VALUE DISTRIBUTION PROPERTIES FOR GAUSS MAPS OF IMMERSED HARMONIC SURFACES RAMIFIED OVER HYPERSURFACES

doi:10.1007/s10473-024-0616-y

Abstract

Abstract: In this paper, we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in $\mathbb{R}^{m}$, which is the case where the generalized Gauss map $\Phi$ is ramified over a family of hypersurfaces $\{Q_{j}\}_{j=1}^{q}$ in $\mathbb{P}^{m-1}(\mathbb{C})$ located in the $N$-subgeneral position. In addition, we investigate the Gauss curvature estimate for the $K$-quasiconformal harmonic surfaces immersed in $\mathbb{R}^{3}$ whose Gauss maps are ramified over a family of hypersurfaces located in the $N$-subgeneral position.

Key words: immersed harmonic surface, generalized Gauss map, hypersurface, ramification, quasiconformal mapping, Gauss curvature

CLC Number:

32H25

Canhui LU, Xingdi CHEN. VALUE DISTRIBUTION PROPERTIES FOR GAUSS MAPS OF IMMERSED HARMONIC SURFACES RAMIFIED OVER HYPERSURFACES[J].Acta mathematica scientia,Series B, 2024, 44(6): 2341-2360.

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