Classification of Lipschitz unimodal function germs

TL;DR

This work develops a Lipschitz-level analogue of Arnold’s singularity classification focused on corank-2 germs with nonzero 4-jets. By introducing the Lipschitz modality ${ m Lmod}$ and relating it to the smooth modality ${ m Smod}$, the authors establish that Lipschitz unimodal corank-2 germs are precisely those deforming to $J_{10}$ but not to $J_{3,0}$, and they prove that germs with vanishing 6-jets have ${ m Lmod}\nobreaker ext{(f)} ext{ }\ge 2$, giving an upper bound on the Lipschitz unimodality order. A central result is that the $J_{3,0}$ family has ${ m Lmod}=2$, which anchors the bimodal boundary and enables a systematic, Newton-diagram-based method to certify Lipschitz triviality of deformations. The main contribution is a detailed, explicit classification of Lipschitz unimodal corank-2 germs with nonzero 4-jets, together with practical criteria (Thom-Levine and Newton polyhedron) for verifying Lipschitz triviality across deformation families, providing tools for future Lipschitz moduli analyses in higher corank cases.

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.

Figures (1)

• Figure 1: Newton diagram of $h$

Theorems & Definitions (36)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1