We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called 'flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded. Moreover, we shall give new examples for which equality holds.
We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other complete maximal surfaces with singularities and we show that they satisfy an Osserman-type inequality.
In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given… (More)
We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is deeply related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails.
A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature −1 has two natural notions of " total curvature " — one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute… (More)
In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4π are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given.
In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H 3. Gálvez, Martínez and Milán showed that when the singular set does not accumulate at an end, then the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called pitch… (More)
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… (More)
In the paper  we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hy-perbolic 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.… (More)