Closed characteristics on non-compact mechanical contact manifolds 1

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1 Closed characteristics on non-compact mechanical contact manifolds Jan Bouwe van den Berg, Federica Pasquotto, Thomas O. Rot and Robert C.A.M. Vandervorst Department of Mathematics, VU Universiteit Amsterdam, the Netherlands. Abstract. This paper is concerned with the existence of closed characteristics for a class of non-compact contact manifolds: mechanical contact manifolds. In [3] it was proved that, provided certain geometric assumptions are satisfied, regular mechanical hypersurfaces in R n, in particular non-compact ones, contain a closed characteristic if one homology group among the top half does not vanish. In the present paper, we extend the above mentioned existence result to the case of non-compact mechanical contact manifolds via embeddings in cotangent bundles of Riemannian manifolds. AMS Subject Class: 37J5, 37J45, 7H. Keywords: Closed characteristic, Weinstein conjecture, Hamiltonian dynamics, linking sets in the free loop space. CONTENTS. Introduction. Mechanical Contact Manifolds 8 3. The Variational Setting 3 4. The Palais-Smale Condition 6 5. Minimax Characterization 6. Some Remarks on the Geometry and Topology of the Loop Space 3 7. Relation of Topology of the Hypersurface to the Topology of its Projection 6 8. The Link 3 9. Estimates 4. Proof of the Main Theorem 43 Jan Bouwe van den Berg supported by NWO Vici grant , Federica Pasquotto supported in part by NWO Meervoud grant 63..9, and Thomas Rot supported by NWO grant 63...

2 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST. Appendix: Construction of the Hedgehog 43. Appendix: Local computations 5 References 5. INTRODUCTION The question of existence of periodic orbits of a Hamiltonian vector field X H on a given regular energy level, i.e. a level set H of the Hamiltonian function H, with dh on, has been a central question in Hamiltonian dynamics and symplectic topology which has generated some of the most interesting recent developments in those areas. The existence of a periodic orbit does not depend on the Hamiltonian itself, but only on the geometry of the energy level that the Hamiltonian defines. For this reason one also speaks of closed characteristics of the energy level. After the first pioneering existence results of Rabinowitz [8] and [9] and Weinstein [3] for star-shaped and convex hypersurfaces respectively, Viterbo [] proved the existence of closed characteristics on all compact hypersurfaces of R n of so called contact type. The latter notion was introduced by Weinstein as a generalization of both convex and star-shaped [4]. These first results were obtained by variational methods applied to a suitable (indefinite) action functional. More recently, Floer, Hofer, Wysocki [6] and Viterbo [], provided an alternative proof of the same results (and much more) using the powerful tools of symplectic homology or Floer homology for manifolds with boundary. Up to now though, very little is known about closed characteristics on non-compact energy hypersurfaces: even the Floer homology type of technique mentioned above breaks down when one drops the compactness assumptions. It is clear that additional geometric and topological assumptions are needed in order to make up for the lack of compactness. In [3] we were able to formulate a set of such assumptions that led to an existence result for the case of mechanical hypersurfaces in R n, that is, hypersurfaces arising as level sets of Hamiltonian functions of the form kinetic plus potential energy. Mechanical hypersurfaces in cotangent bundles are an important class of contact manifolds since they occur naturally in conservative mechanical dynamics. In the case of compact mechanical hypersurfaces Bolotin [4], Benci [], and Gluck and Ziller [7] show the existence of a closed characteristic on via closed geodesics of the Jacobi metric on the configuration manifold. A more general existence result for cotangent bundles is proved by Hofer and Viterbo in [] and improved in []: Any connected compact hypersurface of contact type over a simply connected manifold has a closed characteristic, which confirms the Weinstein Conjecture in cotangent bundles of simply connected manifolds. However, the existence of closed characteristics for non-compact mechanical

3 CLOSED CHARACTERISTICS 3 hypersurfaces is not covered by the result of Hofer and Viterbo and fails without additional geometric conditions. In this paper we address the question for non-compact mechanical hypersurfaces, in cotangent bundles over arbitrary non-compact (smooth) Riemannian manifolds M,g. There is a freedom of choice in the definition of the hypersurface, which we would like to emphasize. This leads to the definition of mechanical contact manifolds... Mechanical contact manifolds. An odd, say n, dimensional manifold with a one form, whose differential d has maximal rank, defines dynamical systems in a variational manner. The characteristic line bundle of, is defined to be L ker d. Critical points of the action functional A, defined on the loop space of, are exactly the periodic orbits of non-zero sections of L. The requirement that d has maximal rank is weaker than the contact condition, but this setting naturally occurs, for example in fourth order systems []. We now introduce a special class of these manifolds, which arise in mechanical Hamiltonian dynamics. Definition. Let be a smooth manifold of dimension n and let be a -form on. The pair, is called a mechanical contact manifold, if it is a regular energy level set of a mechanical Hamiltonian, i.e. (i) There exists an n-dimensional Riemannian manifold N,g, possibly with boundary N, and a potential function V : N R, which satisfies V N N, V N, and grad V N. (ii) There exists an embedding j : T N, such that j q, # q T N H q, # q g # q,# q V q. (iii) The dynamics on is induced by the embedding into T N, thus j is the Liouville form., where The Hamiltonian vector field defined by H, thus i XH d dh, restricted to, corresponds to a non-zero section of the characteristic line bundle defined by. Note that we do not require any compactness of N. In Section 7 we describe the topology of mechanical contact manifolds. In the special case that N, then is a sphere bundle over N. If N then is a pinched sphere bundle. The fibers over the interior points of N are spheres, and at the boundary N the fibers collapse to points, cf. Figure 7.. Condition (iii) implies that the boundary N is diffeomorphic to the degeneracy locus. The nomenclature suggests that the following theorem is true. Theorem. A mechanical contact manifold is a contact manifold.

