Lipschitz stratification of complex hypersurfaces in codimension 2

Abstract

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study the Zariski equisingular families of surface, not necessarily isolated, singularities in ${\mathbb{C}}^3$. We show that a natural stratification of such a family, given by the singular set and the generic family of polar curves, provides a Lipschitz stratification in the sense of Mostowski. In particular such families are bi-Lipschitz trivial, with trivializations obtained by integrating Lipschitz vector fields.