053A15Gauss mapminimal surfaceconstant mean curvature surfacefrontramificationomitted valuethe Ahlfors island theoremunicity theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1411 - 1434
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Ahlfors, L. V., Zur Théorie der Überlagerungsflächen. Acta Math 65(1935), no. 1,157–194.http://dx.doi.Org/10.1007/BF02420945 Google Scholar
[2] Bergweiler, W., The role of the Ahlfors five islands theorem in complex dynamics. Conform. Geom. Dyn. 4(2000), 22–34.http://dx.doi.org/10.1090/S1088–4173-00-00057-6 Google Scholar
[3] Bryant, R. L., Surfaces of mean curvature one in hyperbolic space. Théorie des variétés minimales et applications. (Palaiseau, 1983–1984). Astérisque 154–155(1987), 12, 321–347, 353.Google Scholar
[4] Calabi, E., Improper affine hypersurfaces of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5(1958), 105–126.http://dx.doi.Org/10.1307/mmj71028998055 Google Scholar
[5] Calabi, E., Examples of Bernstein problems for some nonlinear equations. In: Global Analysis (Proc. Sympos. Pure Math., 15, Berkeley, Calif., 1968), American Mathematical Society, Providence, RI, 1970, pp. 223–230.Google Scholar
[6] Chen, B. Y. and Morvan, J. M., Géométrie des surfaces lagrangiennes de C2. J. Math. Pures Appl. 66(1987), no. 3, 321–335.Google Scholar
[7] Cheng, S. Y. and Yau, S. T., Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. of Math. 104(1976), no. 3, 407–419.http://dx.doi.Org/10.2307/1970963 Google Scholar
[8] Chern, S. S., Complex analytic mappings of Riemann surfaces. I. Amer. J. Math. 82(1960), 323–337.http://dx.doi.Org/10.2307/2372738 Google Scholar
[9] Estudillo, F. J. M. and Romero, A., Generalized maximal surfaces in Lorentz–Minkowski space L3. Math. Proc. Cambridge Philos. Soc. 111(1992), no. 3, 515–524.http://dx.doi.Org/10.1017/S0305004100075587 Google Scholar
[10] Fujimoto, H., On the number of exceptional values of the Gauss map of minimal surfaces. J. Math. Soc. Japan 40(1988), no. 2, 235–247.http://dx.doi.org/10.2969/jmsjV04020235 Google Scholar
[11] Fujimoto, H., On the Gauss curvature of minimal surfaces. J. Math. Soc. Japan 44(1992), no. 3, 427–439.http://dx.doi.org/10.2969/jmsj704430427 Google Scholar
[12] Fujimoto, H., Value distribution theory of the Gauss map of minimal surfaces in Rm. Aspects of Mathematics, E21, Friedr. Vieweg & Sohn, Braunschweig, 1993.Google Scholar
[13] Fujimoto, H., Unicity theorems for the Gauss maps of complete minimal surfaces. J. Math. Soc. Japan 45(1993), no. 3, 481–487.http://dx.doi.Org/10.2969/jmsj704530481 Google Scholar
[14] Gálvez, J. A., Martinez, A., and Milan, F., Flat surfaces in hyperbolic 3-space. Math. Ann. 316(2000), no. 3, 419–435.http://dx.doi.Org/10.1007/s002080050337 Google Scholar
[15] Imaizumi, T. and Kato, S., Flux of simple ends of maximal surfaces in R2,1. Hokkaido Math. J. 37(2008), no. 3, 561–610.http://dx.doi.org/10.14492/hokmj71253539536 Google Scholar
[16] Jörgens, K., Uber die Lösungen der differentialgleichung rt−s2 = 1. Math. Ann. 127(1954), 130–134.http://dx.doi.Org/10.1007/BF01361114 Google Scholar
[17] Kawakami, Y., Ramification estimates for the hyperbolic Gauss map. Osaka J. Math. 46(2009), 1059–1076.Google Scholar
[18] Kawakami, Y., On the maximal number of exceptional values of Gauss maps for various classes of surfaces. Math. Z. 274(2013), no. 3–4, 1249–1260.http://dx.doi.org/10.1007/s00209-012-1115-8 Google Scholar
[19] Kawakami, Y., A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space. Geom. Dedicata 171(2014), 387–396.http://dx.doi.Org/10.1007/s10711-013-9904-8 Google Scholar
[20] Kawakami, Y., Kobayashi, R. and Miyaoka, R., The Gauss map of pseudo-algebraic minimal surfaces. Forum Math. 20(2008), no. 6, 1055–1069.http://dx.doi.org/10.1515/FORUM.2008.047 Google Scholar
[21] Kawakami, Y. and Nakajo, D., Value distribution of the Gauss map of improper affine spheres. J. Math. Soc. Japan 64(2012), 799–821.http://dx.doi.Org/10.2969/jmsj706430799 Google Scholar
[22] Klotz, T. and Sario, L., Gaussian mappings of arbitrary minimal surfaces. J. Analyse Math. 17(1966), 209–217.http://dx.doi.org/10.1007/BF02788657 Google Scholar
[23] Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski space L3. Tokyo J. Math. 6(1983), no. 2, 297–309. http://dx.doi.org/10.3836/tjm/ 1270213872 Google Scholar
[24] Kobayashi, R., Toward Nevanlinna theory as a geometric model for Diophantine approximation. Sugaku Expositions 16(2003), no. 1, 39–79.Google Scholar
