Gap type results for spacelike submanifolds with parallel mean curvature vector
DOI:
https://doi.org/10.7146/math.scand.a-133368Abstract
We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $\mathbb L_q^{n+p}(c)$ of index $1\leq q\leq p$ and constant sectional curvature $c\in \{-1,0,1\}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).
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