Maximally tangent complex curves for germs of finite type C∞ pseudoconvex domains in C3

In this paper we discuss D'Angelo finite type pseudoconvex domains Omega in C-3. We are interested in complex curves tangent to higher order. Our main result is that there are only finitely many curves of maximal type. Maximal type has to be taken in a micro-local sense since the maximal type can be different in different directions. And of course to get finiteness we have to ignore higher order irrelevant terms which can be added without restriction. In the process of describing such a curve we find a singular change of coordinates which reduces the curve to a complex line.