#### Vol. 309, No. 2, 2020

 Recent Issues Vol. 311: 1 Vol. 310: 1  2 Vol. 309: 1  2 Vol. 308: 1  2 Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
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 $\Phi$ 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