Bi-Lipschitz triviality of function-germs on singular varieties
Raúl Oset Sinha, Maria Aparecida Soares Ruas
TL;DR
This work tackles the bi-Lipschitz classification of deformations of a function germ $f$ on a singular variety $(X,0)$ by introducing strongly rational $\mathscr{R}_X$-bi-Lipschitz trivial families and an infinitesimal criterion. It proves a Thom-Levine type result for $\mathscr{R}_X$-analytical triviality and provides a practical sufficient condition for strong rational bi-Lipschitz triviality via control functions and tangent vector fields, with emphasis on weighted-homogeneous settings. In the homogeneous case, deformations of $f$ of the same or higher degree are bi-Lipschitz trivial, and rigidity results show that, under weight conditions, strong bi-Lipschitz triviality implies analytic triviality. Collectively, the results bridge topological and analytic classifications for deformations on singular varieties and furnish concrete criteria to verify bi-Lipschitz triviality and rigidity.
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.