4 4 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST This theorem is well known [] when is compact, and we will discuss its proof in the non-compact case in Section. The form needs to be perturbed by an exact form to make a contact manifold, but the existence of closed characteristics does not change under this perturbation... Conformal Geometry. The dynamics on does not change under specific conformal transformations, which we discuss below. We recall some notions of conformal geometry. Two Riemannian metrics g and h on a manifold M are said to be (locally) conformally equivalent if there exists a smooth function f : M R such that h e f g. This defines an equivalence relation on the space on Riemannian metrics on M and a class is called a conformal structure on M, which we denote by g. A conformal structure is also referred to as a conformal metric or conformal class on M. A manifold M with a conformal structure g is called a conformal (Riemannian) manifold and is denoted by M, g. A conformal structure on M is conformally flat if for each point q M there exists a representative h g which is flat on a neighborhood of q, i.e. the curvature tensor vanishes identically. Thus a metric g is conformally flat, if for each q M, there exists a neighborhood U q, and a representative h g such that h is flat on U. The above mentioned conformally flat structures are also called locally conformally flat as opposed to globally conformally flat structures. For the latter a representative h g exists for which the curvature tensor vanishes globally. We are interested in a special class of conformal structures, which are conformally flat outside a compact set. A conformal structure g on M has conformal flat ends if there exists a compact set K M, such that there exists a representative h g whose curvature tensor vanishes on M K. A representative h g for which the curvature vanishes on M K is said to have flat ends. Manifolds that admit conformally flat structures are for instance generalized cylinders S k R`, and in general connected sums of manifolds with conformal flat ends, when the gluing operations are performed in a compact set, admit asymptotically conformally flat structures. Remark. Suppose a conformal structure g has a representative h g, which is complete and has flat ends. If M has only a finite number of components, then M,h is a Riemannian manifold of bounded geometry, i.e. the injectivity radius of M is positive and the curvature, and its covariant derivatives, are uniformly bounded. A proof of this fact is given by S. Ivanov on MathOverflow []..3. Conformal freedom. After N is fixed, there is a conformal freedom in choosing the metric g and the potential V. A different choice of metric h in the same conformal class h g, thus h e f g for some function f, induces the same energy level if the potential V is replaced by the potential e f V. The existence of closed characteristics does not depend on this choice.

5 CLOSED CHARACTERISTICS 5 For the variational approach we need to embed N into a manifold without boundary M, and we require that the metric and potential function extend to M in a compatible manner. Definition. Let N be a n-dimensional manifold with boundary, with metric g N, and potential V N, and M a complete n-dimensional orientable manifold with without boundary, with metric g M, and potential V M. An embedding i : N M, is called admissible if there exists a function f : M R such that g N i e f g M, and V N i e f V M. Outside of i N we demand the potential V M to be positive. We will also call the induced embedding ı : T M admissible. An admissible embedding always exists, by attaching a collar along N, and extending the metric g M and potential V M via partitions of unity. In [3] the geometric assumption asymptotic regularity was introduced for mechanical hypersurfaces in R n. This geometric assumption controls the asymptotic behavior of the potential at infinity. This condition can be formulated for mechanical contact manifolds. Definition 3. Let be a mechanical contact manifold. An admissible embedding ı : T M, is asymptotically regular if i) The metric g on M has flat ends. ii) The potential function V on M satisfies grad V q V, for q M K and Hess V q grad V q for a compact set K M and a constant V., as d M q, K, We will say that is asymptotically regular if an asymptotically regular embedding exists. We will also call the potential V asymptotically regular. In the above notation grad V is the gradient vector field of V and Hess V is the Hessian -tensor field defined by Hess V x, y : r x dv y x y V r x y V, for vector fields x, y on M, where r is the Levi-Civita connection. The norm grad V q on T q M is defined via g and Hess V q sup xq y q Hess V q x q,y q. When we consider mechanical contact manifolds we have a number of parameters that we can choose, which have no influence on the existence of closed characteristics: the embedding i : N M; the extension of the metric g N to a compatible complete structure on M with conformal flat ends; the choice of extension of the potential V N to the potential V to all of M. We are now able to state the main result of this paper: Theorem. Let, be a mechanical contact manifold. Assume there exists an asymptotically regular embedding ı : T M, and an integer k n, such that

6 6 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST (i) H k M and H k M, and (ii) H k n. Then contains a closed characteristic, which is contractible in M. In this theorem M denotes the free loop space of H loops into M. Functional analytical properties of the loop space are discussed in Section 3 and some topological and geometrical properties of the loop space are discussed in Section 6. The proof of Theorem follows the scheme of the proof of the existence result for non-compact hypersurfaces in R n presented in [3]. We regard closed characteristics as critical points of a suitable action functional A. The functional does not satisfy the Palais- Smale condition. Therefore we introduce a sequence of approximating functionals A ", for ", and we prove that these functionals satisfy the Palais-Smale condition, by using the assumptions of asymptotic regularity. Next, based on the assumptions on the topology of and M, we construct linking sets in M and lift these to linking sets in the free loop space, where we apply a linking argument to produce critical points for the approximating functionals. Finally, we show that these critical points converge to a critical point of A as ". Because we construct the linking sets in the component of the loop space containing the contractible loops, we find contractible loops. In this paper we choose not to consider non-contractible loops. One of the main issues in cotangent bundles (as opposed to R n ) is that the functional is defined on a Hilbert manifold rather than a Hilbert space. Another difficulty is that curvature terms appear in the analysis of the functional, which require some care. Theorem directly generalizes the results of [3]. Suppose that admits an asymptotically regular embedding into T M, with M R n. The standard metric on R n is flat, and the the loop space is a Hilbert space, hence contractible. Thus H k M H k M for all k, and we obtain the theorem as stated in [3]. In [3] examples are given that show that both topological and geometric assumptions on are necessary. Theorem also improves the result in the R n case, because it requires weaker assumptions on the metric. Often is directly given as the zero level set of a Hamiltonian H. In this we can state Theorem reduces to: Theorem 3. Let M,g be an n-dimensional complete orientable Riemannian manifold and H : T M R the Hamiltonian defined by H q, # q g # q,# q V q. Assume dh on, and assume V is asymptotically regular, i.e. there exist a compact K M, and a constant V, such that Hess V q grad V q V, for q M K and, as d q, K. grad V q

