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

Abstract

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs ( C 2 , 0 ) → ( C , 0 ) (\mathbb {C}^2,0)\rightarrow (\mathbb {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.