We 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 , and 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)
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)
The papers [Gálvez et al. 2000, Kokubu et al. 2003, Kokubu et al. 2005] gave a method of constructing flat surfaces in the three-dimesnional hyperbolic space. Such surfaces have generically singularities, since any closed nonsigular flat surface is isometric to a horosphere or a hyperbolic cylinder. In the paper [Sasaki et al. 2006], we defined a map,… (More)
Using the Bryant representation, we define a new flux on homology classes of CMC-1 surfaces in H 3 , satisfying a balancing formula which is useful to show nonexistence of certain kinds of complete CMC-1 surfaces.
Introduction This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and for studying constant mean curvature (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu, and is often called the DPW method after them. There already exist a number of other introductions to this method, but all of them… (More)