Aldir Brasil

INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their r-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their r-mean curvatures are linearly related or one of them is constant.

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.

A gap theorem for complete constant scalar curvature hypersurfaces in the de Sitter space

To each immersed complete space-like hypersurfaceM with constant normalized scalar curvature R in the de Sitter space S nC1 1 , we associate sup H 2 , where H is the mean curvature of M .I t is proved that the condition sup H 2 Cn. N R/, where N R D .R 1 /> 0 and Cn. N R/ is a constant depending only on R and n, implies that either M is totally umbilical or M is a hyperbolic cylinder. It is also proved the sharpness of this result by showing the existence of a class of new rotation constant scalar curvature hypersurfaces in S nC1 1 such that sup H 2 >C n. N R/.

On the stability index of hypersurfaces with constant mean curvature in spheres

In (2) Barbosa, do Carmo and Eschenburg characterized the to- tally umbilical spheres as the only weakly stable compact constant mean cur- vature hypersurfaces in the Euclidean sphere Sn+1. In this paper we prove that the weak index of any other compact constant mean curvature hyper- surface Mn in Sn+1 which is not totally umbilical and has constant scalar curvature is greater or equal to n + 2, with equality if and only if M is a constant mean curvature Cliord torus

Hypersurfaces with constant mean curvature and two principal curvatures in Sn+1

In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) and Wang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.

Minimal Submanifolds in Sp+3 with Constant Scalar Curvature

Let M be a compact, minimal 3-dimensional submanifold with constant scalar curvature R immersed in the standard sphere S3+p. In codimension 1, we know from the work that has been done on Chern’s conjecture that M is isoparametric and R = 3D0, R = 3D3 or R = 3D6. In this paper we extend this result from codimension one to compact submanifolds with a flat normal bundle and give a complete classification.