Conditions for finite existence time of maximal tubes and bands in Lorentzian warped products

Let $H$ be an $n$-dimensional Riemannian manifold, $\delta>0$ a smooth function on $H$, and $\widehat R$ the interval $(-\infty, +\infty)$ furnished with a negative definite metric $(-dt^2)$. Let $H\times_\delta\widehat R$ be the corresponding Lorentzian warped product [1, § 2.6]. We investigate the spacelike tubes and bands $\mathscr M$ with zero mean curvature in $\Omega\subset H$. It is shown that if $\mathscr M$ projects one-to-one onto some domain $\Omega\subset H$ of $\delta$-hyperbolic type, then $\mathscr M$ has a finite existence time. Examples are considered of maximal tubes and bands in Schwarzschild and de Sitter spaces. Geometric criteria are obtained for $\Omega$ to be of $\delta$-hyperbolic type.