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arXiv · 2311.09461

Outer Lipschitz Classification of Normal Pairs of H\"older Triangles

Lev Birbrair·Andrei Gabrielov
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Abstract

A normal pair of H\"older triangles is the union of two normally embedded H\"older triangles satisfying some natural conditions on the tangency orders of their boundary arcs. It is a special case of a surface germ, a germ at the origin of a two-dimensional closed semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) subset of $R^n$. Classification of normal pairs considered in this paper is a step towards outer Lipschitz classification of definable surface germs. In the paper \cite{BG} we introduced a combinatorial invariant of the outer Lipschitz equivalence class of normal pairs, called $\sigma\tau$-pizza, and conjectured that it is complete: two normal pairs of H\"older triangles with the same $\sigma\tau$-pizzas are outer Lipschitz equivalent. In this paper we prove that conjecture and define realizability conditions for the $\sigma\tau$-pizza invariant. Moreover, only one of the two pizzas in the $\sigma\tau$-pizza invariant, together with some admissible permutations related to $\sigma$ and $\tau$, is sufficient for the existence and uniqueness, up to outer Lipschitz equivalence, of a normal pair of H\"older triangles.

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Lev Birbrair, Andrei Gabrielov. 2023-11-15. Outer Lipschitz Classification of Normal Pairs of H\"older Triangles. https://arxiv.org/abs/2311.09461

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