Multiple Closed Characteristics in the Cotangent Bundle of S^{1}-Manifolds

by

Frank W. Josellis

Wilhelm Klingenberg

Abstract

Let P be a compact smooth manifold equipped with a free S^{1}-structure and let Σ ⊂ T^{∗}P be a compact hypersurface which is invariant under the induced S^{1}-action on T^{∗}P and bounds a neighborhood of the 0-section. In addition, Σ is assumed to be of contact type where the corresponding contact 1-form is related to the canonical Liouville form. Then Σ is shown to carry at least 1+cuplength(P/S^{1}) geometrically distinct closed characteristics that coincide with S^{1}-orbits on T^{∗}P. If, moreover, all these particular closed characteristics on Σ are non-degenerate then their number is greater or equal to the sum of the Betti numbers of P/S^{1}.

1991 Mathematics Subject Classification Primary: 53C15, 58F05, 58F22 Secondary: 58E05, 58F25

1. Introduction and preliminaries Let P denote a compact connected oriented smooth manifold carrying the structure of a free smooth 𝕊^{1}-space. That is, we assume that on P there is given a free smooth Lie group action 𝕊^{1} ×P → P where the circle group 𝕊^{1} is parametrized by ℝ/τℤ for some τ >0. The group structure on 𝕊^{1} is induced from (ℝ, +). We shall use the notation 𝕊_{τ}^{1} := ℝ/τℤ when we refer to a distinguished period τ.

Denoting the 𝕊^{1}-action on P by (t, x) ↦ c_{t}(x), or (t, x) ↦ t·x for brevity, we write 𝕊^{1}·x = { t·x | t ∈ 𝕊_{τ}^{1} } for the 𝕊^{1}-orbit of x∈P and P/𝕊^{1} = { 𝕊^{1}·x | x∈P } for the corresponding orbit space.

We are interested in the study of closed characteristics on suitable hypersurfaces Σ in the cotangent bundle T^{∗}P. The closed characteristics we have in mind are special in the sense that they are related to the 𝕊^{1}-orbit structure of P. We shall derive lower bounds for the number of these particular closed characteristics on Σ employing homotopy invariants of the orbit space P/𝕊^{1}. Before we can state the precise formulation of our results we need some preparations.

Using the canonical projection p: P → P/𝕊^{1} given by p(x) := 𝕊^{1} ·x, we may consider P as a (locally trivial) principal 𝕊^{1}-bundle over P/𝕊^{1}, cf. [7]. A local trivialization of P at p(x) consists of an open neighborhood U ⊂ P/𝕊^{1} of p(x) and an 𝕊^{1}-equivariant diffeomorphism φ_{U} : p^{−1}(U) → U × 𝕊^{1} such that the following diagram commutes

[figure: p^{−1}(U); U × 𝕊^{1}; U; φ_{U}; p; p_{U}]

where p_{U} : U × 𝕊^{1} → U is the projection onto the first component. In the context of principal G-bundles one usually refers to a right action of the structure group G rather than a left group action (t, x) ↦ t · x as above. However, since 𝕊^{1} is commutative, the left 𝕊^{1}-manifold P can be identified with a right 𝕊^{1}-space by setting x·t := t·x.

We point out that the 𝕊^{1}-action on P is proper, i.e., the map 𝕊^{1} × P →P ×P defined by (t, x) ↦ (x, t·x) is proper. As a consequence, the orbit space P/𝕊^{1} can be endowed with a unique structure of a smooth manifold such that the canonical projection p is a submersion, cf. [4, Prop. 5.2].

The free 𝕊^{1}-action c_{t} on P induces a free 𝕊^{1}-action γ_{t} on T^{∗}P. We note that for each fixed t ∈ 𝕊^{1} the map x ↦ c_{t}(x) is a diffeomorphism of P, having the inverse c_{t}^{−1} = c_{−t}. The induced 𝕊^{1}-action on T^{∗}P is given by the push-forward of cotangent vectors, i.e., the map γ_{t} : T_{x}^{∗}P → T_{t·x}^{∗}P , (1)

is given by (γ_{t} ξ_{x}) (v_{t·x}) = ξ_{x}(T c_{−t} v_{t·x}) (2)

where ξ_{x} ∈ T_{x}^{∗}P and v_{t·x} ∈ T_{t·x}P. In particular, for each fixed t ∈ 𝕊^{1} the map ξ ↦ γ_{t} ξ is a point transformation and, therefore, leaves the canonical forms on T^{∗}P invariant. The 𝕊^{1}-symmetry thus gives rise to a distinguished class of loops in T^{∗}P:

Definition 1 A smooth loop u : 𝕊_{τ}^{1} → T^{∗}P will be called a principal loop in T^{∗}P if it is of the form u(t) = γ_{t} ξ for some ξ ∈ T^{∗}P.

