1. CJM 2015 (vol 67 pp. 1411)
||Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces|
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.
Categories:53C42, 30D35, 30F45, 53A10, 53A15