[Du raccourcissement des courbes à la stabilité des entrelacs plats et aux sections de Birkhoff des flots géodésiques]
We employ the curve shortening flow to establish three new results on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for closed connected orientable Riemannian surfaces: for surfaces of positive genus, the existence of a contractible simple closed geodesic $\gamma $ forces the existence of infinitely many closed geodesics intersecting $\gamma $ in every primitive free homotopy class of loops; for the $2$-sphere, the existence of two disjoint simple closed geodesics forces the existence of a third one intersecting both. The final result asserts the existence of Birkhoff sections for the geodesic flow of any closed connected orientable Riemannian surface.
Nous utilisons le flot de raccourcissement des courbes pour établir trois nouveaux résultats concernant la dynamique des flots géodésiques sur les surfaces riemanniennes fermées. Le premier concerne la stabilité, sous des perturbations $C^0$-petites de la métrique riemannienne, de certains entrelacs plats formés de géodésiques fermées. Le deuxième est un théorème d’existence forcée pour les surfaces riemanniennes fermées, connexes et orientables : sur les surfaces de genre strictement positif, l’existence d’une géodésique fermée simple et contractile $\gamma $ entraîne l’existence d’une infinité de géodésiques fermées intersectant $\gamma $ dans chaque classe d’homotopie libre primitive ; sur la 2-sphère, l’existence de deux géodésiques fermées simples et disjointes implique celle d’une troisième géodésique intersectant les deux premières. Le troisième résultat établit l’existence de sections de Birkhoff pour le flot géodésique de toute surface riemannienne fermée, connexe et orientable.
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Keywords: Curve shortening flow, closed geodesics, flat links, Birkhoff sections
Mots-clés : Flot de raccourcissement des courbes, géodésiques fermées, entrelacs plats, sections de Birkhoff
Marcelo R. R. Alves 1 ; Marco Mazzucchelli 2
CC-BY 4.0
@article{JEP_2025__12__801_0,
author = {Marcelo R. R. Alves and Marco Mazzucchelli},
title = {From curve shortening to flat link stability and {Birkhoff} sections of geodesic flows},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {801--851},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.301},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.301/}
}
TY - JOUR AU - Marcelo R. R. Alves AU - Marco Mazzucchelli TI - From curve shortening to flat link stability and Birkhoff sections of geodesic flows JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 801 EP - 851 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.301/ DO - 10.5802/jep.301 LA - en ID - JEP_2025__12__801_0 ER -
%0 Journal Article %A Marcelo R. R. Alves %A Marco Mazzucchelli %T From curve shortening to flat link stability and Birkhoff sections of geodesic flows %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 801-851 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.301/ %R 10.5802/jep.301 %G en %F JEP_2025__12__801_0
Marcelo R. R. Alves; Marco Mazzucchelli. From curve shortening to flat link stability and Birkhoff sections of geodesic flows. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 801-851. doi : 10.5802/jep.301. https://jep.centre-mersenne.org/articles/10.5802/jep.301/
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