On the manifold structure of the set of unparameterized embeddings with low regularity

Given manifolds $M$ and $N$, with $M$ compact, we study the geometrical structure of the space of embeddings of $M$ into $N$, having less regularity than $\mathcal C^\infty$, quotiented by the group of diffeomorphisms of $M$.Comment: To appear in the Bulletin of the Brazilian Mathematical Societ

Hypersurfaces of constant higher order mean curvature in warped products

In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.Comment: To appear in the Transactions of the American Mathematical Society. See http://www.ams.org/cgi-bin/mstrack/accepted_papers?jrnl=tra

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature $S$ of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of $S$. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli and Setti \cite{PRS}.Comment: Final version (August 2009). To appear in Journal of Mathematical Analysis and Applications. Dedicated to Professor Marcos Dajczer on the occasion of his 60th birthda

Geometric analysis of Lorentzian distance function on spacelike hypersurfaces

Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sectional or Ricci) curvatures of the ambient spacetime, we obtain sharp estimates for the mean curvature of those hypersurfaces. Moreover, we also give a suficient condition for its hyperbolicity.Comment: Final version (January 2009). To appear in the Transactions of the American Mathematical Societ