Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction

Abstract

In this work, we investigate the bi-Lipschitz invariance of two fundamental local invariants in singularity theory: the {\L}ojasiewicz exponent and the local Euler obstruction. We draw inspiration from Bivi\`a-Ausina and Fukui, whose framework we extend to ideals in rings of analytic functions defined on affine toric varieties. We establish conditions under which these invariants remain unchanged under bi-Lipschitz equivalence. We also provide an answer, to a particular case, to the open question of whether the local Euler obstruction is a bi-Lipschitz invariant. For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions. These results contribute to the understanding of metric invariants in complex analytic geometry.