Clearly, for ξ ∈ T_{x}^{∗}P we get π^{∗}(γ_{t} ξ) = c_{t}(x) by the projection π^{∗} : T^{∗}P → P so that the principal loops in T^{∗}P cover the fibres of the principal 𝕊^{1}-bundle P ⟶^{p} P/𝕊^{1}.

Given a smooth hypersurface Σ ⊂ T^{∗}P that is invariant under the 𝕊^{1}-action γ_{t}, one may ask for the existence and multiplicity of special closed characteristics on Σ which can be represented by principal loops. This question is motivated by the study of the orbit structure of symmetric Hamiltonian systems at a fixed energy level, i.e., Σ is a regular level surface of a Hamiltonian function H : T^{∗}P → ℝ which satisfies H(γ_{t} ξ) = H(ξ) for all t ∈ 𝕊^{1}, ξ ∈ T^{∗}P. The closed characteristics on Σ correspond to the periodic trajectories of the Hamiltonian flow on the energy surface Σ. Thus we deal with the problem of finding symmetric periodic solutions of an 𝕊^{1}-symmetric Hamiltonian system at a prescribed energy level.

As an example we mention the n-dimensional torus 𝕋^{n}, represented by

𝕋^{n} = { (z_{1}, …, z_{n}) ∈ ℂ^{n} | |z_{i}|=1 for i=1, …, n } . (3)

If j = (j_{1}, …, j_{n}) ∈ ℤ^{n} is any relatively prime integer vector, a free action of 𝕊_{τ}^{1} on 𝕋^{n} is defined by

t·(z_{1}, …, z_{n}) := (exp (\frac{2πi}{τ} j_{1}t) z_{1} , …, exp (\frac{2πi}{τ} j_{n}t) z_{n}) . (4)

The orbit space 𝕋^{n}/𝕊^{1} is diffeomorphic to 𝕋^{n−1}. Note that there exist as many different free 𝕊^{1}-structures on 𝕋^{n} as there are relatively prime integer vectors j, where, however, the 𝕊^{1}-actions corresponding to j and −j coincide up to the orientation of the fibres of the principal 𝕊^{1}-bundle.

Multiplicity of closed characteristics on hypersurfaces Σ in T^{∗}𝕋^{n} has been studied in [8] and [10]. This present work can be understood as a step towards a generalization of the results in [10] where the 𝕊^{1}-manifold P replaces 𝕋^{n}. However, we point out that by the above definition of principal loops, which is purely geometrical in nature, we have to restrict ourselves to the symmetric case. This is in contrast to the situation in [10], where the 𝕊^{1}-action on 𝕋^{n} determines a homotopy class of loops in T^{∗}𝕋^{n} and, therefore, provides a topological characterization of the loops related to the given 𝕊^{1}-structure.

2. Statement of the results By λ we denote the canonical 1-form on the cotangent bundle T^{∗}P, also referred to as the Liouville form. Its exterior derivative ω = dλ is called the canonical symplectic form on T^{∗}P. Locally, in the domain of a chart, we may introduce coordinates (x_{1}, …, x_{n}) on P. In associated local coordinates (x_{1}, …, x_{n}, y_{1}, …, y_{n}) of T^{∗}P the canonical forms can be represented by

λ = ∑_{i=1}^{n} y_{i} dx_{i} , ω = ∑_{i=1}^{n} dy_{i} ∧ dx_{i} . (5)

Let D ⊂ T^{∗}P be a relatively compact neighborhood of the 0-section with smooth boundary Σ := ∂D. We shall assume that the hypersurface Σ is 𝕊^{1}-invariant and satisfies an additional hypothesis, requiring that Σ admits a contact structure whose contact form is compatible with the Liouville form. For this purpose we consider the pull-back of the forms λ and ω to Σ, denoted by λ_{Σ} and ω_{Σ} = dλ_{Σ}, respectively.

Definition 2 A smooth 𝕊^{1}-invariant hypersurface Σ ⊂ T^{∗}P will be called an invariant Liouville hypersurface if there exists a 1-form β on Σ such that the following conditions are satisfied:

(i) γ_{t}^{∗}β = β, i.e., β is 𝕊^{1}-invariant.

(ii) β−λ_{Σ} is exact.

(iii) β ∧ ω_{Σ}^{n−1} is a volume form on Σ.

