A characterization of domains in $\C^n$ with locally Levi-flat   boundaries
  A domain in $\C^n$ with Levi-flat boundary near a given point is characterized in terms of the boundary behavior of the Kobayashi or Bergman metrics, or of the Bergman kernel. Some results are given in the case of intermediate values of the rank of the Levi form. 
Nikolai Nikolov and Pascal J. Thomas
math.CV