On a global estimate of the Diederich--Fornaess index of Levi-flat real   hypersurfaces
  In this expository paper, we review a recent progress of the study of the Diederich--Fornaess index of complex domains with emphasis on the case of domains with Levi-flat boundary. It is exhibited that for any compact Levi-flat real hypersurface, the norm of its infinitesimal holonomy must exceed the curvature of its normal bundle at a point. 
Masanori Adachi
math.CV math.DS