mu-constancy does not imply constant bi-Lipschitz type

TL;DR

This paper demonstrates that constant Milnor number in complex hypersurface singularities does not guarantee constant bi-Lipschitz type, providing a specific counterexample within the Briac con--Speder family.

Contribution

It constructs a family of complex surface germs with constant Milnor number but varying bi-Lipschitz types, challenging previous assumptions.

Findings

Milnor number remains constant across the family.

Bi-Lipschitz type varies despite constant Milnor number.

Counterexample to the equivalence of Milnor number and bi-Lipschitz type.

Abstract

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0, 0)$ is not bi-Lipschitz homeomorphic with respect to the inner metric to the germ $(X_t,0)$ for $t\ne 0$.