Furthermore, we call Σ an invariant strict Liouville hypersurface if it is an invariant Liouville hypersurface with β = λ_{Σ}.

Let us point out that condition (ii) implies dβ = ω_{Σ}. Recall that Σ is said to be a hypersurface of contact type if this latter identity and condition (iii) both hold true, cf. A. Weinstein [14]. The 1-form β defines a contact structure on Σ in the sense that ω_{Σ} is non-degenerate on the null spaces of β.

If β − λ_{Σ} = df with a smooth function f on Σ, we can extend f to an 𝕊^{1}- invariant function F on some open 𝕊^{1}-invariant neighborhood U ⊂ T^{∗}P of Σ. Then α := λ + dF is an 𝕊^{1}-invariant extension of β to U. Therefore, we shall from now on refer to the pull-back α_{Σ} as the contact form on Σ.

Associated to the 1-form α there is a non-vanishing smooth 𝕊^{1}-equivariant vector field X_{α} on U, defined by the identity

α = X_{α} ⌟ ω . (6)

Using Cartan’s formula, we infer L_{X_{α}}ω = ω for the Lie derivative of ω along X_{α}, i.e., X_{α} is a so-called symplectic dilation. Moreover, X_{α} is transverse to Σ. To see this, let us introduce the characteristic line bundle 𝔏_{Σ} ⊂ T Σ which is defined to be the kernel of ω_{Σ},

(7) 𝔏_{Σ} := ker ω_{Σ} = ⋃_{ξ∈Σ} { X ∈T_{ξ}Σ | ω(X, Y) = 0 ∀ Y ∈T_{ξ}Σ } .

A nowhere vanishing section X : Σ → 𝔏_{Σ} is called a characteristic vector field on Σ. The condition that α_{Σ} ∧ ω_{Σ}^{n−1} be a volume form on Σ implies that α_{Σ}(X) ≠ 0 for every characteristic vector field X on Σ. Then

0 ≠ α(X) = (X_{α} ⌟ ω)(X) = ω(X_{α}, X) = −ω(X, X_{α}) . (8)

Consequently, the symplectic dilation X_{α} is transverse to Σ.

Clearly, an invariant hypersurface Σ ⊂ T^{∗}P is Liouville if there exists a transverse 𝕊^{1}-equivariant symplectic dilation X_{∗} defined on some neighborhood of Σ such that the 1-form (X_{∗} ⌟ ω) − λ is exact. Then the pull-back (X_{∗} ⌟ ω)_{Σ} is the desired contact form on Σ.

A closed characteristic on Σ can now be defined as an embedding u : 𝕊_{τ}^{1} → Σ whose tangent map is an embedding u_{∗} : T 𝕊_{τ}^{1} → 𝔏_{Σ}. Any two closed characteristics u, v on Σ are said to be (geometrically) distinct if u(𝕊_{τ}^{1}) ∩ v(𝕊_{τ}^{1}) = ∅. Considering symmetric hypersurfaces, we are particularly interested in the study of symmetric characteristic embeddings of the circle:

Definition 3 A closed characteristic u : 𝕊_{τ}^{1} → Σ will be called a principal closed characteristic on Σ if i∘u : 𝕊_{τ}^{1} → T^{∗}P is a principal loop according to Definition 1, where i : Σ → T^{∗}P is the inclusion.

Our first result is the following

Theorem 1 Let P be a compact connected free 𝕊^{1}-manifold and suppose Σ ⊂ T^{∗}P is an invariant Liouville hypersurface bounding a relatively compact neighborhood of the 0-section in T^{∗}P. Then

(9) # { distinct principal closed characteristics on Σ } ⩾ CL (P/𝕊^{1})

where CL (P/𝕊^{1}) := 1 + cuplength (P/𝕊^{1}).

To speak of closed characteristics means to speak of periodic Hamiltonian orbits without referring to a particular parametrization. However, for a compact Liouville hypersurface Σ, we shall introduce appropriate admissible Hamiltonian functions, defined on some neighborhood of Σ with the aid of the flow ψ_{s} of the symplectic dilation X_{α}. Postponing the details to subsequent sections, let us call the Hamiltonian H admissible if it is of the form H(ψ_{s}(ξ)) = h(s) for ξ ∈ Σ, where h : (−δ, δ) → ℝ is a smooth function satisfying h′(s) ≠ 0. If X_{α} is 𝕊^{1}-equivariant, so is its flow ψ_{s}, and consequently the admissible Hamiltonian H is 𝕊^{1}-invariant. The associated Hamiltonian vector field X_{H}, defined by the identity

− dH = X_{H} ⌟ ω , (10)

gives rise to the characteristic vector field X_{H}|_{Σ} on Σ. Note that the choice of H induces an orientation of 𝔏_{Σ}, according to α(X_{H}) > 0 or α(X_{H}) < 0. This orientation corresponds to the sign of h′(s).