7 CLOSED CHARACTERISTICS 7 N N M FIGURE.. The manifold M viewed as two sheets of R 6 connected by a tube S 5,. The projection N of to the base manifold M is shown in red. In the picture N is depicted as two copies of S, but is diffeomorphic to two copies of S. Assume moreover that there exists an integer k n such that (i) H k M and H k M, and (ii) H k n. Then there is a periodic orbit, contractible in M, with energy..4. Some examples. We give an example of a manifold with non-trivial topology that satisfies the conditions of Theorem. Let M R 6 pt S 5 R. Let ', r be coordinates on S 5 R, and define the metric on M via: g M f r g S 5 dr, with f equal to r outside, and positive everywhere, and g S 5 is the round metric on the 5-sphere. This manifold is asymptotically conformally flat and admits an asymptotically flat and complete representative via g M itself. This manifold can also be visualized as two copies of R 6, with a ball of radius cut out. These are then connected via a tube S 5,, cf. Figure.4. In these coordinates we take the potential function q V q q q q3 q4 q5 q6 C, with equal to zero within the ball of radius and equal to outside the ball of radius. For C chosen large enough, we can study the topology of. The topology of the projection of to the base manifold, and its boundary N q M V q, and N q M V q, are easily visualized, N deformation retracts to S 5, and the boundary N deformation retracts to two disjoint circles S. From this, using Proposition, we can compute the the homology of. From the long exact sequence of the pair we read off (for k ) that H N, N Z, and hence H 7. The free loop space of M is homotopic to the

8 8 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST M M N N N N FIGURE.. Admissible embeddings of N in two different manifolds M. On the left hand side we embed N in T M with M S R, and on the right hand we embed N in M R. free loop space of the 5-sphere. This is well studied, and from the path and loop space fibrations, and a spectral sequence argument, we see that H M H 3 M. Thus contains a contractible closed characteristic. We now study another example. Consider the question of existence of periodic orbits of the Hamiltonian system on the cylinder M S R, with an asymptotically regular potential V, which is negative on N, a set homeomorphic to S,, R, where the corners are sufficiently smoothed, cf. Figure.5. The boundary N consists of two components, one diffeomorphic to R and one component diffeomorphic to S. Using Proposition we find that H k n, for k. The free loop space of M is homotopic to S Z, and we see that H M. It seems Theorem is not applicable. However, it is not hard to find an admissible embedding of into T R, cf. Figure.5, which shows the existence of closed characteristics on..5. Convention. For the remainder of the paper, we assume an admissible embedding ı : T M is given, which always exists by the remark after Definition, and we identify with its image ı. After the proof of Theorem in Section we will always assume that this embedding is asymptotically regular. The Hamiltonian defined by the embedding is H q, # q g q # q,# q V q. Furthermore, without loss of generality, we assume M to be connected, so that it is of bounded geometry. If M is not connected, each component of M is of bounded geometry. The arguments in the proofs go through ad verbatim.. MECHANICAL CONTACT MANIFOLDS In this section we prove Theorem. We give two proofs. One proof shows that a mechanical contact manifold always is of contact type. If the mechanical contact manifold

9 CLOSED CHARACTERISTICS 9 is asymptotically regular, it is possible to write down an explicit contact form which has good asymptotic properties at infinity. This is the content of Proposition 4. Let us fix some notation. We will use vectors and covectors on the base manifold, as well as vectors and covectors on the cotangent bundle. In an attempt to reduce possible confusion, we denote elements of these bundles by different fonts. We use lowercase roman for vectors, thus a vector on M is denoted by x q, x q TM, where x q T q M. A covector on M is denoted in lowercase greek, i.e. # q, # q T M with # q T q M. Vectors on the cotangent bundle, e.g. the Hamiltonian vector field X H, are denoted in uppercase roman. In uppercase greek we denote covectors on T M, e.g. the Liouville form. The Riemannian metric g on M gives the musical isomorphism : TM T M, which is defined by q, x q q, g q x q,, where g q x q, T q M. Its inverse is #:T M TM. We also write and # for the fiber-wise isomorphisms. The Riemannian metric g induces a non-degenerate symmetric bilinear form g on T M via g q # q,# q g q x q,x q, where q, x q # q, # q and q, x q # q, # q. By abuse of notation, both for q, x q, q, x q TM and for q, # q, q, # q T M we write x q g q x q,x q, x q,x q g q x q,x q, # q g q # q,# q, # q,# q g q # q,# q. We need some standard constructions in symplectic geometry, which can be found for example in Hofer and Zehnder []. The tangent map of the bundle projection : T M M is a mapping T : TT M TM. The Liouville form, a one-form on T M, is defined by q,#q # q T q,#q. The cotangent bundle T M has a canonical symplectic structure via the Liouville form given by the -form d. Recall the definition of a contact type hypersurface. Definition 4. A hypersurface ı : on such that (i) d ı, (ii) X, for all X L with X, T M is of contact type if there exists a -form where L is the characteristic line bundle for in T M,, defined by L ker ı. The Hamiltonian vector field X H on, is defined by the relation i XH dh, and is a smooth non-vanishing section in the bundle L.

