Bi-Lipschitz triviality of function-germs on singular varieties

TL;DR

This paper investigates the conditions under which deformations of analytic function germs on singular varieties are bi-Lipschitz trivial, introducing new criteria and proving rigidity results for homogeneous cases.

Contribution

It introduces the notion of strongly rational i-Lipschitz trivial families and provides an infinitesimal criterion for bi-Lipschitz triviality of deformations.

Findings

Homogeneous deformations of the same or higher degree are bi-Lipschitz trivial.

A rigidity result for weighted homogeneous deformations is established.

A sufficient infinitesimal criterion for bi-Lipschitz triviality is provided.

Abstract

In this paper we study the bi-Lipschitz triviality of deformations of an analytic function germ $f$ defined on a germ of an analytic variety $(X, 0)$ in $\mathbb C^n$. We introduce the notion of strongly rational $\mathscr R_X$-bi-Lipschitz trivial families and give an infinitesimal criterion which is a sufficient condition for the bi-Lipschitz triviality of deformations of $f$ on $(X,0).$ As a corollary it follows that when $X$ and $f$ are homogeneous of the same degree, all deformation of $f$ of the same or higher degrees are bi-Lipschitz trivial. We then prove a rigidity result for deformations of $f$ on $X$ when both are weighted homogeneous with respect to the same set of weights.