Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

Viktor Ginzburg, Başak Gürel
2009 Commentarii Mathematici Helvetici  
In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain
more » ... d meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation. Mathematics Subject Classification (2000 ). 53D40, 37J10, 37J45. Keywords. Twisted geodesic flows, periodic orbits, Floer homology, Sturm theory. V. Ginzburg and B. Z. Gürel CMH concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott nondegenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a (partial) generalization of the Weinstein-Moser theorem, [Mo], [We1] , asserting that a certain number of distinct periodic orbits exist on every energy level near a non-degenerate extremum. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument utilizing convexity of the Hamiltonian in the direction normal to the critical submanifold and of a Floer-homological calculation that guarantees "dense existence" of periodic orbits with certain index. The existence of periodic orbits for a twisted geodesic flow with symplectic magnetic field is then an immediate consequence of the generalized Weinstein-Moser theorem. The results proved in this paper are announced in [GG3]. The generalized Weinstein-Moser theorem. Throughout the paper, M will stand for a closed symplectic submanifold of a symplectic manifold .P; !/. We denote by OE! the cohomology class of ! and by c 1 .TP / the first Chern class of P equipped with an almost complex structure compatible with !. The integrals of these classes over a 2-cycle u will be denoted by h!; ui and, respectively, hc 1 .TP /; ui. Recall also that P is said to be spherically rational if the integrals h!; ui over all The key result of the paper is Theorem 1.1 (Generalized Weinstein-Moser theorem). Let K W P ! R be a smooth function on a symplectic manifold .P; !/, which attains its minimum K D 0 along a closed symplectic submanifold M P . Assume in addition that the critical set M is Morse-Bott non-degenerate and one of the following cohomological conditions is satisfied: (i) M is spherically rational and c 1 .TP / D 0, or Then for every sufficiently small r 2 > 0 the level K D r 2 carries a contractible in P periodic orbit of the Hamiltonian flow of K with period bounded from above by a constant independent of r. When M is a point, Theorem 1.1 turns into the Weinstein-Moser theorem (see [We1] and [Mo]) on the existence of periodic orbits near a non-degenerate extremum, albeit without the lower bound dim P =2 on the number of periodic orbits. Remark 1.2. The assertion of the theorem is local and concerns only a neighborhood of M in P . Hence, in (i) and (ii), we can replace c 1 .TP / by c 1 .TP j M / D c 1 .TM / C c 1 .TM ? / and OE! by OE!j M . Also note that in (ii) we do not require Vol. 84 (2009) Weinstein-Moser theorem 867 to be positive, i.e., M need not be monotone. (However, this condition does imply that M is spherically rational.) We also emphasize that we do need conditions (i) and (ii) in their entirety -the weaker requirements c 1 .TP /j 2 .P / D 0 or c 1 .TP /j 2 .P / D OE!j 2 .P / , common in symplectic topology, are not sufficient for the proof. Although conditions (i) and (ii) enter our argument in an essential way, their role is probably technical (see Section 7.2), and one may expect the assertion of the theorem to hold without any cohomological restrictions on P . 1 For instance, this is the case whenever codim M D 2; see [Gi2] . Furthermore, when codim M > 2 the theorem holds without (i) and (ii), provided that the normal direction Hessian d 2 M K and ! meet a certain geometrical compatibility requirement; [GK1], [GK2], [Ke1]. On the other hand, the condition that the extremum M is Morse-Bott non-degenerate is essential; see [GG2]. 1.2. Periodic orbits of twisted geodesic flows. Let M be a closed Riemannian manifold and let be a closed 2-form on M . Equip T M with the twisted symplectic structure ! D ! 0 C , where ! 0 is the standard symplectic form on T M and W T M ! M is the natural projection. Denote by K the standard kinetic energy Hamiltonian on T M corresponding to a Riemannian metric on M . The Hamiltonian flow of K on T M describes the motion of a charge on M in the magnetic field and is referred to as a magnetic or twisted geodesic flow; see, e.g., [Gi3] and references therein for more details. Clearly, c 1 .T .T M // D 0, for T M admits a Lagrangian distribution (e.g., formed by spaces tangent to the fibers of ), and M is a Morse-Bott non-degenerate minimum of K. Furthermore, M is a symplectic submanifold of T M when the form symplectic. Hence, as an immediate application of case (i) of Theorem 1.1, we obtain Theorem 1.3. Assume that is symplectic and spherically rational. Then for every sufficiently small r 2 > 0 the level K D r 2 carries a contractible in T M periodic orbit of the twisted geodesic flow with period bounded from above by a constant independent of r. Remark 1.4. The proof of Theorem 1.1 is particularly transparent when P is geometrically bounded and symplectically aspherical (i.e., !j 2 .P / D 0 D c 1 .TP /j 2 .P / ). This particular case is treated in Section 4, preceding the proof of the general case. The twisted cotangent bundle .T M; !/ is geometrically bounded; see [AL], [CGK], [Lu1]. Furthermore, .T M; !/ is symplectically aspherical if and only if .M; / is weakly exact (i.e., j 2 .M / D 0). Note also that, as the example of the horocycle flow shows, a twisted geodesic flow with symplectic magnetic field need not have periodic orbits on all energy levels; V. Ginzburg and B. Z. Gürel CMH see, e.g., [CMP], [Gi3] for a detailed discussion of this example and of the resulting transition in the dynamics from low to high energy levels. Similar examples also exist for twisted geodesic flows in dimensions greater than two, [Gi4, Section 4]. 1.3. Related results. To the best of the authors' knowledge, the existence problem for periodic orbits of a charge in a magnetic field was first addressed by V.I. Arnold in the early 1980s; [Ar2], [Ko]. Namely, V. I. Arnold established the existence of at least three periodic orbits of a twisted geodesic flow on M D T 2 with symplectic magnetic field for all energy levels when the metric is flat and low energy levels for an arbitrary metric. (It is still unknown if the second of these results can be extended to all energy levels.) Since then the question has been extensively investigated. It was interpreted (for a symplectic magnetic field) as a particular case of the generalized Weinstein-Moser theorem in [Ke1]. Referring the reader to [Gi3], [Gi6], [Gi7] for a detailed review and further references, we mention here only some of the results most relevant to Theorems 1.1 and 1.3. The problems of almost existence and dense existence of periodic orbits concern the existence of periodic orbits on almost all energy levels and, respectively, on a dense set of levels. In the setting of the generalized Weinstein-Moser theorem or of twisted geodesic flows, these problems are studied for low energy levels in, e.g., [CGK][St]. In particular, almost existence for periodic orbits near a symplectic extremum is established in [Lu2] under no restrictions on the ambient manifold P . When P is geometrically bounded and (stably) strongly semi-positive, almost existence is proved for almost all low energy levels in [Gü] under the assumption that !j M does not vanish at any point, and in [Sc] when M has middle-dimension and !j M ¤ 0. These results do not require the extremum M to be Morse-Bott non-degenerate. Very strong almost existence results (not restricted to low energy levels) for twisted geodesic flows with exact magnetic fields and also for more general Lagrangian systems are obtained in [Co], [CIPP]. The dense or almost existence results established in [CGK], [GG2], [Ke3] follow from Theorem 1.1. However, the proof of Theorem 1.1 relies on the almost existence theorem from [GG2] or, more precisely, on the underlying Floer homological calculation. As is pointed out in Section 1.1, in the setting of the generalized Weinstein-Moser theorem without requirements (i) and (ii), every low energy level carries a periodic orbit whenever codim M D 2 or provided that the normal direction Hessian d 2 M K and ! meet certain geometrical compatibility conditions, which are automatically satisfied when codim M D 2 or M is a point; see [Gi1], [Gi2], [GK1], [GK2], [Ke1], [Mo], [We1] and references therein. Moreover, under these conditions, non-trivial lower bounds on the number of distinct periodic orbits have also been obtained. The question of existence of periodic orbits of twisted geodesic flows on (low) energy Vol. 84 (2009) Weinstein-Moser theorem
doi:10.4171/cmh/184 fatcat:grzhcl33hbcozkublu3ibipjgu