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

We 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 φ in general relativity, with an arbitrary potential V (φ) , not necessarily positive-definite. It is shown that a variable scalar field adds nothing to the list of possible structures with a constant φ field, namely, Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild — de Sitter. It follows, in particular, that, whatever is V (φ) , 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 (φ) . Extension of the results to more general field models is discussed.