https://neckpinch.com/authors/marcelo-r-r-alves/New arXiv papers by Marcelo R. R. Alves, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2408.11938/arxiv:2408.11938Wed, 21 Aug 2024 00:00:00 GMTMarcelo R. R. Alves, Marco Mazzucchelli — 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 C⁰-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 γ forces the existence of infinitely many closed geodesics intersecting γ 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.Curve shorteningStability & uniqueness