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| Autors principals: | , |
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| Format: | Preprint |
| Publicat: |
2023
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| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2311.09461 |
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| _version_ | 1866917872042770432 |
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| author | Birbrair, Lev Gabrielov, Andrei |
| author_facet | Birbrair, Lev Gabrielov, Andrei |
| contents | A normal pair of Hölder triangles is the union of two normally embedded Hölder 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 $στ$-pizza, and conjectured that it is complete: two normal pairs of Hölder triangles with the same $στ$-pizzas are outer Lipschitz equivalent. In this paper we prove that conjecture and define realizability conditions for the $στ$-pizza invariant. Moreover, only one of the two pizzas in the $στ$-pizza invariant, together with some admissible permutations related to $σ$ and $τ$, is sufficient for the existence and uniqueness, up to outer Lipschitz equivalence, of a normal pair of Hölder triangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09461 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Outer Lipschitz Classification of Normal Pairs of Hölder Triangles Birbrair, Lev Gabrielov, Andrei Metric Geometry Algebraic Geometry 51F30 (Primary) 14P10, 03C64 (Secondary) A normal pair of Hölder triangles is the union of two normally embedded Hölder 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 $στ$-pizza, and conjectured that it is complete: two normal pairs of Hölder triangles with the same $στ$-pizzas are outer Lipschitz equivalent. In this paper we prove that conjecture and define realizability conditions for the $στ$-pizza invariant. Moreover, only one of the two pizzas in the $στ$-pizza invariant, together with some admissible permutations related to $σ$ and $τ$, is sufficient for the existence and uniqueness, up to outer Lipschitz equivalence, of a normal pair of Hölder triangles. |
| title | Outer Lipschitz Classification of Normal Pairs of Hölder Triangles |
| topic | Metric Geometry Algebraic Geometry 51F30 (Primary) 14P10, 03C64 (Secondary) |
| url | https://arxiv.org/abs/2311.09461 |