2,084 research outputs found
Value distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space
We give an effective estimate for the totally ramified value number of the
hyperbolic Gauss maps of complete flat fronts in the hyperbolic three-space. As
a corollary, we give the upper bound of the number of exceptional values of
them for some topological cases. Moreover, we obtain some new examples for this
class.Comment: 14 pages, to appear in Houston Journal of Mathematic
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.Comment: 26 pages. arXiv admin note: substantial text overlap with
arXiv:1205.478
Ramification estimates for the hyperbolic Gauss map
We give the best possible upper bound on the number of exceptional values and
the totally ramified value number of the hyperbolic Gauss map for
pseudo-algebraic constant mean curvature one surfaces in the hyperbolic
three-space and some partial results on the Osserman problem for algebraic
case. Moreover, we study the value distribution of the hyperbolic Gauss map for
complete constant mean curvature one faces in de Sitter three-space.Comment: 16 pages, corrected some typos. OCAMI Preprint Series 08-1, to appear
in Osaka Journal of Mathematic
On the maximal number of exceptional values of Gauss maps for various classes of surfaces
The main goal of this paper is to reveal the geometric meaning of the maximal
number of exceptional values of Gauss maps for several classes of immersed
surfaces in space forms, for example, complete minimal surfaces in the
Euclidean three-space, weakly complete improper affine spheres in the affine
three-space and weakly complete flat surfaces in the hyperbolic three-space.
For this purpose, we give an effective curvature bound for a specified
conformal metric on an open Riemann surface.Comment: 13 pages, to appear in Mathematische Zeitschrif
The Gauss map and total curvature of complete minimal Lagrangian surfaces in the complex two-space
The purpose of this paper is to reveal the relationship between the total
curvature and the global behavior of the Gauss map of a complete minimal
Lagrangian surface in the complex two-space. To achieve this purpose, we show
the precise maximal number of exceptional values of the Gauss map for a
complete minimal Lagrangian surface with finite total curvature in the complex
two-space. Moreover, we prove that if the Gauss map of a complete minimal
Lagrangian surface which is not a Lagrangian plane omits three values, then it
takes all other values infinitely many times.Comment: 11 page
A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space
We provide an effective ramification theorem for the ratio of canonical forms
of a weakly complete flat front in the hyperbolic three-space. Moreover we give
the two applications of this theorem, the first one is to show an analogue of
the Ahlfors islands theorem for it and the second one is to give a simple proof
of the classification of complete nonsingular flat surfaces in the hyperbolic
three-space.Comment: 11 pages, no figure, to appear in Geometriae Dedicata. arXiv admin
note: substantial text overlap with arXiv:1004.148
The Gauss images of complete minimal surfaces of genus zero of finite total curvature
This paper aims to present a systematic study on the Gauss images of complete
minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and
Euclidean 4-space. We focus on the number of omitted values and the total
weight of a number of totally ramified values of their Gauss maps. In
particular, we construct new complete minimal surfaces of finite total
curvature whose Gauss maps have 2 omitted values and 1 totally ramified value
of order 2, that is, the total weight of a number of totally ramified values of
their Gauss maps are 2.5 in Euclidean 3-space and Euclidean 4-space,
respectively. Moreover we discuss several outstanding problems in this study.Comment: 22 pages, 1 figur
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