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The value distribution of Gauss maps of immersed harmonic surfaces with ramification

Abstract

Motivated by the result of Chen-Liu-Ru [1], we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in ℝn with ramification, which can be seen as a generalization of the results in the case of the minimal surfaces. In addition, we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.

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References

  1. Chen X D, Liu Z X, Ru M, Value distribution properties for the Gauss maps of the immersed harmonic surfaces. Pacific J Math, 2021, 309(2): 267–287

    MathSciNet  Article  Google Scholar 

  2. Chern S S. Minimal surfaces in an Euclidean space of N dimensions//Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse). Princeton, NJ: Princeton University Press,1965

    Google Scholar 

  3. Chern S S, Osserman R, Complete minimal surfaces in euclidean n-space. J Anal Math, 1967, 19: 15–34

    MathSciNet  Article  Google Scholar 

  4. Osserman R, Global properties of minimal surfaces in E3 and En. Ann of Math, 1964, 80: 340–364

    MathSciNet  Article  Google Scholar 

  5. Xavier F, The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere. Ann of Math, 1981, 113(2): 211–214

    MathSciNet  Article  Google Scholar 

  6. Xin Y L, Stable harmonic maps from complete manifolds. Acta Mathe Sci, 1989, 9(4): 415–420

    MathSciNet  Article  Google Scholar 

  7. Ru M, On the Gauss map of minimal surfaces immersed in ℝn. J Differential Geom, 1991, 34(2): 411–423

    MathSciNet  Article  Google Scholar 

  8. Ru M, Gauss map of minimal surfaces with ramification. Trans Amer Math Soc, 1993, 339(2): 751–764

    MathSciNet  MATH  Google Scholar 

  9. Fujimoto H. Value Distribution Theory of the Gauss Map of Minimal Surface in ℝm. Aspects of Mathematics, Vol E21. Friedr. Vieweg and Sohn, Braunschweig, 1993

  10. Osserman R, Ru M, An estimate for the Gauss curvature of minimal surfaces in ℝm whose Gauss map omits a set of hyperplanes. J Differential Geom, 1997, 45: 578–593

    MathSciNet  MATH  Google Scholar 

  11. Wang J S, Degree 3 algebraic minimal surfaces in the 3-sphere. Acta Math Sci, 2012, 32B(6): 2065–2084

    MathSciNet  Article  Google Scholar 

  12. Osserman R, Minimal surfaces in the large. Comment Math Helv, 1961, 35: 65–76

    MathSciNet  Article  Google Scholar 

  13. Fujimoto H, On the number of exceptional values of the Gauss maps of minimal surfaces. J Math Soc Japan, 1988, 40(2): 235–247

    MathSciNet  Article  Google Scholar 

  14. Fujimoto H. Modified defect relations for the Gauss map of minimal surfaces, II. J Differential Geom, 1990, 31(2): 365–385

    MathSciNet  Article  Google Scholar 

  15. Ros A. The Gauss map of minimal surfaces//Differential Geom, Valencia 2001. Singapore: World Scientific, 2002: 235–252

    Chapter  Google Scholar 

  16. Klotz T, Surfaces harmonically immersed in E3. Pacific J Math, 1967, 21: 79–87

    MathSciNet  Article  Google Scholar 

  17. Klotz T, A complete RΛ-harmonically immersed surface in E3 on which H ≠ 0. Proc Amer Math Soc, 1968, 19: 1296–1298

    MathSciNet  MATH  Google Scholar 

  18. Alarcon A, López F J, On harmonic quasiconformal immersions of surfaces in ℝ3. Trans Amer Math Soc, 2013, 365(4): 1711–1742

    MathSciNet  Article  Google Scholar 

  19. Milnor K T, Mapping surfaces harmonically into En. Proc Amer Math Soc, 1980, 78: 269–275

    MathSciNet  MATH  Google Scholar 

  20. Milnor K T. Are harmonically immersed surfaces at all like minimally immersed surafces?//Seminar on Minimal Submanifolds. Princeton University Press, 1984: 99–110

  21. Jensen G R, Rigoli M, Harmonically immersed surfaces of ℝn. Trans Amer Math Soc, 1988, 307(1): 363–372

    MathSciNet  MATH  Google Scholar 

  22. Campana F, Winkelmann J, A Brody theorem for orbifolds. Manuscripta Math, 2009, 128(2): 195–212

    MathSciNet  Article  Google Scholar 

  23. Milnor K T, Restrictions on the curvatures of ϕ-bounded surfaces. J Differential Geom, 1976, 11: 31–46

    MathSciNet  Article  Google Scholar 

  24. Kalaj D, The Gauss map of a harmonic surface. Indag Math (NS), 2013, 24(2): 415–427

    MathSciNet  Article  Google Scholar 

  25. Nochka E I, Uniqueness theorems for rational functions on algebraic varieties. Bul Akad Shtiintsa RSS Moldoven, 1979, 3: 27–31 (Russian)

    MathSciNet  MATH  Google Scholar 

  26. Sakai F, Degeneracy of holomorphic maps with ramification. Invent Math, 1974, 26: 213–229

    MathSciNet  Article  Google Scholar 

  27. 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, 2014, 32: 130–138.

    MathSciNet  Article  Google Scholar 

  28. Nochka E I, On the theory of meromorphic functions. Soviet Math, Dokl, 1983, 27(2): 377–381

    MATH  Google Scholar 

  29. Chen W. Cartan conjecture: Defect Relation for Merommorphic Maps from Parabolic Manifold to Projective Space [D]. University of Notre Dame, 1987

  30. Chen X D, Li Y Z, Liu Z X, Ru M, Curvature estimate on an open Riemann surface with the induced metric. Math Z, 2020, 298: 451–467

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We are grateful to Professor Min Ru for his useful advice and conversation.

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Correspondence to Zhixue Liu.

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The first author was supported by the Fundamental Research Funds for the Central Universities (500421360). The second author was supported by NNSF of China (11571049, 12071047). The third named author was supported by NNSF of China (11971182), NSF of Fujian Province of China (2019J01066).

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Liu, Z., Li, Y. & Chen, X. The value distribution of Gauss maps of immersed harmonic surfaces with ramification. Acta Math Sci 42, 172–186 (2022). https://doi.org/10.1007/s10473-022-0109-9

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  • DOI: https://doi.org/10.1007/s10473-022-0109-9

Key words

  • value distribution
  • harmonic surfaces
  • quasiconformal mappings
  • conformal metric
  • Gauss map

2010 MR Subject Classification

  • 32H25
  • 30D35
  • 53C42
  • 30C65