Peak curves in weakly pseudoconvex boundaries in C2

LetD be a pseudoconvex domain with real analytic boundary in C². A subsetE of ∂D is a local peak set for if for everyp ∈ ∂D, there exist a neighborhoodU ofp and a holomorphic functionf onU such thatf = 1 onE∩U and |f| < 1 on . We give conditions for the existence of real analytic LPι curves in ∂D through a point of finite type. On the other hand, we give examples showing that: (a) there exist a domainD and a real analytic curve γ in ∂D such that the complexification of γ intersectsD only along γ, but γ is not LPι, and (b) there exist a domain D and a pointp ∈ ∂D, which is LPι, of finite type, but such that ∂D contains no real analytic LP∂ curve throughp.