In this article we study self-gravitating static solutions of the
Einstein-ScalarField system in arbitrary dimensions. We discuss the existence
and the non-existence of geodesically complete solutions depending on the form
of the scalar field potential V(ϕ), and provide full global geometric
estimates when the solutions exist. Our main results are summarised as follows.
For the Klein-Gordon field, namely when V(ϕ)=m2∣ϕ∣2, it is proved
that geodesically complete solutions have Ricci-flat spatial metric, have
constant lapse and are vacuum, (that is ϕ is constant and equal to zero if
m=0). In particular, when the spatial dimension is three, the only such
solutions are either Minkowski or a quotient thereof (no nontrivial solutions
exist). When V(ϕ)=m2∣ϕ∣2+2Λ, that is, when a vacuum energy
or a cosmological constant is included, it is proved that no geodesically
complete solution exists when Λ>0, whereas when Λ<0 it is
proved that no non-vacuum geodesically complete solution exists unless
m2<−2Λ/(n−1), (n is the spatial dimension) and the spatial
manifold is non-compact. The proofs are based on techniques in comparison
geometry \'a la Backry-Emery.Comment: Introduction changed and small application to geons remove