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Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

Published online by Cambridge University Press:  20 November 2018

Yu Kawakami
Yu Kawakami*
Affiliation:
Graduate School of Natural Science and Technology, Kanazawa university, Kanazawa, 920-1192, Japan. e-mail: [email protected]
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Abstract

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We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Keywords

53C4230D3530F4553A1053A15Gauss mapminimal surfaceconstant mean curvature surfacefrontramificationomitted valuethe Ahlfors island theoremunicity theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1411 - 1434
Copyright
Copyright © Canadian Mathematical Society 2015

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