On the minimality of pancake decomposition of surface germs

TL;DR

This paper introduces algorithms for minimal pancake decompositions of special surface germs called snakes and circular snakes, establishing their canonical nature under weak outer Lipschitz equivalence.

Contribution

It provides the first algorithms to obtain minimal pancake decompositions for these surface germs and proves their canonical form under weak outer Lipschitz equivalence.

Findings

Algorithms for minimal pancake decomposition of snakes and circular snakes.

Minimal decompositions are weakly equivalent under outer Lipschitz transformations.

Minimal pancake decompositions are canonical up to weak outer bi-Lipschitz equivalence.

Abstract

The abnormal surfaces called snakes and circular snakes, defined in \cite{GabrielovSouza}, are special types of surface germs capturing the outer Lipschitz phenomena relevant to the outer classification problem. We provide algorithms to obtain a minimal pancake decomposition, i.e., where the number of pancakes is minimal, for snakes and circular snakes. We call a pancake decomposition obtained from our algorithm a greedy pancake decomposition. We also prove that greedy pancake decompositions of weakly outer Lipschitz equivalent snakes (or circular snakes) are weakly equivalent, in the sense that there is a weakly outer bi-Lipschitz homeomorphism between the surfaces mapping each greedy pancake to a greedy pancake. This implies that such minimal decompositions are also canonical up to weakly outer bi-Lipschitz equivalence.