| Résume | We study outer Lipschitz geometry of surface germs definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). By the finiteness theorems of Mostowski, Parusinski and Valette, any definable family has finitely many outer Lipschitz equivalence classes. Our goal is classification of definable surface germs with respect to the outer Lipschitz equivalence. The inner Lipschitz classification of definable surface germs was described by Birbrair. The outer Lipschitz classification is much more complicated. There is also a third, even more complicated, ambient Lipschitz classification problem.
Some initial results in this were obtained by Birbrair, Brandenbursky and Gabrielov. Using the contact equivalence classification of Lipschitz functions ("pizza decomposition") by Birbrair et al. and the theory of abnormal surface germs ("snakes") by Gabrielov and Souza, we obtain a decomposition of a surface germ
into normally embedded Holder triangles, unique up to outer Lipschitz equivalence. This triangulation, with some additional data ("pizza toppings") is a complete discrete invariant of an outer Lipschitz equivalence class of surface germs.
Joint work with L. Birbrair, A. Fernandes, R. Mendes and E. Souza (UFC Fortaleza, Brazil). |
