Abstract. Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the belicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space H3; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in//3.

Austere submanifolds of Euclidean space were introduced in 1982 by Harvey and Lawson in their foundational work on calibrated geometries. In general, the austerity condition is much stronger than minimality since it express that the nonzero eigenvalues of the shape operator of the submanifold appear in opposite pairs for any normal vector at any point. Thereafter, the challenging task of finding non-trivial explicit examples, other than minimal immersions of Kaehler manifolds, only turned out submanifolds of rank two, and these are of limited interest in the sense that in this special situation austerity is equivalent to minimality. In this paper, we present the first explicitly given family of austere non-Kaehler submanifolds of higher rank, and these are produced from holomorphic data by means of a Weierstrass type parametrization.

We describe the space Σ H of all surfaces in R z that have constant mean curvature HφO and are invariant by helicoidal motions, with a fixed axis, of R Ά . Similar to the case Σ o of minimal surfaces Σ H behaves roughly like a circular cylinder where a certain generator corresponds to the rotation surfaces and each parallel corresponds to a periodic family of isometric helicoidal surfaces.1. Introduction. 1.1. Rotation surfaces in the Euclidean space ϋ! 3 with constant mean curvature have been known for a long time (Delaunay [3]). A natural generalization of rotation surfaces are the helicoidal surfaces that can be defined as follows.Let R 3 have coordinates (x, y, z). Consider the one-parameter subgroup g t :R3 -> R 3 of the group of rigid motions of R 3 given by 9t(%> V, z) = (& cos t + y sin t, -x sin t + y cos t, z + ht) , te(-oo f oo).The motion g t is called a helicoidal motion with axis Oz and pitch h.A helicoidal surface with axix Oz and pitch pitch h is a surface that is invariant by g t , for all t. When h = 0, they reduce to rotation surfaces.The helicoidal minimal surfaces have also been known for quite a long time (see e.g. [6] for details). It is therefore mildly surprising that we do not find in the literature the helicoidal surfaces with constant nonzero mean curvature; in this paper we want to determine explicitly all of them.Our interest in this question comes (aside its naturality) from the fact that there are very few explicit examples of surfaces with nonzero constant mean curvature. To understand certain aspects of such surfaces (behaviour of the Gauss map, stability, etc.) it might prove convenient to have at hand a reasonable supply of explicit examples. It should be mentioned that the techniques used here can also give a complete description of helicoidal surfaces with constant Gaussian curvature (Remark 3.16); this is, however, very simple and probably known.

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

The main purpose of this paper is to answer the question: To what extent is a euclidean submanifold determined by its Gauss map? More precisely, let /,/: M" -> R n+P be two isometric immersions of a connected riemannian manifold whose Gauss maps into the Grassmannian G np are congruent. When are / and / congruent? Classical examples of isometric noncongruent deformations with the same Gauss map are the associated families of minimal surfaces in R 3 . They are a special case of associated families which can be defined for certain real isometric immersions of Kaehler manifolds which we call circular. It will turn out that locally, all isometric immersions with the same Gauss map can be described in terms of circular submanifolds, whereas globally, additional phenomena arise.In §1, we discuss circular submanifolds in spaces of constant curvature. This is related to work of Calabi, Lawson, and others on minimal surfaces. §2 deals with circular hypersurfaces. More generally, we classify all Kaehler submanifolds of real codimension 1, which is of independent interest. In the remaining two sections we show that all isometric immersions M n -> R n+ ? with congruent Gauss maps form a compact abelian group, and we compute its structure.We would like to thank K. Nomizu for suggesting to look at the above question in the light of some of our previous work. The associated family of a circular immersionLet M n be a connected riemannian manifold of dimension n, and /: M n -» Q" +p an isometric immersion into a complete simply connected space of constant curvature c. We will always assume that / is substantial, i.e. f(M)