Rigidity of hypersurfaces satisfying an Okumura type inequality, height estimates in warped product spaces and stability in weighted manifolds.

Abstract

This work is divided into three parts. In the first part, we study constant mean or scalar curvature hypersurfaces immersed into Riemannian or Lorentzian space forms satisfying a suitable Okumura type inequality. Precisely, we obtain sharp upper and lower estimates for the traceless part of the second fundamental form of these hypersurfaces. In particular, rigidity results are proved. In the second part, we are interested in generalized linear Weingarten hypersurfaces immersed into Riemannian or Lorentzian warped products. In this part, we proved interesting height estimates as well as half-space theorems for these hypersurfaces. As application of these results, we provide informations regarding the topology at infinity of such hypersurfaces. Finally, the third part is dedicated to the study of the stability of hypersurfaces with zero f-mean curvature immersed into weighted semi-Riemannian warped products. In particular, we give a sufficient condition for these hypersurfaces be stable with respect to the its standard Jacobi operator.