Hipersuperfícies de curvatura média constante em espaços produto do tipo Warped

Abstract

In this work we will study hypersurfaces of constant mean curvature immersed in warped product spaces of the form Mn+1 = R × f Pn, where Pn is a complete Riemannian manifold. In particulary, our study includes that of constant mean curvature hypersurfaces in product ambient spaces. It also includes constant mean curvature hypersurfaces in the so called pseudohyperbolic spaces. We will present conditions that allow us to conclude when such a hypersurface is a leaf Pt = {t} × Pn of the foliation t ∈ R 7−→ Pt fully umbilic of M through complete hypersurfaces. If the hypersurface is compact, we show that the immersion must be a leaf Pt , which generalizes some results due to Montiel into [21]. We will also extend a result due to Guan and Spruck on [7] from the hyperbolic ambient space Hn+1 to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel in [21], and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.