From curve shortening to flat link stability and Birkhoff sections of geodesic flows
| Autor: | Alves, Marcelo R. R., Mazzucchelli, Marco |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 orientable closed 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 orientable Riemannian surface. Comment: 48 pages, 9 figures; version 2: the existence of Birkhoff sections (Theorem D) now holds for all closed orientable Riemannian surfaces, including spheres |
| Databáze: | arXiv |
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