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Abstract
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We study the value distribution theory for the immersed harmonic surfaces and
-QC
harmonic surfaces. We first investigate the value distribution properties for the generalized
Gauss map
of an immersed harmonic surface, similar to the result of Fujimoto and
Ru in the minimal surfaces case. After building a relation between
and the classical
Gauss map
for the
-QC
harmonic surfaces, we derive that, for a complete harmonic and
-quasiconformal surface
immersed in
, if its
unit normal
omits
seven directions in
and any three of which are not contained in a plane in
,
then the surface must be flat. In the last section, under an additional
condition, we give an estimate of the Gauss curvature for the
-QC
harmonic surfaces, generalizing the result of the minimal surfaces in the case that the unit
normal
omits a neighborhood of some fixed direction.
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Keywords
harmonic immersion, quasiconformal mapping, value
distribution theory, Hopf differential, conformal metric,
Gauss map
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Mathematical Subject Classification
Primary: 53C42, 53C43
Secondary: 30C65, 32H25
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Milestones
Received: 16 December 2019
Revised: 5 March 2020
Accepted: 30 July 2020
Published: 14 January 2021
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