Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T22:39:19.555Z Has data issue: false hasContentIssue false
  • English
  • Français

GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES

Part of: Global differential geometry Classical differential geometry Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  18 March 2019

PHAM HOANG HA
PHAM HOANG HA*
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136, XuanThuy str., Hanoi, Vietnam email ha.ph@hnue.edu.vn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

MSC classification

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Secondary: 30D35: Distribution of values, Nevanlinna theory 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 275 - 297
Check for updates
Copyright
© 2019 Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2018.03.

References

Ahlfors, L. V., An extension of Schwarzs lemma, Trans. Amer. Math. Soc. 43 (1938), 359–364.Google Scholar
Bryant, R. L., Surfaces of mean curvature one in hyperbolic space, Astérisque 154–155 (1987), 321–347.Google Scholar
Calabi, E., “Examples of Bernstein problems for some nonlinear equations”, in Global Analysis, Proceedings of Symposia in Pure Mathematics 15, (eds. Chern, S.-S. and Smale, S.) American Mathematical Society, Providence, RI, 1968, 223–230.Google Scholar
Chen, B.-Y. and Morvan, J.-M., Géométrie des surfaces lagrangiennes de C2, J. Math. Pures Appl. 66 (1987), 321–325.Google Scholar
Chern, S. S. and Osserman, R., Complete minimal surface in Euclidean n-space, J. Analyse Math. 19 (1967), 15–34.10.1007/BF02788707Google Scholar
Estudillo, F. J. M. and Romero, A., Generalized maximal surfaces in Lorentz–Minkowski space L 3, Math. Proc. Cambridge Philos. Soc. 111 (1992), 515–524.10.1017/S0305004100075587Google Scholar
Forster, O., Lectures on Riemann Surfaces, Springer, NewYork, 1981.10.1007/978-1-4612-5961-9Google Scholar
Fujimoto, H., On the number of exceptional values of the Gauss map of minimal surfaces, J. Math. Soc. Japan 40 (1988), 237–249.Google Scholar
Fujimoto, H., On the Gauss curvature of minimal surfaces, J. Math. Soc. Japan 44 (1992), 427–439.10.2969/jmsj/04430427Google Scholar
Fujimoto, H., Unicity theorems for the Gauss maps of complete minimal surfaces, J. Math. Soc. Japan 45 (1993), 481–487.10.2969/jmsj/04530481Google Scholar
Fujimoto, H., Unicity theorems for the Gauss maps of complete minimal surfaces II, Kodai Math. J. 16 (1993), 335–354.10.2996/kmj/1138039844Google Scholar
Fujimoto, H., “Value distribution theory of the Gauss map of minimal surfaces in Rm”, in Aspect of Math. Vol. E21, Vieweg, Wiesbaden, 1993.Google Scholar
Gálvez, J. A., Martínez, A. and Milán, F., Flat surfaces in hyperbolic 3-space, Math. Ann. 316 (2000), 419–435.Google Scholar
Ha, P. H., An estimate for the Gaussian curvature of minimal surfaces in Rm whose Gauss map is ramified over a set of hyperplanes, Differential Geom. Appl. 32 (2014), 130–138.10.1016/j.difgeo.2013.11.005Google Scholar
Ha, P. H. and Kawakami, Y., A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space, Canad. Math. Bull. 61(2) (2018), 292–300.10.4153/CMB-2017-015-0Google Scholar
Imaizumi, T. and Kato, S., Flux of simple ends of maximal surfaces in R2, 1, Hokkaido Math. J. 37 (2008), 561–610.10.14492/hokmj/1253539536Google Scholar
Jin, L. and Ru, M., Algebraic curves and the Gauss map of algebraic minimal surfaces, Differential Geom. Appl. 25 (2007), 701–712.10.1016/j.difgeo.2007.06.014Google Scholar
Kawakami, Y., On the maximal number of exceptional values of Gauss maps for various classes of surfaces, Math. Z. 274 (2013), 1249–1260.10.1007/s00209-012-1115-8Google Scholar
Kawakami, Y., A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space, Geom. Dedicata 171 (2014), 387–396.10.1007/s10711-013-9904-8Google Scholar
