891 research outputs found
Applications of a completeness lemma in minimal surface theory to various classes of surfaces
We give several applications of a lemma on completeness used by Osserman to
show the meromorphicity of Weierstrass data for complete minimal surfaces with
finite total curvature. Completeness and weak completeness are defined for
several classes of surfaces which admit singular points. The completeness lemma
is a useful machinery for the study of completeness in these classes of
surfaces. In particular, we show that a constant mean curvature one (i.e.
CMC-1) surface in de Sitter 3-space is complete if and only if it is weakly
complete, the singular set is compact and all the ends are conformally
equivalent to a puntured disk.Comment: 9 page
Hyperbolic Schwarz map for the hypergeometric differential equation
The Schwarz map of the hypergeometric differential equation is studied since
the beginning of the last century. Its target is the complex projective line,
the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is
the hyperbolic 3-space. This map can be considered to be a lifting to the
3-space of the Schwarz map. This paper studies the singularities of this map,
and visualize its image when the monodromy group is a finite group or a typical
Fuchsian group. General cases will be treated in a forthcoming paper.Comment: 16 pages, 8 figure
Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts
In the previous paper (math.CA/0609196) we defined a map, called the
hyperbolic Schwarz map, from the one-dimensional projective space to the
three-dimensional hyperbolic space by use of solutions of the hypergeometric
differential equation, and thus obtained closed flat surfaces belonging to the
class of flat fronts. We continue the study of such flat fronts in this paper.
First, we introduce the notion of derived Schwarz maps of the hypergeometric
differential equation and, second, we construct a parallel family of flat
fronts connecting the classical Schwarz map and the derived Schwarz map.Comment: 15 pages, 12 figure
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