Let χ : Σ×ℝ → Σ denote the flow of X_{H}|_{Σ} and suppose that χ(ξ, t) is a τperiodic integral curve. Then 1 is obviously a relative Floquet multiplier of χ(ξ, t), i.e., 1 is an eigenvalue of T χ(ξ, τ) ∈ Hom (T_{ξ}Σ). We say χ(ξ, t) is non-degenerate mod 𝕊^{1} if it has only one simple relative Floquet multiplier equal to 1. This definition is independent of the particular admissible Hamiltonian H and, therefore, we may speak of non-degeneracy of closed characteristics on Σ.

Theorem 2 Let Σ satisfy the hypotheses of Theorem 1 and assume, in addition, that all the principal closed characteristics on Σ are non-degenerate. Then

(11) # { distinct principal closed characteristics on Σ } ⩾ SB (P/𝕊^{1})

where SB (P/𝕊^{1}) stands for the sum of the Betti numbers of P/𝕊^{1}.

A special situation is given if P is endowed with a Riemannian metric g. Recall that g defines an isomorphism of vector bundles T P → T^{∗}P by ξ_{∗} ↦ ξ := g(ξ_{∗}, ·). Correspondingly, there is a scalar product g^{∗} on the fibres of T^{∗}P defined by g^{∗}(ξ, η) := g(ξ_{∗}, η_{∗}). The hypersurface Σ^{∗} = { ξ ∈ T^{∗}P | g^{∗}(ξ, ξ) = 1 } is called the unit cosphere bundle of (P, g).

In particular, let us assume that g^{∗} is 𝕊^{1}-invariant, i.e., g^{∗}(γ_{t} ξ, γ_{t} η) = g^{∗}(ξ, η) for all ξ, η ∈ T^{∗}P. This is equivalent to saying that g is an 𝕊^{1}-invariant metric, meaning that g(T c_{t} ξ_{∗}, T c_{t} η_{∗}) = g(ξ_{∗}, η_{∗}). If g^{∗} is 𝕊^{1}-invariant then Σ^{∗} is a strict Liouville hypersurface that meets the assumptions of Theorem 1, see Lemma 4 below.

It is well known that the closed characteristics on Σ^{∗} correspond uniquely to closed geodesics on (P, g), cf. [11, Ch. 3.1]. Hence we may define the principal closed geodesics on P as the projections of the principal closed characteristics on an 𝕊^{1}-invariant cosphere bundle Σ^{∗}. Moreover, non-degenerate closed characteristics on Σ^{∗} correspond to non-degenerate closed geodesics on P. For the unit cosphere bundle the above results thus can be reformulated in terms of closed geodesics:

Corollary 1 Let g be an 𝕊^{1}-invariant Riemannian metric on P. Then

(12) # { distinct principal closed geodesics on P } ⩾ CL (P/𝕊^{1}) .

If all the principal closed geodesics on P are non-degenerate then

(13) # { distinct principal closed geodesics on P } ⩾ SB (P/𝕊^{1}) .

3. The strategy of the proof Evidently it is sufficient to carry out the proofs under the assumption that there is only a finite number of distinct principal closed characteristics on Σ. This assumed number is allowed to be zero, i.e., the existence of a principal closed characteristic on Σ is not presupposed. We emphasize that additional closed characteristics which are not principal with respect to the 𝕊^{1}-action in question may exist on Σ. In particular, by the above finiteness assumption we do not impose any restrictions on the possible number of these additional closed characteristics.

As already mentioned in the previous section, the closed characteristics on Σ can be represented as periodic trajectories of a Hamiltonian flow. The choice of a Hamiltonian function H : T^{∗}P → ℝ with X_{H}|_{Σ} ≠ 0 provides a parametrization of each integral curve of X_{H} and, therefore, assigns a distinguished minimal period to every periodic orbit. Let us introduce the following denotation:

Definition 4 Suppose u(t) is a periodic integral curve of X_{H} having the minimal period T >0. We call u a principal periodic Hamiltonian orbit if the map 𝕊_{τ}^{1} → T^{∗}P, defined by t ↦ u(\frac{T}{τ} t), is a principal loop according to Definition 1.

