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 φ in general relativity, with an arbitrary potential $V\left(\phi \right),$ 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 anti–de Sitter (AdS)], Schwarzschild, de Sitter, and Schwarzschild–de Sitter structures. It follows in particular that whatever $V\left(\phi \right)$ 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 $V\left(\phi \right).\mathrm{}$ Extension of the results to more general field models is discussed.