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

DOI: 10.2140/pjm.2020.309.267

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 $\Phi$ of an immersed harmonic surface, similar to the result of Fujimoto and Ru in the minimal surfaces case. After building a relation between $\Phi$ and the classical Gauss map $\mathfrak{𝔫}$ for the $K$-QC harmonic surfaces, we derive that, for a complete harmonic and $K$-quasiconformal surface immersed in ${ℝ}^3$, if its unit normal $\mathfrak{𝔫}$ omits seven directions in $S^2$ and any three of which are not contained in a plane in ${ℝ}^3$, 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 $\mathfrak{𝔫}$ 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