In general, the distinct principal periodic X_{H}-orbits on Σ have distinct minimal periods. However, using the symplectic dilation X_{α}, we can achieve an 𝕊^{1}-equivariant blow up of Σ which allows us to transform the problem of counting principal periodic X_{H}-orbits on Σ into the problem of counting principal Hamiltonian orbits which have a fixed minimal period T > 0. This latter problem will be treated by a variational approach on a suitable space of T -periodic loops u : 𝕊_{T}^{1} → T^{∗}P.

The blow up of Σ along the flow of X_{α} is described in section 4. Recall that the symplectic dilation X_{α} is solely defined on some neighborhood of Σ. Together with the blow up we introduce an admissible 𝕊^{1}-invariant Hamiltonian H : U → ℝ. Thus we use the flow of X_{α} to generate a tower of periodic X_{H}-orbits over each closed characteristic on Σ. Consequently, the required counting of the distinct principal closed characteristics can be done by counting the corresponding principal towers.

If T > 0 is sufficiently large, we can derive a priori estimates which guarantee that each principal tower contains exactly two periodic trajectories of X_{H} of minimal period T . Furthermore, in each principal tower the orbits of minimal period T can be distinguished by means of the Hamiltonian action. More precisely, the i-th principal tower contains a unique Hamiltonian orbit, represented by u_{i}(t), which has both the minimal period T and a positive Hamiltonian action integral, i.e.,

A_{H}(u_{i}) := ∫_{𝕊_{T}^{1}} u_{i}^{∗}λ − ∫_{0}^{T} H(u_{i}(t)) dt > 0 . (14)

Corresponding to the choice of T > 0 we fix an adapted parametrization for the 𝕊^{1}- action γ on T^{∗}P. Hence u_{i} is represented by u_{i}(t) = γ_{t} ξ_{i} for some ξ_{i} ∈ U. Loosely speaking, by the variational formulation the problem is reduced to counting those distinct critical 𝕊^{1}-orbits of A_{H} in U which satisfy the inequality (14).

We point out that from our construction of the Hamiltonian H : U → ℝ the intended multiplicity result for critical 𝕊^{1}-orbits does not follow immediately. The situation is similar to that of the Poincaré-Birkhoff fixed point theorem on an annulus where the twist of the boundaries is well known to be an indispensable hypothesis. So far we have included no twist condition, and there is indeed no topological evidence for the existence of critical 𝕊^{1}-orbits. For this reason we extend the Hamiltonian suitably to a function H : T^{∗}P → ℝ such that H is a metric outside a compact neighborhood of the 0-section. For a more detailed discussion of the role of this asymptotic boundary condition we refer to the recent paper by C. Golé [9].

The asymptotic twist condition allows a global variational characterization of critical 𝕊^{1}-orbits, but it does not guarantee that these are found on a positive level of A_{H}. Moreover, the extension engenders additional critical 𝕊^{1}-orbits describing principal Hamiltonian orbits outside U. These will be called technical critical 𝕊^{1}- orbits and have to be distinguished from the significant critical 𝕊^{1}-orbits in the towers over closed characteristics on Σ. As a consequence, our demands on the extension of H are two-fold: firstly, the technical critical 𝕊^{1}-orbits have to lie on a negative A_{H}-level and, secondly, the extended variational problem has to allow an analysis of the significant critical set on a positive A_{H}-level.

The latter requirement does not depend on the asymptotic twist and will be settled by defining H ≡ 0 in a small neighborhood of the 0-section. By this construction we can establish a sort of topological linking which leads to the claimed multiplicity of critical 𝕊^{1}-orbits.

In sections 5 and 6 we describe the global extension of the Hamiltonian function as indicated above. The variational setup for principal T -periodic 𝕊^{1}-orbits and the associated free Morse complex are described in section 7. We emphasize that the structure of the problem allows to separate a finite dimensional manifold Ω_{prin} of principal periodic Hamiltonian orbits on which 𝕊^{1} acts freely. In particular, the iterate orbits, i.e., the principal periodic Hamiltonian orbits having a minimal period T /m with m = 2, 3, … do not contribute to the critical set.

Section 8 contains the equivariant cuplength estimate for the positive part of the free Morse complex. This is the essential topological ingredient in the proof of Theorem 1, which is completed in section 9 by the use of Lyusternik-Schnirelman theory for flows. The proof of the non-degenerate case of Theorem 2, given in section 10, is based on an 𝕊^{1}-equivariant version of Morse theory for flows in the spirit of Conley-Zehnder.

