2023/08/21 by Zhixue Liu, Liu, Zhixue, Li, Yezhou
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2308.10507
openalex publication_date 2023/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the estimation of Gauss curvature for K-quasiconformal harmonic surface in \mathbb R3 and present an accurate improvement of the previous result in [6, Theorem 5.2]. Let X:M→\mathbb R3 denote a K-quasiconformal harmonic surface and let \mathfrakn be the unit normal map of M. We define d(p) as the distance from point p to the boundary of M and K(p) as the Gauss curvature of M at p. Assuming that the Gauss map (i.e., the normal \mathfrakn) omits 7 directions d1,⋯,d7 in S2 with the property that any three of these directions are not contained in a plane in \mathbb R3. Then there exists a positive constant C depending only on d1,⋯,d7 such that |K(p)|≤ C/d(p)2 for all points p∈ M. Furthermore, a modified defect relation for the generalized Gauss map of the immersed harmonic surfaces in ℝn(n≥ 3) is verified.