Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane

TL;DR

This paper proves that in the plane, bi-Lipschitz equivalence of weighted homogeneous function-germs implies their analytical equivalence, revealing a rigidity property in their classification.

Contribution

It establishes that bi-Lipschitz equivalence for weighted homogeneous function-germs in the plane implies analytical equivalence, a new rigidity result in singularity theory.

Findings

Bi-Lipschitz equivalence implies analytical equivalence for weighted homogeneous function-germs in the plane.

The result applies specifically to non-homogeneous weighted homogeneous function-germs.

The work extends understanding of classification invariants in singularity theory.

Abstract

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs $(\C^2,0)\to(\C,0)$ are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent.