Spherically symmetric false vacuum: No-go theorems and global structure

I enumerate all possible types of spacetime causal structures that can appear in static, spherically symmetric configurations of a self-gravitating, real, nonlinear, minimally coupled scalar field in general relativity, with an arbitrary potential V (), not necessarily positive definite. It is shown that a variable scalar field adds nothing to the list of possible structures with a constant field, namely, Minkowski or anti--de Sitter (AdS), Schwarzschild, de Sitter, and Schwarzschild--de Sitter structures. It follows in particular that whatever V () is, this theory does not admit regular black holes with flat or AdS asymptotics. It is concluded that the only possible globally regular, asymptotically flat solutions are solitons with a regular center, without horizons and with at least partly negative potentials V (). Extension of the results to more general field models is discussed.