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

TL;DR

The paper addresses curvature control for the Gauss map of immersed $K$-quasiconformal harmonic surfaces in ${\mathbb R}^3$ and establishes a refined curvature bound when the Gauss map omits seven directions with no three coplanar, via a link between the intrinsic metric and a conformal Klotz-type metric. It advances value distribution theory for harmonic surfaces by proving a modified defect relation for the generalized Gauss map in ${\mathbb R}^n$, using a pseudo-metric constructed from Nochka weights and derived curves. A key improvement over prior work eliminates a restrictive condition and recovers Chen-2021's six-direction bound for complete surfaces, while also providing a broad framework that extends to the generalized Gauss map and higher dimensions. The results integrate curvature estimates, metric comparisons, and defect theory to yield a unified approach to curvature control and direction-omission phenomena for harmonic and $K$-QC surfaces, with potential implications for geometric analysis in higher codimensions.

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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Theorem 1.4
• Remark 1.2
• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• proof : The proof of Theorem \ref{['mainthm-2']}