From curve shortening to flat link stability and Birkhoff sections of geodesic flows

2024/08/21 by Alves, Marcelo R. R., Mazzucchelli, Marco · 1 citation
#37D40 #53C22 #53D25 #53E10 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2408.11938

Abstract

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.

Cited by

• Uniqueness of the measure of maximal entropy for geodesic flows on surfaces

Related

• The Dirichlet problem for curve shortening flow
• Curve shortening flow on Riemann surfaces with possible ambient conic singularities
• Integrable geodesic flows on surfaces
• Non-uniqueness of curve shortening flow
• Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary