Periodic orbits of conservative systems below the Mañé critical energy value

• Applicant: Professor Dr. Alberto Abbondandolo
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 273417880

• Final Report Year: 2019

Final Report Abstract

Most of the questions that were listed in the research proposal for the present project were successfully addressed, and some unforseen results were also obtained. The main results concern the following topics: (i) Existence of periodic orbits for general Tonelli systems on surfaces. (ii) Existence of infinitely many periodic orbits for non-exact magnetic flows on surfaces. (iii) Existence of periodic orbits of general non-exact Tonelli systems on non-aspherical closed manifolds of arbitrary dimension. (iv) Existence of waists and of infinitely many periodic orbits for Tonelli systems on closed manifolds of arbitrary dimension having finite fundamental group. (v) Existence and non-existence results for conormal boundary conditions. (vi) Existence of infinitely many closed geodesics on non-compact Finsler surfaces.

Publications

• “Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level”, Calc. Var. Partial Differential Equations 54 (2015), 1525–1545
  L. Asselle, G. Benedetti
  (See online at https://doi.org/10.1007/s00526-015-0834-1)
• “On the existence of orbits satisfying periodic or conormal boundary conditions for Euler-Lagrange flows”, Ph. D. Thesis, Ruhr-Universität Bochum, 2015
  L. Asselle
  
• “On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions”, J. Funct. Anal. 271 (2016), 3513–3553
  L. Asselle
  (See online at https://doi.org/10.1016/j.jfa.2016.08.023)
• “The Lyusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles”, J. Topol. Anal. 8 (2016), 545–570
  L. Asselle, G. Benedetti
  (See online at https://doi.org/10.1142/S1793525316500205)
• Minimal boundaries in Tonelli Lagrangian systems
  L. Asselle, G. Benedetti, M. Mazzucchelli
  (See online at https://doi.org/10.1093/imrn/rnz246)
• “Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level”, J. Eur. Math. Soc. (JEMS) 19 (2017), 551–579
  A. Abbondandolo, L. Macarini, M. Mazzucchelli, G. P. Paternain
  (See online at https://doi.org/10.4171/JEMS/674)
• “On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus”, Math. Z. 286 (2017), 843–859
  L. Asselle, G. Benedetti
  (See online at https://doi.org/10.1007/s00209-016-1787-6)
• “The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere”, Adv. Nonlinear Stud. 17 (2017), 17-30
  A. Abbondandolo, L. Asselle, G. Benedetti, M. Mazzucchelli, I. Taimanov
  (See online at https://doi.org/10.1515/ans-2016-6003)
• “On Tonelli periodic orbits with low energy on surfaces”, Trans. Amer. Math. Soc. 371 (2019), 3001–3048
  L. Asselle, M. Mazzucchelli
  (See online at https://doi.org/10.1090/tran/7185)
• “Waist theorems for Tonelli systems in higher dimension”, Manuscripta Mathematica, Online first: October 1, 2019
  L. Asselle, M. Mazzucchelli
  (See online at https://doi.org/10.1007/s00229-019-01154-5)