10 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST Theorem states that a mechanical contact manifold always is of contact type. The proof is an adaptation of [, Lemma 3.3], which is in turn inspired by a contact type theorem for mechanical hypersurfaces in R n in []. Proof of Theorem. The restriction of the Liouville form ı to vanishes on S q, # q # q, which is a submanifold of of dimension n. The form ı satisfies condition (i) of Definition 4, but does not satisfy condition (ii). Any exact perturbation ı df, for functions f : R, obviously satisfies condition (i) as well. We now construct an f such that the condition (ii) is also satisfied. This is first done locally. We observe that i X ı for all positive multiples X of X H. We therefore only need to prove that i XH. Let x S. Because is a regular hypersurface, that is dh on, the Hamiltonian vector field is transverse to S, i.e. X H TS. Thus there exists a chart U x, with coordinates y y,...,y n, with S U y,...,y n,, and X H y n. Construct a smooth and positive bump function h around x, supported in U, which is a product of different bump functions of the form h y h y n h y,...,y n. Assume h is equal to on a neighborhood W U of x. The function f x : R defined by then satisfies f x y n h y for y U otherwise, i XH df x h y n h y n h on U. Because y n on S, this shows that i XH df x on S, and equal to at x, hence positive on a small neighborhood around x. By the product structure of the bump function i XH df x only on a compact set outside a neighborhood of S. The quantity A x sup S max i XH df x, i XH ı is therefore finite and non-zero. Now we patch the local constructions together. Take a countable and dense collection of points x k k in S, and construct f x k as above. Define f : R via, f k f xk. k kf xk k C k A k

11 CLOSED CHARACTERISTICS This series convergences in C to a smooth function. On S we have i XH df xk for all k, and for any x S, there exists a k such that i XH f xk x by the density of the sequence x k k. Thus i X H f S. We compute this quantity outside of S i XH df k i XH df xk, k kf xk k C k A k k max i X H df xk,, k A k k k i XH i X H. On S, we therefore have that i XH df i X H. From this we see that i XH i XH ı df on. Thus is a contact form. Theorem shows that the contact type condition is a general property of mechanical contact manifolds, regardless of asymptotic behavior at infinity. In the case that is asymptotically regular, and an asymptotically regular embedding has been chosen, an explicit contact form can be constructed and a stronger contact type condition holds. Consider the vector field: grad V q v q grad V q, (.) and the function f : T M R defined by f x # q v q, for all x q, # q T M. For apple, define the -form appledf on T M. Clearly, d. Define the energy surfaces " x T M H x ". Proposition 4. Let T M be an asymptotically regular embedding. There exists ",apple, such that for every " " ", appledf restricts to a contact form on ", for all apple apple. Moreover, for every apple, there exists a constant a apple such that X H a apple, for all x " and for all " " ". The energy surfaces " are said to be of uniform contact type. Proof. A tedious, but straightforward, computation, which we have moved to an appendix, Section, reveals that grad V q Hess V q ## q, ## q X H f q, # q grad V q grad V q # q grad V q Hess V q grad V,## q grad V q. (.)

12 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST The reverse triangle inequality, and Cauchy-Schwarz directly give X H f grad V q Hess V q # q Hess V q grad V q # q grad V q grad V q grad V q grad V q 3 Hess V q # q grad V q grad V q. By asymptotic regularity there exists a constant C such that X H f This yields the following global estimate x X H q, # q applec # q grad V q grad V q C # q. grad V q apple grad V q 3 Hess V q grad V q C, hence # q grad V q apple, for all x T M, grad V q for all apple apple C. The final step is to establish a uniform positive lower bound on a apple for q, # q ", independent of q, # q and ". If d M q, K R is sufficiently large, then asymptotic regularity gives that grad V q V. Thus, in this region, # q grad V q grad V q applev apple apple grad V q grad V q V. On d M q, K R we can use standard compactness arguments. For q, # q ", we have the energy identity # q V q ". Suppose that # q ", then V q " ". If " is sufficiently small, this implies that grad V q V for some constant V, because grad V at V q. Therefore in this case # q grad V q applev apple. grad V q V If # q ", then obviously We have exhausted all possibilities and is a uniform lower bound for i XH. # q grad V q apple " grad V q. a apple min applev V, applev," V,

13 CLOSED CHARACTERISTICS 3 3. THE VARIATIONAL SETTING Closed characteristics on M can be regarded as critical points of the action functional for mappings q :,T B q, T T q t V q t dt, M. Via the coordinate transformation we obtain the rescaled action functional A c, q t,t c s, q st, log T. e c s ds e V c s for mappings c :, M and R. For closed loops we impose the boundary condition c c. By defining the parametrized circle S,,, the mappings c : S M satisfy the appropriate boundary condition. In order to treat closed characteristics as critical points of the action functional A we need an appropriate functional analytic setting. We briefly recall this setting. Details can be found in the books of Klingenberg [3] and [4]. A map c : S M is called H if it is absolutely continuous and the derivative is square integrable with respect to the Riemannian metric g on M, i.e. c s ds. The space of H maps is denoted by H S,M, or M, the free loop space of H -loops. An equivalent way to define M is to consider continuous curves c : S M such that for any chart U, ' of M, the function ' c : J c U R n is in H J, R n. There is a natural sequence of continuous inclusions C S,M H S,M C S,M, and the free loop space C S,M is complete with respect to the metric d C c, e sup s S d M c s, ẽ s, where d M is the metric on M induced by g. The smooth loops C S,M are dense in C S,M. The free loop space M H S,M can be given the structure of a Hilbert manifold. Let c C S,M and denote by C c TM the space of smooth sections in the pull-back bundle c : c TM S. For sections, C c TM consider the norms and inner products: ds, C, L, H sup s, s S s, s s, s ds, ds r s s, r s s ds,

