On the minimality of pancake decomposition of surface germs

2025/05/29 by Medeiros, Davi Lopes, da Silva, Euripedes Carvalho, Souza, Emanoel
#03C64 #14P10 #51F30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2505.23976

Abstract

The abnormal surfaces called snakes and circular snakes, defined in \citeGabrielovSouza, 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.

Citations

• Lipschitz geometry and combinatorics of circular snakes
• Outer Lipschitz Classification of Normal Pairs of Hölder Triangles
• Lipschitz Geometry of Real Semialgebraic Surfaces
• Lipschitz geometry and combinatorics of abnormal surface germs
• Lipschitz contact equivalence and real analytic functions
• Arc criterion of normal embedding
• Normal embeddings of semialgebraic sets.

Related

• Lipschitz geometry and combinatorics of circular snakes
• Lipschitz geometry and combinatorics of abnormal surface germs
• Lipschitz geometry of pairs of normally embedded Hölder triangles
• Lipschitz Geometry of Real Semialgebraic Surfaces
• An introduction to Lipschitz geometry of complex singularities