Kawakami, Y., Function-theoretic properties for the Gauss maps for various classes of surfaces, Canad. J. Math. 67(6) (2015), 1411–1434.10.4153/CJM-2015-008-5Google Scholar
Kawakami, Y., Value distribution for the Gauss maps of various classes of surfaces, Sūgaku 69(1) (2017), 56–69.Google Scholar
Kawakami, Y., Kobayashi, R. and Miyaoka, R., The Gauss map of pseudo-algebraic minimal surfaces, Forum Math. 20(6) (2008), 1055–1069.10.1515/FORUM.2008.047Google Scholar
Kawakami, Y. and Nakajo, D., Value distribution of the Gauss map of improper affine spheres, J. Math. Soc. Japan 64 (2012), 799–821.10.2969/jmsj/06430799Google Scholar
Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski space L 3, Tokyo J. Math. 6 (1983), 297–309.10.3836/tjm/1270213872Google Scholar
Kokubu, M., Rossman, W., Saji, K., Umehara, M. and Yamada, K., Singularities of flat fronts in hyperbolic space, Pacific J. Math. 221 (2005), 303–351.10.2140/pjm.2005.221.303Google Scholar
Kokubu, M., Rossman, W., Umehara, M. and Yamada, K., Flat fronts in hyperbolic 3-space and their caustics, J. Math. Soc. Japan 59 (2007), 265–299.10.2969/jmsj/1180135510Google Scholar
Kokubu, M., Rossman, W., Umehara, M. and Yamada, K., Asymptotic behavior of flat surfaces in hyperbolic 3-space, J. Math. Soc. Japan 61 (2009), 799–852.10.2969/jmsj/06130799Google Scholar
Kokubu, M., Umehara, M. and Yamada, K., An elementary proof of Smalls 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
Kokubu, M., Umehara, M. and Yamada, K., Flat fronts in hyperbolic 3-space, Pacific J. Math. 216 (2004), 149–175.10.2140/pjm.2004.216.149Google Scholar
Liu, X. and Pang, X., Normal family theory and Gauss curvature estimate of minimal surfaces in Rm, J. Differential Geom. 103(2) (2016), 297–318.10.4310/jdg/1463404120Google Scholar
Martínez, A., Improper affine maps, Math. Z. 249 (2005), 755–766.10.1007/s00209-004-0728-yGoogle Scholar
Nakajo, D., A representation formula for indefinite improper affine spheres, Results Math. 55 (2009), 139–159.10.1007/s00025-009-0399-4Google Scholar
Nevanlinna, R., Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen, Acta Math. 48 (1926), 367–391.10.1007/BF02565342Google Scholar
Osserman, R. and Ru, M., An estimate for the Gauss curvature on minimal surfaces in Rm whose Gauss map omits a set of hyperplanes, J. Differential Geom. 46 (1997), 578–593.10.4310/jdg/1214459977Google Scholar
Park, J. and Ru, M., Unicity results for Gauss maps of minimal surfaces immersed in Rm, J. Geom. 108 (2017), 481–499.Google Scholar
Ros, A., “The Gauss map of minimal surfaces”, in Differential geometry, Valencia (2001), World Sci. Publ., River Edge, NJ, 2002, 235–252.10.1142/9789812777751_0022Google Scholar
Ru, M. and Ugur, G., Uniqueness results for algebraic and holomorphic curves into Pn(C), Internat. J. Math. 28(9) (2017), 1740003, 27 pp.10.1142/S0129167X17400031Google Scholar
Saji, K., Umehara, M. and Yamada, K., The geometry of fronts, Ann. of Math. 169 (2009), 491–529.10.4007/annals.2009.169.491Google Scholar
Umehara, M. and Yamada, K., Complete surfaces of constant mean curvature one in the hyperbolic 3-space, Ann. of Math. 137 (1993), 611–638.10.2307/2946533Google Scholar
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), 229–237.10.21099/tkbjm/1496163174Google Scholar
Umehara, M. and Yamada, K., Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006), 13–40.10.14492/hokmj/1285766302Google Scholar
Umehara, M. and Yamada, K., Applications of a completeness lemma in minimal surface theory to various classes of surfaces, Bull. Lond. Math. Soc. 43 (2011), 191–199; Corrigendum, Bull. Lond. Math. Soc. 44 (2012), 617–618.10.1112/blms/bdq094Google Scholar
Yu, Z., Value distribution of hyperbolic Gauss maps, Proc. Amer. Math. Soc. 125 (1997), 2997–3001.10.1090/S0002-9939-97-03937-3Google Scholar