Periodic orbits in virtually contact structures

Abstract

<jats:p> We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three. </jats:p>

Two constructions of virtually contact structures

“…In fact, the results in [10,11,12,13,14] motivated the definition of the covering contact connected sum. Extending the work of Geiges-Zehmisch [12] the existence of periodic orbits for virtually contact structures addressed by Theorem 1.1 that in addition admit a C 3 -bounded contact form on the total space of the covering is shown in [2].…”

“…Let D 2n−1 i , i = 1, 2, be a closed embedded disc contained in U i such that a neighbourhood of the disc is equipped with Darboux coordinates for the contact form α Ui . We perform contact index-1 surgery as described in [9] identifying ∂D 2n−1 i with the boundary {i} × S 2n−2 of the upper boundary of [1,2] × S 2n−2 the 1-handle [1,2] × D 2n−1 . The resulting contact form on the connected sum U 1 #U 2 is denoted by α U1 #α U2 .…”

“…asked by G. P. Paternain whether virtually contact manifolds have to admit periodic orbits. The question was answered positively in many instances by Cieliebak-Frauenfelder-Parternain [7] and, more recently, by Bae-Wiegand-Zehmisch [2]. The content of the following theorem is to give a large class of examples to which the existence theory developed in [2] applies.…”

Diffeomorphism type via aperiodicity in Reeb dynamics

“…points that correspond to single intersections, form a submanifold diffeomorphic to a (2n − d − 1)-sphere. In view of condition (1) we remark that the hypersurface S bounds a bounded domain D inside C, whose closure is diffeomorphic to the closed unit disc bundle D T * Q ⊕ R 2n+1−2d . Condition (2) will play an important role in Section 3.1.…”

“…This will result in a more advanced analysis for the holomorphic discs. The essential point here will be a target rescaling argument in Section 7, which was invented by Bae-Wiegand-Zehmisch [1] in the context of virtually contact structures, to ensure C 0 -bounds on holomorphic discs in the situation of general manifolds Q. Furthermore in order to obtain C 0bounds of holomorphic discs along their boundaries in T * Q-direction we develop an integrated maximum principle in Sections 5 and 6.5.…”