14 4 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST with r s the induced connection on the pull-back bundle (from the Levi-Civita connection on M). The spaces C c TM, L c TM and H c TM are the completions of C c TM with respect to the norms C, L and H, respectively. The loop space M is a smooth Hilbert manifold locally modeled over the Hilbert spaces H c TM, with c any smooth loop in M. For each smooth loop c, the space H c TM is a separable Hilbert space, hence these are all isomorphic. The tangent space T c M at a smooth loop c consists of H vector fields along the loop and is canonically isomorphic to H c TM. For fixed R the functional is well-defined on the loop space M. The kinetic energy term E c : ds is well-defined and continuous. The embedding of M into C S,M and the continuity of the potential V imply that the potential energy is also well-defined and continuous. The latter implies the continuity of the functional A : M R R. We now discuss differentiability and the first variation formula. The details are in the book of Klingenberg [4]. We also denote the weak derivative of c M by c c. It is possible to extend the Levi-Civita connection on M to differentiate H c TM with respect to L c TM. We denote this connection with the same symbol r, and we write, for T c M, O r c. In general O L c TM. If and c are smooth, then O s r s s. For c M, and, T c M, the metric on M can be written as,, L O,O L. Observe that the kinetic energy is given by E c c L. Let T c M and : ", " M a smooth curve with c and. Then, d dt E t t t, r t t t L t c, O L. For the function W c V c s ds a similar calculation gives: d dt W t t grad V c, L. For convenience we write A c, e E c e W c.

15 CLOSED CHARACTERISTICS 5 Lemma 5. The action A : M R R is continuously differentiable. For any, T c, M R the first variation is given by da c,, e e c s, O s ds e c s e V c s ds d c A c, e E c e W c, where the gradient grad V is taken with respect to the metric g on M. grad V c s, s ds Proof. The first term in A is e E c of which the derivative is given by e E c e de c e E c e c, O L. Writing out the L -metric gives the desired result. The second and third term follow directly from the definition of derivative. A critical point of A is a pair c, M R such that da c,. Critical points are contained in C S,M R and satisfy the second order equation e r s c s e grad V c s. One can see this by taking variations of the form,. Consider e the expression H s c s V c s, then d ds H s e c s, r s c s grad V c s,c s, which implies that H s is constant. From the first variation formula, with variations, it follows that H s ds, and therefore H for critical points of A. Since M R has a Riemannian metric we can define the gradient of A. The metric on M R is denoted by, H R, H, R. The gradient grad A c, is the unique vector such that grad A c,,, H R da c,,, for all, T c, M R, and grad A defines a vector field on M R. We conclude with some basic inequalities for the various metrics which will be used in this analysis. The proofs are in Klingenberg [4]. Let c, e M, then d M c s,c s s s E c, (3.) d C c, e d M c, e, (3.) where d M is the metric induced by the Riemannian metric on M, and d C c, e sup s S d M c s,e s. Furthermore, for T c M, we have L C H. (3.3)

16 6 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST 4. THE PALAIS-SMALE CONDITION The functional A does not satisfy the Palais-Smale condition. We therefore approximate this functional by functionals A ", and show that the approximating functionals do satisfy PS. We then find critical points of the approximating functionals using a linking argument. Finally we show that these critical points converge to a critical point of A as ". The approximating, or penalized, functionals are defined by A " c, A c, " e e. The term "e penalizes orbits with short periods, and "e penalizes orbits with long periods. Recall, that for " fixed, a sequence c n, n M R is called a Palais- Smale sequence for A ", if: (i) there exist constants a,a such that a A " c n, n a ; (ii) da " c n, n as n tends to. Condition (ii) can be equivalently rewritten as da " c n, n, grad A " c n, n,, H R o H, (4.) as n and, T cn, n M R. Condition (i) implies that, by passing to a subsequence if necessary, A " c n, n a ", with a a " a. In what follows we tacitly assume we have passed to such a subsequence. Remark. We will only consider Palais-Smale sequences that are positive, thus a. The functionals A " satisfy the Palais-Smale condition for critical levels a " a. The relation between A and A " gives: da " c n, n, da c n, n, " e e. Lemma 6. A Palais-Smale sequence c n, n satisfies e n E c n " e n e n a " o, (4.) e n W c n " 3 4 e n a " o, as n. (4.3) Proof. Consider variations of the form,,. From the variation formula and (4.) we then derive that e n E c n e n W c n " e n e n o as n. On the other hand, A " c n, n a " means that e n E c n e n W c n " e n e n a " o. Combining these two estimates completes the proof.

17 We obtain the following a priori bounds on n. CLOSED CHARACTERISTICS 7 Lemma 7. Let c n, n be a Palais-Smale sequence. There are constants T T (depending on ") such that T n T for sufficiently large n. Proof. From Equation (4.) it follows that " e n e n a " for sufficiently large n, which proves the lemma. From this we also obtain a priori bounds on the energy. Lemma 8. Let c n, n be a Palais-Smale sequence. Then c n L E c n C ", independent of n. Proof. The a priori bounds on n can be used in Equation (4.), which yields which proves the lemma. E c n e n C " " e n e n o C ", The following proposition establishes the Palais-Smale condition for the action A ", ". Proposition 9. Let c n, n be a Palais-Smale sequence for A ". Then c n, n has an accumulation point c ", " M R that is a critical point, i.e. da " c ", " and the action is bounded a A " c ", " a " a. Proof. From Lemmas 7 and 8 we have that E c n C and n C, with the constants C, C depending only on ". Fix s S, then by Eq. (3.) we have d M c n s,c n s s s C C, and therefore c n s B C c n s, for all s S and all n. We argue that c n s must remain in a compact set as n. From this, and the energy bound on c n, it follows that c n s remains in a compact set for all s S as n. The sequence c n is equicontinuous and the generalized Arzela-Ascoli Theorem shows that c n must have a convergent subsequence in the metric d C. We then argue that this is actually an accumulation point for the metric d M, which shows that A " satisfies the Palais-Smale condition. We argue by contradiction. Suppose d M c n s,k as n. Since K M is compact, its diameter is finite and there exists a constant C K such that d M q, q C K, for all q, q K. If K B C c n s? for all n, then there exist points q n K B C c n s, with d M q, q n C K for all q K and all n. This implies: C K C d M q, q n d M q n,c n s d M q, c n s d M c n s,k, as n, which is a contradiction, and thus there exists an N such that K B C c n s?, for n N.

