Sobre la geometría de hipersuperficies en espacios producto con la misma curvatura media riemanniana y lorentziana

Resumo

A hypersurface in the Lorentz-Minkowski space Ln+1 is said to be spacelike if its induced metric from the Lorentzian metric of Ln+1 is a Riemannian one. We can endow a spacelike hypersurface in the Lorentz-Minkowski space with two different Riemannian metrics: the one inherited from Ln+1, gL, and the one induced from the Euclidean space Rn+1, gR. Consequently, we can consider two different mean curvature functions on the hypersurface, HL and HR respectively. Recall that a hypersurface in Rn+1 is said to be minimal when HR = 0, while a spacelike hypersurface in Ln+1 is maximal if HL = 0. From the Bernstein and Calabi-Bernstein theorems it is concluded that the only complete spacelike hypersurfaces in Ln+1 being simultaneously minimal and maximal are the spacelike hyperplanes. This uniqueness result can be extended to spacelike hypersurfaces in Ln+1 with the same constant mean curvature HL = HR thanks to a Heinz, Chern and Flanders result which states that entire graphs with constant mean curvature in Rn+1 are minimal. On the other hand, from a local point of view, Kobayashi proved that the only spacelike surfaces in L3 being simultaneously minimal and maximal are open pieces of a spacelike plane or of a helicoid, in the region where it is spacelike. Recently, A. L. Albujer and M. Caballero considered a more general case in which a spacelike surface in L3 satisfies HL = HR not necessarily constant, and they showed several geometric properties that these surfaces verify. The first purpose of this thesis will be to generalize the results of Albujer and Caballero, and we will also obtain some geometric properties for the spacelike hypersurfaces in Ln+1 with HR = HL. Specifically, it will be shown that they do not have elliptic points. This, jointly with a classical argument on the existence of elliptic points from Osserman, will give rise to several consequences on the geometry of these hypersurfaces. On the other hand, since every spacelike hypersurface in Ln+1 is locally a graph over any spacelike hyperplane, the spacelike hypersurfaces with HR = HL will be locally determined by the solutions of a partial differential equation which will be studied, and some uniqueness results for it will be provided. Next, the second objective will be to study the results which can occur in the product spaces with arbitrary dimension. To do this, we will study the extrinsic geometry of non-degenerate hypersurfaces immersed in the product space Mn x R, with (Mn, < , >M) a Riemannian manifold, and which will be given two metrics: the standard Riemannian metric < , >M + dt2 and the Lorentzian metric < , >M - dt2. Therefore two normal vectors, two shape operators, two mean curvatures and two Gaussian curvatures will be considered, one associated with each metric. Then, assuming that the hypersurface under study has mean curvature equal to zero with respect to both metrics, it will be shown that it is foliated by hypersurfaces that are minimal submanifolds of the ambient space. Finally, we will also study the case in which the non-degenerate surfaces of the Lorentzian product space M2(c) x R1 (where M2(c) is the Euclidean plane when c = 0, the Euclidean sphere when c = 1 and the plane hyperbolic when c = -1) have the same Gaussian curvature with respect to both metrics.