On the bifurcation of spacelike hypersurfaces with constant mean curvature in spacetimes

Abstract

In a time-oriented Lorentzian manifold M‾n+1, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants or the local rigidity of family {Ωτ} of open sets of M‾n+1 whose boundaries ∂Ωτ are closed (compact without boundary) spacelike H-hypersurfaces, this is, closed spacelike hypersurfaces with constant mean curvature H. In particular, we do this study for a certain family of open sets in a spatially closed generalized Robertson-Walker spacetime, namely, Lorentz warped products (−I×fMn,−dt2+f2〈,〉M) with 1-dimensional negative definite base (I,−dt2), closed Riemannian fiber (Mn,〈,〉M) and warping function f:I→(0,+∞). In this case, the results are obtained considering some appropriate hypotheses that depend of f and of the behavior of the eigenvalues of Laplacian operator of Mn. Applications to such families in some spacetimes, like the de Sitter space, are also given.