4. The Hamiltonian near the hypersurface We begin by introducing an almost complex structure on (T^{∗}P, ω), i.e., we fix a complex operator J ∈ Hom(T T^{∗}P), J(ξ)^{2} = −id_{T_{ξ}T^{∗}P}, which is adapted to ω in the sense that ⟨ X, Y ⟩ := ω(X, JY) (15)

defines a scalar product with corresponding norm ∥·∥ on the fibres of T T^{∗}P, cf. [13]. The operator J will be specified with regard to our particular purposes in section 5 below. Here we only use the fact that J can be chosen in such a way that the metric is 𝕊^{1}-invariant, i.e., ⟨ T γ_{t} X, T γ_{t} Y ⟩ = ⟨ X, Y ⟩.

The outward unit normal vector N(ξ) at ξ ∈ Σ with respect to ⟨ · , · ⟩ is characterized by the following properties: ∥N(ξ)∥ = 1, ⟨N(ξ), X(ξ)⟩ = 0 for all X(ξ) ∈ T_{ξ}Σ, and ⟨N(ξ), X_{α}(ξ)⟩ > 0 where X_{α} is the 𝕊^{1}-equivariant transverse symplectic dilation defined in (6). We note that JN is a characteristic vector field on Σ and that the choice of N determines a distinguished orientation of 𝔏_{Σ} in view of α(JN) = ω(X_{α}, JN) = ⟨ X_{α}, N ⟩ > 0.

By ψ_{s} we denote the flow of X_{α}. Since Σ is compact there exists ŝ > 0 such that ψ_{s}(ξ) is defined for all ξ ∈ Σ and |s| ⩽ ŝ. The flow yields a parametrized family of 𝕊^{1}-invariant hypersurfaces Σ_{s} = ψ_{s}(Σ) modelled on Σ, and we set

U := ⋃_{|s|<ŝ}Σ_{s} ⊂ T^{∗}P . (16)

Each Σ_{s} is diffeomorphic to Σ and has the outward unit normal vectors N(ψ_{s}(ξ)). Choosing ŝ > 0 sufficiently small, we can assume that ⟨N(ψ_{s}(ξ)), X_{α}(ψ_{s}(ξ))⟩ > 0 for all ξ ∈ Σ and |s| ⩽ ŝ so that, in particular, each Σ_{s} is an invariant Liouville hypersurface. Moreover, we can assume that ŝ is chosen small enough such that for all |s| < ŝ the hypersurface Σ_{s} bounds a neighborhood of the 0-section in T^{∗}P.

For a Hamiltonian function H : T^{∗}P → ℝ the gradient of H with respect to the metric (15) is given by dH = ⟨ grad H , · ⟩ and satisfies X_{H}(ξ) = J(ξ) grad H(ξ), where X_{H} is the Hamiltonian vector field introduced in (10). The Hamiltonian equations thus have the form

u̇ = X_{H}(u) = J(u) grad H(u) . (17)

If H is 𝕊^{1}-invariant, H(γ_{t} ξ) = H(ξ), then the gradient of H is 𝕊^{1}-equivariant, that is, grad H(γ_{t} ξ) = T γ_{t} grad H(ξ).

Now we introduce a special Hamiltonian on the neighborhood U of Σ:

Proposition 1 Let the invariant Liouville hypersurface Σ ⊂ T^{∗}P satisfy the hypotheses of Theorem 1 and, moreover, assume that Σ carries at most finitely many distinct principal closed characteristics. Then there exist δ ∈ (0, ŝ), T > 0, and a smooth Hamiltonian H : U → ℝ such that the following conditions are satisfied:

(i) Σ_{s} is a regular level surface of H if |s| < δ.

(ii) Assume that u_{0} is a principal periodic orbit of X_{H} on Σ. Then there exist s∈(−δ, δ) and a principal T -periodic orbit u_{s} of X_{H} on Σ_{s} such that s and u_{s} are uniquely determined by the following properties:

(a) u_{s} has the minimal period T .

(b) u_{s}(0) = (ψ_{s} ∘ u_{0})(0).

(c) The Hamiltonian action integral of u_{s} is positive, i.e.,

A_{H}(u_{s}) = ∫_{𝕊_{T}^{1}} u_{s}^{∗}λ − ∫_{0}^{T} (H ∘u_{s}) dt > 0

where λ is the Liouville form.

