On static solutions of the Einstein-Scalar Field equations

In this note we study the Einstein-ScalarField static equations in arbitrary dimensions. We discuss the existence of geodesically complete solutions depending on the form of the scalar field potential $V(\phi)$, and provide full global geometric estimates when the solutions exist. As a special case it is shown that when $V(\phi)$ is the Klein-Gordon potential, i.e. $V(\phi)=m^{2}|\phi|^{2}$, geodesically complete solutions are necessarily Ricci-flat, have constant lapse and are vacuum, (that is $\phi=\phi_{0}$ with $\phi_{0}=0$ if $m\neq 0$). Hence, if the spatial dimension is three, the only such solutions are either Minkowski or a quotient thereof. For $V(\phi)=m^{2}|\phi|^{2}+2\Lambda$, that is, including a vacuum energy or a cosmological constant, it is proved that no geodesically complete solution exists when $\Lambda>0$, whereas when $\Lambda<0$ it is proved that no non-vacuum geodesically complete solution exists unless $m^{2}<-2\Lambda/(n-1)$, ($n$ is the spatial dimension) and the manifold is non-compact