18 8 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST As a consequence c n s M K, for all s S and all n N. Next we show it is impossible that d M c n s,k. Suppose there does not exist a constant C such that d M c n s,k C, i.e. d M c n s,k as n. By the previous considerations there exists an N such that c n s M K, for all s S and all n N, and thus, using the asymptotic regularity, we have that For q grad V c n s c n s V, for all s S, and for all n N. M K we can define the smooth vector field on M by x q grad V q grad V q. For a loop c C S,M the vector field along c is given by s x c s grad V c s grad V c s which is a smooth section in the pull-back bundle C c TM. We now investigate the H -norm of in H c TM. For a vector field y on TM we have that r y x q r y grad V q grad V q 3 r ygrad V q grad V q, r ygrad V q, grad V q grad V q 4 3 Hess V q grad V q y, grad V q where we used the identity r y grad V q Hess V q y, r y grad V q. Now set n s x c n s. Using the asymptotic regularity of V, we obtain that r s n s 3 Hess V c s grad V c s c s o c s, and n s as d V M c n s,k. Thus the H norm of n is n H o c L. V By the first variation formula in Lemma 5 da " c n, n n, e n c n, O n L e n grad V c n, n L o c n L e n o e n. (4.4) On the other hand, since c n s is a Palais-Smale sequence, we have da " c n, n n, o n H o Co o, V which contradicts (4.4), since n is bounded by Lemma 7. This now shows that d M c n s,k C and therefore there exists an R such that c n s B R K for all s S and all n N. Since M,g is complete, the Hopf-Rinow Theorem implies

19 CLOSED CHARACTERISTICS 9 that B R K is compact and thus c n s M is pre-compact for any fixed s S. The sequence c n s is point wise relatively compact and equicontinuous by Eq. (3.). Therefore, by the generalized version of the Arzela-Ascoli Theorem [6] there exists a subsequence c nk converging in C S,M (uniformly) to a continuous limit c " C S,M. It remains to show that c " is an accumulation point in M, thus in H sense. Due to the above convergence in C S,M, the sequence c n can be assumed to be contained in a fixed chart U c, expc, for a fixed c C S,M. Following [3] it suffices to show that expc c n is a Cauchy sequence in T c M H c TM. This final technical argument is identical to Theorem.4.7 in [3], which proves that c n has an accumulation point in c ", " M R, proving the Palais-Smale condition for A ". The limit points satisfy da " c ", ", and A " c ", " a ". For critical points of A " we prove additional a priori estimates on ". The latter imply a priori estimates on c ". This allows us to pass to the limit as ". We start with pointing out that critical points of the penalized action A " satisfy the following Hamiltonian identity e " H " s c " s V c " s " e " e " ". (4.5) Thus the critical point c ", " corresponds to a closed characteristic on ". Via the transformation q " t c " te and the Legendre transform of q ",q " to a curve " on the cotangent bundle we see that the Hamiltonian action is A H " ", " " " e " e ", The following a priori bounds are due to the uniform contact type of, cf. Lemma 4. Lemma. Let c ", " be critical points of A ", with a A " c ", " a. Then there is a constant T, independent of ", such that " T for sufficiently small ". Proof. We start with the case ". The Hamiltonian action satisfies A H " ", " " " e " e " a, and thus a. Since is of uniform contact type it holds for " ", with " " " ", that a " " e " " X H a apple e ". We conclude that always " max, log a a apple, which proves the lemma. We can also establish a lower bound on " under the condition that M is asymptotically flat. This is the only estimate that requires flat ends. All other estimates carry through under the weaker assumption of bounded geometry.

20 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST Lemma. Let c ", " be critical points of A ", with a A " c ", " a. If M,g is asymptotically flat, then there is a constant T 3, independent of ", such that " T 3 for sufficiently small ". Proof. Assume, by contradiction that " as ". Then Equation (4.) gives " E c " e " a " " e3 ", as ". (4.6) Fix s S. Then the previous equation implies, using Equation 3., that c " s B " c " s, where " e " " a " e3 ". We distinguish two cases: (i) There exists an R such that d M c " s,k R for all ". Then c " s B " R K, and therefore V c " s C for all s S and all ". This implies V c e " " s ds, which contradicts (4.3), as a ", and thus " T 3. (ii) Now we assume no such R exists, and assume thus that d M c " s,k as " to derive a contradiction. By bounded geometry of M, every point q M has a normal charts U q, expq and constants,r such that B q U q and ` k ij q R. This implies that c " s U c" s for sufficiently small ". We assume M has flat ends, and since d c " s,k the metric on the charts U c" s is flat. We identify these charts with open subsets of R n henceforth. The differential equation c " satisfies is e " r s c " s grad V c " s. (4.7) Take the unique geodesic from c " s to c " s parameterized by arc length, i.e. c " s, d M c " s,c " s c " s, and t. Then, by asymptotic regularity, Hess V t C grad V t for some constant C, and d dt grad V t Hess V t grad V t, t Hess V t grad V t C grad V t Gronwall s inequality therefore implies that grad V t grad V e Ct. We identify U c" s with an open subset of R n, and we write Hence grad V t grad V grad V t grad V t t d d grad V d. Hess V d C t grad V grad V e Ct. (4.8). d