Proof We fix 0 < δ < ŝ and introduce the smooth function g : (−ŝ, ŝ) → ℝ_{0}^{+} by setting

g(s) := \{ \begin{matrix}exp (\frac{−δ^{2}}{δ^{2}−s^{2}} ) & if & 0 ⩽ |s| < δ \\ 0 & if & δ ⩽ |s| < ŝ . \end{matrix} (18)

Let h : (−ŝ, ŝ) → ℝ be given by

h(s) := ∫_{−ŝ}^{s} g(t) dt . (19)

Now we define the Hamiltonian H|_{U} : U → ℝ by

H|_{U}(ψ_{s}(ξ)) = H(ψ_{s}(ξ)) := h(s) if ξ ∈ Σ , |s| < ŝ . (20)

The condition (i) means exactly that X_{H} is a characteristic vector field on Σ_{s} for |s| < δ. Clearly, since the value of H remains constant along integral curves of X_{H} it follows that X_{H}(ψ_{s}(ξ)) ∈ T_{ψ_{s}(ξ)}Σ_{s} for all |s| < ŝ.

We claim that X_{H}(ψ_{s}(ξ)) ∈ ker ω_{Σ_{s}}. Let X ∈ T_{ψ_{s}(ξ)}Σ_{s} be a tangent vector and let w : (−a, a) → Σ_{s} be a smooth path in Σ_{s} with w(0) = ψ_{s}(ξ) and ẇ(0) = X. Because H is constant along w we find

ω(X_{H}, X) = −dH(X) = −\frac{d}{dt}|_{t=0}H(w(t)) = 0 . (21)

Moreover, we claim that X_{H}(ψ_{s}(ξ)) ≠ 0 if |s| < δ. Indeed,

\begin{aligned}α(X_{H})|_{ψ_{s}(ξ)} & = & dH(X_{α})|_{ψ_{s}(ξ)} = L_{X_{α}}H|_{ψ_{s}(ξ)} = \frac{d}{dt}|_{t=0}ψ_{t}^{∗}H(ψ_{s}(ξ)) \\ & = & \frac{d}{dt}|_{t=s}h(t) = h′(s) . \end{aligned} (22)

Consequently, X_{H} is a characteristic vector field on Σ_{s} if |s| < δ, and (i) of the proposition is herewith verified.

The equations (21) and (22) can be interpreted in terms of grad H. From (21) it follows that grad H is parallel to the normal vector N, so that

⟨ grad H(ψ_{s}(ξ) , X_{α}(ψ_{s}(ξ)) ⟩ = γ(ψ_{s}(ξ)) ⟨ N(ψ_{s}(ξ)) , X_{α}(ψ_{s}(ξ)) ⟩ (23)

for some function γ : U → ℝ. In view of (22) the left hand side is equal to h′(s). Therefore, the gradient of H at ψ_{s}(ξ) with ξ ∈ Σ, |s| < ŝ, can be represented by

grad H(ψ_{s}(ξ)) = \frac{h′(s)}{⟨ N(ψ_{s}(ξ)) , X_{α}(ψ_{s}(ξ)) ⟩} N(ψ_{s}(ξ)) . (24)

The next lemma shows how the Hamiltonian orbits on Σ_{s} are related to the Hamiltonian orbits on Σ_{0}. We merely state the assertion and refer to [10] for a proof, which is literally the same for the situation at hand.

Lemma 1 Let u_{0} : ℝ → Σ be a principal periodic Hamiltonian orbit on Σ, having the minimal period τ_{0} > 0. If |s| < δ then there is an associated principal periodic Hamiltonian orbit u_{s} on Σ_{s} given by

u_{s}(t) = ( ψ_{s} ∘ u_{0} ) (\frac{t}{σ(s)}) , (25)

where

σ(s) = e^{s} \frac{h′(0)}{h′(s)} , |s| < δ . (26)

The minimal period of u_{s} is given by

τ(s) = σ(s) τ_{0} . (27)

Furthermore, there exists T_{0} > 0 depending on u_{0} such that for every T > T_{0} the equation T = τ(s) has exactly two solutions s^{+}< s^{−} in (−δ, δ). These solutions s^{±} depend smoothly on T and satisfy s^{+}→ −δ, s^{−}→ +δ as T → ∞. Indeed, one has T_{0} = σ(s_{0})τ_{0} where s_{0} is uniquely determined by σ′(s_{0}) = 0.

If u_{0} is any periodic orbit of X_{H} on Σ_{0}, the collection { u_{s} | |s| < δ } constitutes a tower of periodic Hamiltonian orbits over u_{0}, where u_{s} has the minimal period τ(s). Each of these towers corresponds to a closed characteristic on Σ, while distinct towers represent distinct closed characteristics. We have presupposed that Σ carries only finitely many distinct principal closed characteristics which, subsequently, are represented by the periodic integral curves u_{0}^{1}, …, u_{0}^{ℓ} of X_{H} on Σ_{0}.

