Marcos Dajczer

Rotation Hypersurfaces in Spaces of Constant Curvature

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 helicoid 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 H3.

Compact Conformally Flat Hypersurfaces

Roughly speaking, a conformal space is a differentiable manifold \(M^n\)in which the notion of angle of tangent vectors at a point \(p \in M^n\)makes sense and varies differentiably with p; two such spaces are (locally) equivalent if they are rela ted by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space R n. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure: in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, as~ume the ambient space to be R n+1. Then, if \(n \geqslant 4\), a conformally flat hypersurface \({M}^{n} \subset {R}^{n+1}\) 1 can be described as follows. Diffeomorphically, M n is a sphere S n with h1( M) handles attached, where h1 ( M) is the first Betti number of M. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of R n+1 that are foliated by complete round (n – 1 )-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of R n+1: (a) an open piece of an n-sphere or an n-plane bounded by round ( n – 1 )-sphere, (b) a round ( n – 1 )-sphere, (c) a point.

CONSTANT MEAN CURVATURE HYPERSURFACES IN WARPED PRODUCT SPACES

We study hypersurfaces of constant mean curvature immersed into warped product spaces of the form

Commuting Codazzi tensors and the Ribaucour transformation for submanifolds

\mathbb{R}\times_\varrho\mathbb{P}^n

Uniqueness of constant mean curvature surfaces properly immersed in a slab

, where

The mean curvature of cylindrically bounded submanifolds

\mathbb{P}^n

On deformable hypersurfaces in space forms

is a complete Riemannian manifold. In particular, our study includes that of constant mean curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant mean curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation

An extension of a theorem of Serrin to graphs in warped products

t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n

Submanifolds of codimension two attaining equality in an extrinsic inequality

, generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.

An interior gradient estimate for the mean curvature equation of Killing graphs and applications

In this paper we develop a general theory for the Ribaucour transformation of sub-manifolds. In the process, a beautiful correspondence between Ribaucour transforms of a submanifold and Codazzi tensors that commute with its second fundamental form is revealed.