Classification of Lipschitz unimodal function germs

Abstract

In this paper, we introduce the notion of Lipschitz modality for isolated singularities $f:({\mathbb{C}}^n,0)\to (\mathbb{C},0)$ and provide a complete classification of Lipschitz unimodal singularities of corank~2 with non-zero $4$-jets. As a consequence, such singularities are Lipschitz unimodal if they deform to $J_{10}$ but not to $J_{3,0}$. Furthermore, we show that singularities with vanishing $6$-jets have Lipschitz modality at least~$2$, thus establishing an upper bound for the order of Lipschitz unimodality.

40 pages