Lipschitz Geometry of Mixed Pham-Brieskorn Singularities

TL;DR

This paper establishes criteria for topological and bi-Lipschitz equivalences of mixed Pham-Brieskorn singularities, constructing examples with trivial topology but different Lipschitz geometries, and introduces invariants based on exponents.

Contribution

It provides new conditions for bi-Lipschitz classification of mixed singularities and constructs infinite families with distinct Lipschitz types despite topological triviality.

Findings

Constructed infinite families with topologically trivial but bi-Lipschitz distinct singularities.

Derived conditions for inner, outer, and ambient bi-Lipschitz equivalences in two-variable cases.

Identified an invariant of the outer geometry determined by exponents.

Abstract

We give conditions for topological and bi-Lipschitz equivalences within a class of mixed singularities of Pham-Brieskorn type. As a consequence, we construct infinite families that are topologically trivial but have distinct bi-Lipschitz types. We also investigate this problem in the context of mixed surfaces defined by these singularities in the case of two complex variables, deriving conditions for inner, outer, and ambient bi-Lipschitz equivalences. In particular, we obtain an invariant of the subanalytic outer geometry of the associated mixed surfaces, which is determined by the exponents.