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Florent Tallerie

An universal triangulation for flat tori
Jeudi, 9 Mars, 2023 - 17:00
Résumé : 

A famous theorem of Nash, improved by Kuiper, implies that every Riemannian surface admits a C1 isometric embedding in the three dimensional Euclidean space. A similar result due to Burago and Zalgaller states that every polyhedral surface, that is a surface obtained from gluing Euclidean polygons, admits a piecewise linear (in short PL) isometric embedding in the three dimensional Euclidean space.  In particular, any flat torus, that is a quotient of the Euclidean plane by a rank two lattice, admits such a PL isometric embedding. Nevertheless, the proof of Burago and Zalgaller is partially not constructive, relying on the Nash-Kuiper process. We will see how to make it effective in the case of flat tori.

In practice, the construction of Burago and Zalgaller gives embeddings with a huge number of faces, and distinct for each flat torus. Thanks to another construction of Zalgaller, and recent works of Arnoux, Lelièvre and Malaga we present a universal triangulation T for flat tori made of less than 6000 triangles, that admits for every flat torus an isometric embedding, linear in restriction to any triangle of T.
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