21 For any solution to Equation (4.7), we compute CLOSED CHARACTERISTICS d ds e " grad V c " s,c " s e " grad V c " s, r s c " s grad V c " s, grad V c " s grad V c " s, grad V c " s By asymptotic regularity and Estimate (4.8), we find that grad V c " s, grad V c " s grad V c " s d ds e " grad V c " s,c " s V V e Cd M c " s,c " s. We see that as " that e " grad V c " s,c " s is monotonically decreasing in s. We conclude that c " cannot be periodic. This is a contradiction, therefore there exists a constant T 3 such that " T 3, for sufficiently small ". Proposition. Let c ", ", " be a sequence satisfying da " c ", ", and a A " c ", " a. If M,g has flat ends, then there exists a convergent subsequence c ", " c, in M R, ". The limit satisfies da " c,, and a A c, a. Proof. From Lemmas and we obtain uniform bounds on ". We can now repeat the arguments of the proof of Proposition 9 on the sequence c ", from which we draw the desired conclusion. 5. MINIMAX CHARACTERIZATION We prove that bounds of the action functional on certain linking homology classes give rise to critical values. Set E M R. For d R define the sublevel sets Lemma 3. Suppose we have subsets A, B A d " c, E A " c, d. E such that (i) The morphism induced by inclusion i k : H k A H k E B is nontrivial. (ii) The subset A is compact, and the action functional satisfies the bounds A B b A A a b. (iii) The homology of the loop space vanishes in the k H k E. -st degree, and therefore Then there exist a and " such that for all " ", there exist a non-trivial y" a H k E,A" a and C " inf max A ", y" a y" a y" a satisfies the estimates a C " C for a finite constant C, independent of ".

22 OO O / OO O / / O OO J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST Proof. Choose a and b such that a a b b, with a. Define " inf c, A a a, which is finite and positive because A is compact. For all " ", e e and c, A, the following estimate holds For c, A " c, A c, " e e a B we obtain a a A " c, A c, " e e b b. ". One immediately verifies the follow- For the remainder of this proof, we assume " ing inclusions of sublevel sets a, A / A a " / A a " When we pass to homology, the sequence becomes / A b " / E B. H k A / H k A a " / H k A a " / H k A b " / H k E B, (5.) By assumption (i), the morphism induced by the inclusion is non-trivial, and it factors through this sequence. Therefore all the homology groups in Sequence (5.) are nontrivial. Consider the following part of the long exact sequence of the pair H k E,A b " H k A b " H k E. E,A b " By assumption H k E, hence H k E,A" b H k A" b is surjective. Because H k A" b, we conclude H k E,A" b. The same argument shows that the homology groups H k E,A" a are non-zero. Naturality of the boundary map with respect to the inclusions of pairs shows that the following diagram commutes H k A / H k A a H k " E,A a " / H k A a " / H k E,A a " / H k A b H k " E,A b " / H k E B. (5.) The vertical maps are the boundary homomorphisms of the long exact sequences of the pairs. Choose x H k A such that i k x. Because the map i k factors through Sequence (5.), we get non-zero elements in those groups. We can lift the nonzero y a H k A" a to some non-zero y" a H k E,A" a by the surjectivity of the boundary homomorphism. The element x is mapped by the inclusions in equation (5.) as follows x / x a " _ y" a x a " _ / y a " / x b " _ y" b / i k x.

23 CLOSED CHARACTERISTICS 3 This diagram defines the undefined elements. All elements in the above diagram are nonzero. Now define C " inf y a " y a " max A ". y" a Here we abuse notation and write y" a for the representatives of y" a. The infimum runs over all elements representing the class y" a. The support, or image, of y" a is denoted by y" a. The morphism induced by inclusion maps y" a y", b an element in C k E,A" b, which represents the class y" b. The support does not change, hence max A " y" a max A " b. y" b This follows from the following observation. If max y b " A " b, then y" a A" b, hence y" b in H k E,A" b, which is a contradiction. We therefore conclude C " b a. If we now pick a representative y" a y" a, which is mapped to y" a, we have that inf y a " y a " max A " y" a max A " y a " max A " : C. (5.3) y a " We conclude that we have, for each " ", a non-trivial class y" a H k E,A" a, such that The constants that bound C " are independent of ". a C " C. (5.4) A minimax argument now gives, if the functionals satisfy (PS), a critical point of A. Proposition 4. Assume the hypotheses of Lemma 3 are met. There exists constants a a, such that, for all " sufficiently small, A " has a critical point c ", ", satisfying a A c ", " a. Proof. The homology class y" a is homotopy invariant, and the minimax value C " over this class is finite and greater than zero, by Lemma 3. According to the Minimax principle, cf. [5], this gives rise to a Palais-Smale sequence c n, n for A ", with a a and a C. Proposition 9 states that A " satisfies the Palais-Smale condition, hence this produces, for each " sufficiently small, a critical point c ", " of A ". 6. SOME REMARKS ON THE GEOMETRY AND TOPOLOGY OF THE LOOP SPACE The following observation is a simple relation between the homology of the loop space and the base manifold. Proposition 5. There exists an isomorphism H M H M H M,M. (6.)

