Lipschitz geometry of complex surfaces: analytic invariants and equisingularity

Abstract

We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in ${\mathbb{C}}^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.

Added a new section 10 to correct a minor gap and simplify some arguments

References in corpus (1)

• The thick-thin decomposition and the bilipschitz classification of normal surface singularities

Cited by in corpus (6)

• Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms
• Multiplicity of singularities is not a bi-Lipschitz invariant
• Lipschitz Normal Embeddings and Determinantal Singularities
• Moderately Discontinuous Homotopy
• Concentration of curvature and Lipschitz invariants of holomorphic functions of two variables
• Lipschitz Normal Embeddings in the Space of Matrices