  We study the Diederich-Forn{\ae}ss exponent and relate it to non-existence of
Stein domains with Levi-flat boundaries in complex manifolds. In particular, we
prove that if the Diederich-Forn{\ae}ss exponent of a smooth bounded Stein
domain in an $n$-dimensional complex manifold is $>k/n$, then it has a boundary
point at which the Levi-form has rank $\ge k$.
