Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces
Printed: Dec 2015
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.
Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem.
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