[25] Kokubu, M., Rossman, W., Saji, K., Umehara, M., and Yamada, K., Singularities of flat fronts in hyperbolic space. Pacific J. Math. 221(2005), no. 2, 303–351.http://dx.doi.org/10.2140/pjm.2005.221.303 Google Scholar
[26] Kokubu, M., Rossman, W., Umehara, M., and Yamada, K., Flat fronts in hyperbolic 3-space and their caustics. J. Math. Soc. Japan 59(2007), no. 1, 265–299.http://dx.doi.Org/10.2969/jmsj71180135510 Google Scholar
[27] Kokubu, M., Rossman, W., Umehara, M., and Yamada, K., Asymptotic behavior of flat surfaces in hyperbolic 3-space. J. Math. Soc. Japan 61(2009), no. 3, 799–852.http://dx.doi.org/10.2969/jmsj706130799 Google Scholar
[28] Kokubu, M., Umehara, M., and Yamada, K., An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2, C). Osaka J. Math. 40(2003), 697–715.Google Scholar
[29] Kokubu, M., Umehara, M., and Yamada, K., Flat fronts in hyperbolic 3-space. Pacific J. Math. 216(2004), no. 1,149–175.http://dx.doi.org/10.2140/pjm.2004.216.149 Google Scholar
[30] Lawson, H.B Jr., Lectures on minimal submanifolds. Vol. I. Second ed., Mathematics Lecture Series, 9, Publish or Perish, Inc., Wilington, Del., 1980.Google Scholar
[31] Martínez, A., Improper affine maps. Math. Z. 249(2005), no. 4, 755–766.http://dx.doi.org/10.1007/s00209-004-0728-y Google Scholar
[32] Nakajo, D., A representation formula for indefinite improper affine spheres. Results Math. 55(2009), no. 1-2, 139–159.http://dx.doi.org/10.1007/s00025-009-0399-4 Google Scholar
[33] Nevanlinna, R., Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen. Acta Math. 48(1926), no. 3-4, 367–391.http://dx.doi.org/10.1007/BF02565342 Google Scholar
[34] Nevanlinna, R., Analytic functions. Die Grundlehren der mathematischen Wissenschaften, 162, Springer-Verlag, New York-Berlin, 1970.Google Scholar
[35] Noguchi, J. and Ochiai, T., Geometric function theory in several complex variables. Translations of Mathematical Monographs, 80, American Mathematical Society, Providence, RI, 1990.Google Scholar
[36] Noguchi, J. and Winkelmann, J., Nevanlinna theory in several complex variables and Diophantine approximation. Grundlehren der Mathematischen Wissenschaften, 350, Springer, Tokyo, 2014.Google Scholar
[37] Osserman, R., Proof of a conjecture of Nirenberg. Comm. Pure Appl. Math. 12(1959), 229–232.http://dx.doi.Org/10.1OO2/cpa.3160120203 Google Scholar
[38] Osserman, R., Global properties of minimal surfaces in E3 and En. Ann. of Math. 80(1964), 340–364.http://dx.doi.Org/10.2307/1970396 Google Scholar
[39] Osserman, R., A survey of minimal surfaces. Second ed., Dover Publications Inc., New York, 1986.Google Scholar
[40] Ru, M., Gauss map of minimal surface with ramification. Trans. Amer. Math. Soc. 339(1993), no. 2,751–764.Google Scholar
[41] Osserman, R., Nevanlinna theory and its relation to Diophantine approximation. World Scientific Publishing Co, Inc., River Edge, NJ, 2001.Google Scholar
[42] Saji, K., Umehara, M. and Yamada, K., The geometry of fronts. Ann. of Math. 169(2009), no. 2,491–529.http://dx.doi.Org/10.4007/annals.2009.169.491 Google Scholar
[43] Sasaki, S., On complete flat surfaces in hyperbolic 3-space. Kōdai Math Sem. Rep. 25(1973), 449–457.http://dx.doi.org/10.2996/kmj71138846871 Google Scholar
[44] Umehara, M. and Yamada, K., Complete surfaces of constant mean curvature one in the hyperbolic 3-space. Ann. of Math. 137(1993), no. 3, 611–638.http://dx.doi.Org/10.2307/2946533 Google Scholar
[45] Umehara, M. and Yamada, K., A duality on CMC-1 surfaces in hyperbolic space, and a hyperbolic analogue of the Osserman inequality. Tsukuba J. Math. 21(1997), no. 1, 229–237.Google Scholar
[46] Umehara, M. and Yamada, K., Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35(2006),no. 1, 13–40. http://dx.doi.org/10.14492/hokmj71285766302 Google Scholar
[47] Umehara, M. and Yamada, K., Applications of a completeness lemma in minimal surface theory to various classes of surfaces. Bull. London Math. Soc. 43(2011), no. 1,191-199; Corrigendum, Bull. London Math. Soc. 44(2012), no. 3, 617–618.http://dx.doi.Org/10.1112/blms/bdqO94 Google Scholar
[48] Volkov, Ju. A. and Vladimirova, S. M., Isometric immersions of the Euclidean plane in Lobač evskiĭ space. (Russian). Mat. Zametki 10(1971), 327–332.Google Scholar
[49] Xavier, F., The Gauss map of a complete nonflat minimal surface cannot omit 7 points of the sphere. Ann. of Math. 113(1981), no. 1, 211-214; Erratum, Ann. of Math. 115(1982), no. 3, 667.http://dx.doi.Org/10.2307/1971139 Google Scholar
[50] Yu, Z.-H., The value distribution of the hyperbolic Gauss map. Proc. Amer. Math. Soc. 125(1997), no. 10, 2997–3001.http://dx.doi.org/10.1090/S0002-9939-97-03937-3 Google Scholar