Reiko Aiyama

Kenmotsu-Bryant type representation formula for constant mean curvature spacelike surfaces in H31(c2)

Our primary object of this paper is to give a representation formula for constant mean curvature spacelike surfaces (including maximal surfaces) in the anti-de Sitter 3-space H31(c2) of constant negative curvature c2, similar to the Kenmotsu type representation formula for nonzero constant mean curvature spacelike surfaces in the Minkowski 3-space L3. This formula implies that every simply connected spacelike surface M with constant mean curvature in H31(c2) can be represented by a harmonic map from M to the hyperbolic 2-plane H2.

Kenmotsu–Bryant Type Representation Formulas for Constant Mean Curvature Surfaces in ” 3(-c2) and S 31(c2)

Let c be a positive constant and H a constant satisfying |H| > c. Our primary object of this paper is to give representation formulas for branched CMC H (constant mean curvature H) surfaces in the hyperbolic 3-space ” 3(-c2) of constant curvature c2, and for spacelike CMC H surfaces in the de Sitter 3-space S 31(c2) of constant curvature c2. These formulas imply, for example, that every CMC H surface in ” 3(-c2) can be represented locally by a harmonic map to the unit 2-sphere S2.

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.

Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group

In this paper, we give a representation formula for Legendrian surfaces in the 5-dimensional Heisenberg group \(\mathfrak {H}^5\), in terms of spinors. For minimal Legendrian surfaces especially, such data are holomorphic. We can regard this formula as an analogue (in Contact Riemannian Geometry) of Weierstrass representation for minimal surfaces in \(\mathbb {R}^3\). Hence for minimal ones in \(\mathfrak {H}^5\), there are many similar results to those for minimal surfaces in \(\mathbb {R}^3\). In particular, we prove a Halfspace Theorem for properly immersed minimal Legendrian surfaces in \(\mathfrak {H}^5\).

3-manifolds with positive flat conformal structure

In this paper, we consider a closed 3-manifold

On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space

M

Kenmotsu type representation formula for surfaces with prescribed mean curvature in the

with flat conformal structure

3

C

-sphere

. We will prove that, if the Yamabe constant of

Minimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de sitter 3-space

(M, C)