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.

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