Spacelike submanifolds in semi-Riemannian product spaces: an approach via maximum principles, parabolicity and conditions of volume growth

Abstract

The main objective of this thesis is the study of submanifolds immersed in certain semi- Riemannian products. For this, applying a more general Omori-Yau maximum principle due to Chen and Qiu and results due to Alias, Caminha and Nascimento, we obtain new principles of the maximum for the Laplacian drift in Riemannian manifolds with Bakry- ´Emery-Ricci tensor bounded from below by a continuous function or with polynomial volume growth condition. We apply these new maximal principles to obtain various uniqueness results of hypersurface in weighted Lorentzian product spaces of type −R×Mn f and analogous results in weighted product space of the form R × Mn f . In both cases, we also obtain Calabi-Bernstein type results for the entire graph of functions defined in the Riemannian basis Mn. We determined uniqueness and rigidity results to submanifold immersed with parallel Gaussian mean curvature vector in the classical Gaussian and pseudo-Gaussian spaces. Finally, using for parabolicity, we determine various rigidity conditions onto stationary spacelike surface into generalized Roberston-Walker spacetime and we present examples justifying the need for these conditions.