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

  Published:2015-04-15
 Printed: Dec 2015
  • Yu Kawakami,
    Graduate School of Natural Science and Technology, Kanazawa university, Kanazawa, 920-1192, Japan
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Abstract

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: Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem. Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem.
MSC Classifications: 53C42, 30D35, 30F45, 53A10, 53A15 show english descriptions Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Distribution of values, Nevanlinna theory
Conformal metrics (hyperbolic, Poincare, distance functions)
Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Affine differential geometry
53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
30D35 - Distribution of values, Nevanlinna theory
30F45 - Conformal metrics (hyperbolic, Poincare, distance functions)
53A10 - Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
53A15 - Affine differential geometry
 

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