An estimation of the Gauss curvature and the modified defect relation for the Gauss map of immersed harmonic surfaces in $\mathbb{R}^n$

Abstract

In this paper, we study the estimation of Gauss curvature for $K$-quasiconformal harmonic surface in ${\mathbb R}^3$ and present an accurate improvement of the previous result in [6, Theorem 5.2]. Let $X:M\rightarrow{\mathbb R}^3$ denote a $K$-quasiconformal harmonic surface and let $\mathfrak{n}$ be the unit normal map of $M$. We define $d(p)$ as the distance from point $p$ to the boundary of $M$ and $\mathcal{K}(p)$ as the Gauss curvature of $M$ at $p$. Assuming that the Gauss map (i.e., the normal $\mathfrak{n}$) omits $7$ directions $\mathbf{d}_1,\cdots,\mathbf{d}_7$ in $S^2$ with the property that any three of these directions are not contained in a plane in ${\mathbb R}^3$. Then there exists a positive constant $C$ depending only on $\mathbf{d}_1,\cdots,\mathbf{d}_7$ such that \begin{equation*} |\mathcal{K}(p)|\leq C/d(p)^2 \end{equation*} for all points $p\in M$. Furthermore, a modified defect relation for the generalized Gauss map of the immersed harmonic surfaces in $\mathbb{R}^n(n\geq 3)$ is verified.