$L$-parabolic linear Weingarten spacelike submanifolds immersed in an Einstein manifold

Abstract

In this paper, we deal with complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector and flat normal bundle in an Einstein manifold ${\mathcal{E}}_p^{n+p}$ of index $p$. Under some curvature constraints on the ambient space, we establish an $L$-parabolicity criterion related to a suitable modified Cheng–Yau's operator $L$ and we apply it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be an isoparametric hypersurface of ${\mathcal{E}}_1^{n+1}$ with two distinct principal curvatures one of which is simple.