The equivalence theory for infinite type hypersurfaces in $\mathbb C^2$

Abstract

We develop a classification theory for real-analytic hypersurfaces in C 2 in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in C 2 in the Problème local, formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR -DS technique in CR-geometry.