Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized M¯n+1=I×ϕFn Robertson–Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given x:Mn→M¯n+1 a closed, strongly stable spacelike hypersurface of M¯n+1 with constant mean curvature H, if the warping function ϕ   satisfying ϕ″⩾max{Hϕ′,0}ϕ″⩾max{Hϕ′,0} along M  , then MnMn is either maximal or a spacelike slice Mt0={t0}×FMt0={t0}×F, for some t0∈It0∈I.