Geodesic
Conditions for finite existence time of maximal tubes and bands in Lorentzian warped products
Izvestiya. Mathematics, Tome 44 (1995) no. 3, pp. 629-643 
V. A. Klyachin; V. M. Miklyukov. Conditions for finite existence time of maximal tubes and bands in Lorentzian warped products. Izvestiya. Mathematics, Tome 44 (1995) no. 3, pp. 629-643. http://geodesic.mathdoc.fr/item/IM2_1995_44_3_a9/
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     title = {Conditions for finite existence time of maximal tubes and bands in {Lorentzian} warped products},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1995_44_3_a9/}
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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.

[1] Dzh. Bim, P. Erlikh, Globalnaya lorentseva geometriya, Mir, M., 1985 | MR | Zbl

[2] V. A. Klyachin, V. M. Miklyukov, “Maksimalnye giperpoverkhnosti trubchatogo tipa v prostranstve Minkovskogo”, Izv. AN SSSR. Ser. matem., 55:1 (1991), 206–217 | MR | Zbl

[3] V. M. Miklyukov, “Ob odnoi lorents-invariantnoi kharakteristike maksimalnykh trubok v prostranstve Minkovskogo”, DAN SSSR, 332:2 (1992), 233–276 | MR

[4] V. M. Miklyukov, “Maksimalnye trubki i lenty v prostranstve Minkovskogo”, Matem. sb., 1992, no. 12, 45–76 | MR | Zbl

[5] V. M. Goldshtein, Yu. G. Reshetnyak, Vvedenie v teoriyu funktsii s obobschennymi proizvodnymi i kvazikonformnye otobrazheniya, Nauka, M., 1983 | MR

[6] G. D. Suvorov, Semeistva ploskikh topologicheskikh otobrazhenii, Izd-vo SO AN SSSR, Novosibirsk, 1985 | MR

[7] A. G. Losev, “Nekotorye liuvillevy teoremy na rimanovykh mnogoobraziyakh spetsialnogo vida”, Izv. VUZov. Matematika, 1991, no. 12, 15–24 | MR | Zbl

[8] A. A. Tuzhilin, “Indeksy tipa Morsa dvumernykh minimalnykh poverkhnostei $R^3$ i $H^3$”, Izv. AN SSSR. Ser. matem., 55:3 (1991), 581–607 | MR | Zbl

[9] P. Li, S. T. Yau, “Curvature and holomorphic mappings of complete Kähler manifolds”, Composito Mathematica, 73 (1990), 125–144 | MR | Zbl

[10] J. Milnor, “On deciding whether a surface is parabolic or hyperbolic”, Amer. Math. Monthly, 84:1 (1977), 43–46 | DOI | MR | Zbl

[11] R. Bishop, R. Krittenden, Geometriya mnogoobrazii, Mir, M., 1967 | MR | Zbl