A contractible Levi-flat hypersurface in C^2 which is a determining set   for pluriharmonic functions
  We construct a real analytic Levi-flat hypersurface M in a neighborhood of an ellipsoid B in C^2 such that the each leaf of the Levi foliation of M is a complex disc, M intersects the boundary of B transversely, and the intersection A of M with B has the following properties: (i) the closure of A is diffeomorphic to the ball in R^3 and has a basis of Stein neighborhoods in C^2, (ii) any real analytic function on A which is constant on each Levi leaf is constant, (iii) any pluriharmonic function in a connected open neighborhood of A and vanishing on A is identically zero. 
Franc Forstneric