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Lipschitz geometry and combinatorics of abnormal surface germs

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Abstract

We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable Hölder triangle is either Lipschitz normally embedded or contains some “abnormal” arcs. We show that abnormal arcs constitute finitely many “abnormal zones” in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal Hölder triangles.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Andrei Gabrielov

  2. Departamento de Matemática, Universidade Federal do Ceará (UFC), Campus do Pici, Bloco 914, Cep., Fortaleza, CE, 60455-760, Brazil

    Emanoel Souza

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  1. Andrei Gabrielov
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  2. Emanoel Souza
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Correspondence to Andrei Gabrielov.

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A. Gabrielov: Research supported by the NSF Grant DMS-1665115.

E. Souza: Research supported by Grant Print/CAPES/UFC N 88887.312005/2018-00.

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Gabrielov, A., Souza, E. Lipschitz geometry and combinatorics of abnormal surface germs. Sel. Math. New Ser. 28, 1 (2022). https://doi.org/10.1007/s00029-021-00716-4

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