\(E^3\)-complete timelike surfaces in \(E_1^3\) are globally hyperbolic
Let \(S\) be a timelike surface of Minkowski 3-space \(E^3_1\), that is, the induced metric on \(S\) is Lorentzian. First, the authors prove that \(S\) is causally stable (cf. \textit{J. K. Beem}, \textit{P. E. Ehrlich} and \textit{K. L. Easley} [`Global Lorentzian Geometry', 2nd ed. (Pure Appl. Math. 202, Marcel Dekker, New York) (1996; Zbl 0846.53001)], pp. 63-64). Then, in the case the immersion \(Z\) of \(S\) in \(E^3_1\) is supposed to be \(E^3\)-complete, the authors prove the following results: (i) \(S\) is globally hyperbolic. (ii) If \(Z\) is minimal outside a compact subset \(S_0\) of \(S\), then \(S\) is \(C^\infty\)-conformally diffeomorphic to the Lorentz plane \(E^2_1\), provided that \(S\) is simply connected.
scientific article; zbMATH DE number 1127770