Abdênago Barros

A spectral characterization of the H(r)-torus by the first stability eigenvalue

Let M be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere S n+1 . In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of M in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called H(r)-tori in S n+1 , with r 2 < (n - 1)/n. This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

Bounds on volume growth of geodesic balls for Einstein warped products

The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped product. In particular, we present an obstruction for the existence of such a class of manifolds.

Complete CMC hypersurfaces in the hyperbolic space with prescribed Gauss mapping

Our aim in this paper is to show that a complete hypersurface x : Mn → Hn+1 immersed with constant mean curvature into the hyperbolic space Hn+1 is totally umbilical provided that its Gauss mapping ν has some suitable behavior. In this setting, our first result requires that the image ν(M) lies in a totally umbilical spacelike hypersurface of the de Sitter space S 1 , while in our second one we suppose that Mn has scalar curvature bounded from below and that ν(M) is contained in the closure of a domain enclosed by a totally umbilical spacelike hypersurface of S 1 determined by some vector a of the Minkowski space Ln+2, with the tangential component of a with respect to Mn having Lebesgue integrable norm.

A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.

A new class of killing invariant surfaces in 3

The aim of this note is to characterize Clifford tori as the only Killing invariant surfaces, under an additional hypothesis. Moreover, we build a family of Killing invariant minimal surfaces in 𝕊3, that does not contain Clifford tori as well as we present examples of Killing invariant surfaces whose mean curvature is not identically zero.

A new characterization of the Euclidean sphere

In this paper, we obtain a new characterization of the Euclidean sphere as a compact Riemannian manifold with constant scalar curvature carrying a nontrivial conformal vector field which is also conformal Ricci vector field.