Vol. 309, No. 2, 2020

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Value distribution properties for the Gauss maps of the immersed harmonic surfaces

Xingdi Chen, Zhixue Liu and Min Ru

Vol. 309 (2020), No. 2, 267–287
Abstract

We study the value distribution theory for the immersed harmonic surfaces and K-QC harmonic surfaces. We first investigate the value distribution properties for the generalized Gauss map Φ of an immersed harmonic surface, similar to the result of Fujimoto and Ru in the minimal surfaces case. After building a relation between Φ and the classical Gauss map 𝔫 for the K-QC harmonic surfaces, we derive that, for a complete harmonic and K-quasiconformal surface immersed in 3, if its unit normal 𝔫 omits seven directions in S2 and any three of which are not contained in a plane in 3, then the surface must be flat. In the last section, under an additional condition, we give an estimate of the Gauss curvature for the K-QC harmonic surfaces, generalizing the result of the minimal surfaces in the case that the unit normal 𝔫 omits a neighborhood of some fixed direction.

Keywords
harmonic immersion, quasiconformal mapping, value distribution theory, Hopf differential, conformal metric, Gauss map
Mathematical Subject Classification
Primary: 53C42, 53C43
Secondary: 30C65, 32H25
Milestones
Received: 16 December 2019
Revised: 5 March 2020
Accepted: 30 July 2020
Published: 14 January 2021
Authors
Xingdi Chen
School of Mathematical Sciences
Huaqiao University
Quanzhou
China
Zhixue Liu
Department of Mathematics, School of Sciences
Beijing University of Posts and Telecommunications
Beijing
China
Min Ru
Department of Mathematics
University of Houston
Houston, TX
United States