Flat Lorentz surfaces in anti-de Sitter 3-space and Gravitational Instantons

In this paper, we study flat Lorentz surfaces in anti-de Sitter 3-space [Formula: see text] in terms of the second conformal structure. Those flat Lorentz surfaces can be represented in terms of a Lorentz holomorphic and a Lorentz anti-holomorphic data similarly to Weierstraß representation formula. An analogue of hyperbolic Gauß map is considered for timelike surfaces in [Formula: see text] and the relationship between the conformality (or the holomorphicity) of hyperbolic Gauß map and the flatness of a Lorentz surface is discussed. It is shown that flat Lorentz surfaces in [Formula: see text] are associated with a hyperbolic Monge–Ampère equation. It is also known that Monge–Ampére equation may be regarded as a 2-dimensional reduction of the Einstein’s field equation. Using this connection, we construct a class of anti-self-dual gravitational instantons from flat Lorentz surfaces in [Formula: see text].

Hyperbolic submanifolds with finite type hyperbolic Gauss map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

A note on surfaces with prescribed oriented Euclidean Gauss map

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

SINGULARITIES OF HYPERBOLIC GAUSS MAPS

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.