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
2. CJM 1999 (vol 51 pp. 470)
||Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations |
In this article we characterize the univalent harmonic mappings from
the exterior of the unit disk, $\Delta$, onto a simply connected
domain $\Omega$ containing infinity and which are solutions of the system
of elliptic partial differential equations $\fzbb = a(z)f_z(z)$
where the second dilatation function $a(z)$ is a finite Blaschke
product. At the end of this article, we apply our results to
nonparametric minimal surfaces having the property that the image
of its Gauss map is the upper half-sphere covered once or twice.
Keywords:harmonic mappings, minimal surfaces
Categories:30C55, 30C62, 49Q05