Stability of a Generalized Linear Weingarten Spacelike Hypersurface in Robertson–Walker Spacetimes

Abstract

In this article, we study the variational problem of maximizing a certain linear combination of area functionals of a closed generalized linear Weingarten spacelike hypersurface Mⁿ immersed in a foliated spacetime ${\tilde{M}}_1^{n+1}$. Given a Robertson-Walker spacetime ${\tilde{M}}_1^{n+1}\left(c\right)=I{\times }_{\varphi }{F}^n$, we exhibit conditions on Mⁿ which guarantee that (strong) stability is necessary and sufficient for Mⁿ to be totally umbilic. Under a suitable restriction on Mⁿ, we also prove that if a closed generalized linear Weingarten spacelike hypersurface in $S_1^{n+1}$ is (strongly) stable, then it must be a totally umbilic round sphere.