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


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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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).

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Key words

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

2010 MR Subject Classification

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