The Diederich-Forn{\ae}ss exponent and non-existence of Stein domains   with Levi-flat boundaries
  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 $\ge k$. 
Siqi Fu and Mei-Chi Shaw
math.CV