2009/06/30 by Will J. Merry · 2 citations
Mathematics · #Advanced Operator Algebra Research #Charge (physics) #Contractible space #Corollary #Geometric Analysis and Curvature Flows #Homotopy #Homotopy group #Magnetic field #Nonlinear Partial Differential Equations #Orbit (dynamics) #Riemannian manifold #math.DG #math.DS
paper · pdf · doi:10.2140/pjm.2010.247.189
published as Pacific J. Math. 247 (2010) 189-212 · 25 pages. v3 - minor corrections, this version to appear in PJM
arxiv created 2009/10/20 · openalex publication_date 2010/07/01 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let (M, g) denote a closed connected Riemannian manifold and 2 (M) a weakly exact 2form. Let t : T M T M denote the magnetic flow determined by , and let c(g, ) denote the Ma critical value of the pair (g, ). We prove that if k > c(g, ), then for every nontrivial free homotopy class of loops on M there exists a closed orbit of t with energy k whose projection to M belongs to that free homotopy class. We also prove that for almost all k < c(g, ) there exists a closed orbit of t with energy k whose projection to M is contractible. In particular, when c(g, ) = this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that a weakly exact magnetic flow with [ ] = 0 on a manifold with amenable fundamental group (which implies c(g, ) = ) has contractible closed orbits on almost every energy level.