A CR proof for a global estimate of the Diederich--Fornaess index of   Levi-flat real hypersurfaces
  Yet another proof is given for a global estimate of the Diederich--Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. This proof reveals that this kind of estimate makes sense and holds also for abstract compact Levi-flat CR manifolds. 
Masanori Adachi
math.CV math.DS