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It is well-known that the unit cotangent bundle of any Riemann-ian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a… (More)
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.
After Gálvez, Martínez and Milán discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space H 3 , the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H 3. In the present work we broaden the notion of completeness to weak completeness, and of front to… (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)
In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Osserman type inequality for such… (More)
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a certain class of Weingarten surfaces. From… (More)
We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hy-perbolic Gauss map via another hyperbolic Gauss map which is… (More)
1 Forward These notes are about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using… (More)
We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. SL(n, C)/ SU(n)), which contains minimal surfaces in R n and constant mean curvature 1 surfaces in H 3. A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.