Closed characteristics on non-compact hypersurfaces in \({\mathbb {R}^{2n}}\)

Jan Bouwe van den Berg Federica Pasquotto Robert C. Vandervorst Viterbo demonstrated that any (2n 1)-dimensional compact hypersurface M (R2n , ) of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (R2n , ). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces M (R2n , ) coming from mechanical Hamiltonians, and prove an analogue of Viterbo's result. The main result provides a strong connection between the top half homology groups Hi (M ), i = n, . . . , 2n 1, and the existence of closed characteristics in the non-compact case (including the compact case). 1 Introduction It was proven by Rabinowitz [24, 25] that any starshaped and compact hypersurface in R2n , i.e., a hypersurface that occurs as a regular energy surface of the Hamilton equations, contains at least one periodic orbit for the Hamilton equations, also called a J. B. van den Berg is supported by NWO VENI grant 639.031.204. R. C. Vandervorst and F. Pasquotto are supported by NWO VIDI grant 639.032.202. This research is also partially supported by the RTN project Fronts-Singularities. closed characteristic. At the same time a similar result was obtained by Weinstein for convex hypersurfaces [33] and under the assumption of compactness he reformulated this problem in symplectically invariant terms, generalizing the convexity hypothesis. Triggered by these results Weinstein [34] conjectured that compact smooth hypersurfaces M R2n (in fact an arbitrary symplectic manifold) with H1(M ) = 0, that satisfy a specific geometric property, always contain a closed characteristic for the (normalized) Hamilton equations x = J nM . Here nM is the outward pointing normal on M and J = 01 01 the standard symplectic matrix, i.e. (, J ) = , , with the standard symplectic form on R2n, and , the standard inner product. Viterbo [31] proved Weinsteins conjecture in R2n without the condition on the first homology group. The geometric condition in Weinsteins conjecture, known as the contact type condition, can be explained as follows. A hypersurface M R2n is of contact type if there exists a so-called Liouville vector field Y (i.e. a vector field Y such that LY = ) defined on a neighborhood of M , which is transverse to M . Given such a Liouville vector field Y , the associated 1-form = iY is a contact form on M . There are examples by Ginzburg of compact hypersurfaces which are not of contact type and contain no closed characteristics [11,12]. To give some more background, the problem posed by Weinstein can also be phrased in purely geometric terms. The characteristic line bundle of M is defined by A closed characteristic of M is an embedded circle : S1 M such that T = M on . For a hypersurface M (R2n, ) the contact type condition is equivalent to the existence of a 1-form on M such that d = |M , and is non-vanishing on M \{0}. As we mentioned before, Viterbo proved that any compact hypersurface M (R2n, ) of contact type has a closed characteristic for the characteristic line bundle. We note that regular compact and starshaped hypersurfaces, as considered by Rabinowitz, are automatically of contact type [16]. In that sense the results on compact hypersurfaces of contact type are an extension of the results by Rabinowitz. The objective of this paper is to investigate this result in the case of non-compact hypersurfaces. In particular the connection between the existence of closed characteristics and the topology and geometry of a hypersurface. The complications encountered in dealing with the non-compactness of a hypersurface lead to formidable difficulties. Therefore, in this paper, we choose to consider the class of hypersurfaces that occur as energy surfaces of a classical mechanical Hamiltonians. For compact hypersurfaces coming from mechanical Hamiltonians existence of closed characteristics was proven by Weinstein [33]. To be precise about this definition, let ( p, q) be the standard symplectic coordinates on R2n, and consider a hypersurface M R2n given as 0-level set of a Hamiltonian function H ( p, q) = 21 | p|2 + V (q), i.e. M = H 1(0) = ( p, q) R2n 21 | p|2 + V (q) = 0 , where the potential V is a C 2(Rn; R) function (in particular, it is not singular). From now on we will restrict our attention to hypersurfaces of the above type, which we refer to as mechanical hypersurfaces. There is some freedom in the choice of the potential. Let N be the projection of M onto the q-coordinate: where is the projection ( p, q) q. The shape of M only fixes the function V on N Rn, hence on Rn N the potential can be suitably altered. We point out that for mechanical systems the energy surfaces M are non-compact if and only if the configuration space N = (M ) is non-compact. If one were to consider an indefinite kinetic energy term 21 Ap, p , then such systems would allow non-compact energy surfaces with sometimes compact components in N . Problems of that type were considered for example in [8,17,18], and they also occur for second order Lagrangians [1,6,19,30]. Regular energy surfaces of mechanical systems are always of contact type, also in the non-compact case, cf. [1]. Some simple counterexamples show that non-compact hypersurfaces of contact type need not contain any closed characteristics in general. Consider M1 = {| p|2 |1q|2 1 = 0} = Sn1 Rn, which is of contact type by virtue of the contact form = 2 ( pdq qd p), but clearly contains no closed characteristics. The nonzero homology groups in this cases are H0(M1) = Hn1(M1) = Z. The topologically different example M2 = {| p|2 + in=11 qi2 + 2 arctan qn = 1} = S2n2 R also contains no closed characteristics, and its homology is given by H0(M2) = H2n2(M2) = Z, and zero elsewhere. For compact hypersurfaces Poincar duality reveals that the first n Betti numbers are equal to the last n Betti numbers: i = 2n1i , or more precisely H i (M ) = H2n1i (M ). In the non-compact case this result is not true; since M is orientable it holds that M is non-compact if and only if H2n1(M ) = 0. Our main theorem states that the latter n homology groups give information about the existence of closed characteristics. In the above examples, the manifold M1 has nontrivial homology for i < n only, while M2 has nontrivial homology for i = 2n 2. Nevertheless, both examples have no closed characteristics. Topology is thus not the only requirement for existence. An additional geometric condition is needed in the non-compact case. The topological information about M will be used to construct critical values of an appropriate action functional and therefore construct closed characteristics. In Viterbos proof of the Weinstein Conjecture compactness is used analytically for the convergence of PalaisSmale sequences, and topologically to construct critical points of the action functional. We want to replace co (...truncated)