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

Abstract

Let be an -dimensional Riemannian manifold, a smooth function on , and the interval furnished with a negative definite metric . Let be the corresponding Lorentzian warped product [1, §2.6]. We investigate the spacelike tubes and bands with zero mean curvature in . It is shown that if projects one-to-one onto some domain of -hyperbolic type, then has a finite existence time. Examples are considered of maximal tubes and bands in Schwarzschild and de Sitter spaces. Geometric criteria are obtained for to be of -hyperbolic type.