Masatoshi Kokubu

An analogue of minimal surface theory in (

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.

Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space

We introduce a new notion called the extended hyperbolic metrics, as a hyperbolic metric (i.e. metric of constant curvature − 1) with certain kinds of singularities defined on a Riemann surface, and we give several fundamental properties of such metrics. Extended hyperbolic metrics are closely related to space-like surfaces of constant mean curvature one (i.e. CMC-1 surfaces) in de Sitter 3-space S 1 3. For example, the singular set of a given CMC-1 surface in S 1 3 is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids in S 1 3 (i.e. weakly complete constant mean curvature 1 surfaces in S 1 3 of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are called S 1 3-catenoids. Since there is a bijection between the moduli space of S 1 3-catenoids and the moduli space of co-orientable extended hyperbolic metrics with two regular singularities, a classification of such hyperbolic metrics is also given. (Co-orientability of extended hyperbolic metrics is defined in this paper.)

Quadrics and Scherk towers

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.

Linear Weingarten Surfaces in Hyperbolic Three‐space

We report some results on surfaces in three‐dimensional hyperbolic space from the differential geometric viewpoint. We allow the surfaces some kind of singularities.

HYPERBOLIC GAUSS MAPS AND PARALLEL SURFACES IN HYPERBOLIC THREE-SPACE

In the differential geometry of surfaces in hyperbolic three-space H, surfaces of constant mean curvature one (CMC-1 surfaces) and surfaces of constant Gaussian curvature zero (flat surfaces) are well studied. (cf. Refs. 1,2,4,6–13, etc.) These surfaces have representation formulas which turn the complex function theory into the efficacious tool. They play the role such as the Weierstrass-Enneper formula in the minimal surface theory of Euclidean space. Galvez, Martinez and Milan (Ref. 3) also derived a representation formula for a certain class of Weingarten surfaces in H, which contains CMC-1 surfaces and flat surfaces. These Weingarten surfaces are the ones satisfying α(H − 1) = βK for some constants α and β, where K denotes the Gaussian curvature and H the mean curvature. In Ref. 5, the author gave a refinement for their representation formula. In these formulas, the hyperbolic Gauss maps (see Section 2 for the definition) play an important role. Moreover, one of the remarkable thing is that this class of Weingarten surfaces is closed under taking the parallel surfaces. The author thinks it rather important to investigate hyperbolic Gauss maps and parallel surfaces themselves, before representation formulas. For this reason, the purpose of this note is to collect some elementary properties of hyperbolic Gauss maps and parallel surfaces.

Hamiltonian Systems Derived from Constant Mean Curvature Surfaces in Hyperbolic Three-Space

We give a representation formula for surfaces of constant mean curvature in Euclidean or hyperbolic space, which is a natural generalization of Weierstrass-Enneper representation formula. The data (two functions) used in our formula should satisfy a certain system of differential equations. The system can be interpreted as an infinite dimensional Hamiltonian system. We investigate two finite-dimensional reductions in detail.