18 may 2017 -- 14:30
Aula Dal Passo, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
In this talk, which is based on joint work with Luca Asselle and Gabriele Benedetti, I will present a few recent results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an orientable closed surface. I will show that in every level of a suitable low energy range there is a "minimal boundary": a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of the surface. Minimal boundaries satisfy an analogue of the celebrated graph theorem of Mather: in the tangent bundle, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base. I will also present some corollaries of these statements to the existence of simple periodic orbits with low energy on non-orientable closed surfaces, and to the existence of infinitely many closed geodesics on certain Finsler 2-spheres.