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GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES

Part of: Entire and meromorphic functions, and related topics Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  18 March 2019

PHAM HOANG HA
PHAM HOANG HA*
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136, XuanThuy str., Hanoi, Vietnam email [email protected]
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Abstract

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

MSC classification

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Secondary: 30D35: Distribution of values, Nevanlinna theory 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 275 - 297
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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Footnotes

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2018.03.

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