24 4 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST Proof. We show that M is a retract of M. Recall that the topology on M coincides with the compact open topology. Define the inclusion map : M M, by sending a point q M to the constant loop c q, with c q s q for all s S. This map is an embedding [3, Theorem.4.6]. Also the evaluation map ev : S M M, defined by ev s c c s is continuous [6, Theorem 46.]. Pick a fixed s S. Then ev s : M M is continuous, and ev s id is the identity on M. Hence M is a retract (but in general not a deformation retract) of M and the Splitting Lemma [8] gives the desired result. The goal is to construct a link in the function space with right bounds for A on the linking sets. We will in fact construct this link in a tubular neighborhood of M in M. To construct the linking sets in the loop space, we need to have a geometric understanding of the submanifold of constant loops. By Proposition 5 the inclusion : M M sending a point to the constant loop at that point is an embedding. By the assumption of bounded geometry, we can construct a well behaved tubular neighborhood. Let NM be the normal bundle of M in M. Recall that we denote the constant loop at q M, by c q, thus c q s q for all s S. Elements N cq M are characterized by the fact that s, S ds, for all T q M. Proposition 6. Assume that M is of bounded geometry. Then there exists an open neighborhood V of M in M and a diffeomorphism ' : NM V, with the property that it inj M maps NM with H to ' M with d H c q,' H. Proof. Let k : NM NM be a smooth injective radial fiber-wise contraction such that k inj M for H inj M for H H large. Thus k H inj M for all NM, and k is the identity on the disc bundle of radius inj M. Define the map ' : NM M by ' exp cq k, for N cq M. This is a diffeomorphism onto an open subset V M. We use the fact that k is injective, and along with bounded geometry this shows that exp cq is injective for H inj M. We use bounded geometry to globalize this statement to the whole of M. The inclusion of the zero section in the normal bundle is denoted by : M NM. The zero section of the normal bundle is mapped diffeomorphically into M M by '. To form the required linking sets, we need a different, but equivalent, Riemannian structure on the normal bundle.

25 CLOSED CHARACTERISTICS 5 Proposition 7. On the normal bundle NM, the norm defined by is equivalent to the norm S O s, O s ds, (6.) H. To be precise the following estimate holds H. (6.3) Proof. It is obvious that H. We need to show the remaining inequality. We do this using Fourier expansions. For T cq M, a vector field along a constant loop c q, we can write s k ke iks, with k T q M C. Then H k k, k k 4 k k, k. If NM then, by the characterization of the normal bundle given above, hence k k, k k 4 k k, k, which gives that H. We will use the next proposition to show that the linking sets we construct in the normal bundle persist in the loop space. Proposition 8. Let V be the tubular neighborhood of M inside M, constructed in Proposition 6. Let T be any subspace of V whose closure is contained in V. Assume H k M. Then the morphism induced by inclusion is injective, and is surjective. Proof. We know from Proposition 5 that H k V T H k M T, (6.4) H k V T H k M T, (6.5) H k M H k M,M H k M. We assume H k M, thus H k M,M. The tubular neighborhood V deformation retracts to M. If we apply the Five-Lemma to the long exact sequences of the pairs M,M and M,V, where the vertical maps are induced by the deformation retraction, H k V / H k M / H k M,V / H k V / H k M H k M / H k M / H k M,M / H k M / H k M

26 6 J.B. VAN DEN BERG, F. PASQUOTTO, T.O. ROT AND R.C.A.M. VANDERVORST N N N FIGURE 7.. The relation between the topology of and N for the harmonic oscillator. The fibers of over the interior of N are copies of S that get smaller at the boundary, where they are collapsed to a point. we see that H k M,V H k M,M. Since T is contained in the closure of V, we can excise T. This gives an isomorphism H k M T,V T H k M,V. The long exact sequence of the pair M T,V T is H k V T H k M T H k M T,V T H k V T H k M T. The homology group H k M T,V T is zero by the preceding argument, thus the map on the right is injective and the map on the left is surjective. 7. RELATION OF TOPOLOGY OF THE HYPERSURFACE TO THE TOPOLOGY OF ITS PROJECTION We investigate the relation between the topology of and its projection N to the base manifold. In the case N does not have a boundary, this relation is expressed by the classical Gysin sequence for sphere bundles. Recall that we assume H to be mechanical and that the hypersurface H is regular. Thus N and its boundary N are given by N q M V q, and N q M V q, and N is smooth. Let us consider a basic example, the harmonic oscillator on R. The k potential is V q q V for constants k, V. One directly verifies that N V, V is an interval and q, # R k k # k q V is an ellipse. The fibers over the interior of N are copies of S, whose size is determined by the distance to the boundary N. Close to the boundary the fibers get smaller and at the boundary the fibers are collapsed to a point. Topologically the collapsing process can be understood by the gluing of an interval at the boundary of N. The hypersurface is thus homeomorphic

27 CLOSED CHARACTERISTICS 7 to N S N S N D, which Figure 7. illustrates. This picture generalizes to arbitrary hypersurfaces. We have the topological characterization ST N ST N N DT N N. (7.) The characterization is given in terms of the sphere bundle ST N and the disc bundle DT N in the cotangent bundle of N. The vertical bars denote the restriction of the bundles to the boundary. This topological characterization gives a relation between the homology of and N. In this section we identify with this characterization. Recall that a map is proper if preimages of compact sets are compact. In the proof of the next proposition, compactly supported cohomology H c M is used, which is contravariant with respect to proper maps. In singular (co)homology, homotopic maps induce the same maps in (co)homology. For compactly supported cohomology, maps that are homotopic via a homotopy of proper maps, induce the same maps in cohomology. If N the following proposition directly follows from the Gysin sequence. Proposition 9. There exist isomorphisms H i c H i c N for all i n. Proof. Let C be the closure of a collar of N in N, which always exists, cf. [9]. Thus C deformation retracts via a proper homotopy onto N. Now C is the closure of a collar of ST N N ST N in ST N, and therefore it deformation retracts via a proper homotopy onto ST N N. Define D by D C ST N N DT N N. This is a slight enlargement of the disc bundle of M restricted to the boundary N, which Figure 7.3 clarifies. By construction D deformation retracts properly to DT N N, which in turn deformation retracts properly to N. This induces an isomorphism H c D H c N. (7.) Let S ST N. The intersection D S deformation retracts properly to ST N N. Thus the isomorphism holds. The inclusions in the diagram H c D S H c ST N N, (7.3) S D ı 6 S ' 7 ı ( D