Considering the i-th principal tower, we denote by τ_{0,i} > 0 the minimal period of u_{0}^{i}. Let τ_{i}(s) := σ(s)τ_{0,i} be the minimal period of the corresponding orbit u_{s}^{i} on Σ_{s}. From Lemma 1 we infer that for every T > τ_{i}(s_{0}) there are exactly two periodic orbits u_{s_{i}^{+}}^{i} and u_{s_{i}^{−}}^{i} of minimal period T , where s_{i}^{+}, s_{i}^{−} are uniquely determined by the condition T = τ_{i}(s_{i}^{±}). In order to fulfill (ii) of the proposition it has to be shown that for sufficiently large T we can distinguish between u_{s_{i}^{+}}^{i} and u_{s_{i}^{−}}^{i} by their respective values of the Hamiltonian action integral A_{H}(u_{s_{i}^{±}}^{i}).

A periodic Hamiltonian orbit u_{s} on Σ_{s}, having the minimal period T = τ(s), can be seen as an embedding u_{s} : 𝕊_{T}^{1} → Σ_{s} ⊂ U. Note that for an arbitrary smooth loop u : 𝕊_{T}^{1} → U we have

u^{∗}α = u^{∗}(X_{α} ⌟ ω) = ω(X_{α}(u), u̇) dt . (28)

For the T -periodic Hamiltonian orbit u_{s} we have u̇_{s} = X_{H}(u_{s}), and consequently, in view of (22),

u_{s}^{∗}α = ω(X_{α}(u_{s}), X_{H}(u_{s})) dt = α(X_{H})|_{u_{s}(t)} dt = h′(s) dt . (29)

Recalling that α−λ (with λ the Liouville form) is exact, we obtain

∫_{𝕊_{T}^{1}} u_{s}^{∗}λ = ∫_{𝕊_{T}^{1}} u_{s}^{∗}α = T h′(s) . (30)

Therefore, the Hamiltonian action of u_{s} : 𝕊_{T}^{1} → Σ_{s} is given by

A_{H}(u_{s}) = ∫_{𝕊_{T}^{1}} u_{s}^{∗}λ − ∫_{0}^{T}(H ∘u_{s}) dt = T (h′(s) − h(s)) . (31)

In the notation of Lemma 1, we have either s = s^{+} or s = s^{−}, and we have s^{+} → −δ, s^{−} → δ as T → ∞. Then from (31) we conclude (cf. [10]):

Lemma 2 For each periodic orbit u_{0} of X_{H} on Σ there exists T_{1} >0, depending on u_{0}, such that for every T > T_{1} the Hamiltonian action of the associated T -periodic orbits u_{s^{+}}, u_{s^{−}} satisfies

A_{H}(u_{s^{+}}) > 0 and A_{H}(u_{s^{−}}) < 0 . (32)

Applying this result to the finitely many principal periodic Hamiltonian orbits u_{0}^{1}, …, u_{0}^{ℓ} on Σ, we observe that the T_{1} > 0 can be chosen so large that the statement of Lemma 2 holds for all u_{0}^{i} simultaneously. Hence we obtain

Lemma 3 Let u_{0}^{1}, …, u_{0}^{ℓ} be the principal periodic Hamiltonian orbits on Σ_{0}. Then there exists T_{1} > 0 such that for every T ⩾ T_{1} the periodic orbits u_{s_{i}^{+}}^{i} and u_{s_{i}^{−}}^{i} of minimal period T satisfy

A_{H}(u_{s_{i}^{+}}^{i}) > 0 and A_{H}(u_{s_{i}^{−}}^{i}) < 0 , (33)

for i=1, …, ℓ.

To summarize the above arguments, we note that the assumption of at most finitely many principal closed characteristics on Σ implies that a principal periodic Hamiltonian orbit on Σ_{0} is of the form u_{0}^{i}. If T ⩾ T_{1} as in Lemma 3, the associated orbit u_{s_{i}^{+}}^{i} is the unique orbit satisfying the conditions of (ii) of the proposition, and the proof is complete. □

We end this section with a few remarks on the case of non-degenerate closed characteristics on Σ. Using the denotation from [10], the Hamiltonian function H : U → ℝ introduced in (20) is admissible for the pair (Σ, X_{α}) in the sense that it is of the form H(ψ_{s}(ξ)) = h(s) with h′(s) > 0 if |s| < δ. Non-degeneracy mod 𝕊^{1} relative to Σ of periodic X_{H}-orbits is then understood with respect to the flow of the characteristic vector field X_{H}|_{Σ}. If the principal periodic X_{H}-orbit u_{0}^{i} is nondegenerate mod 𝕊^{1} relative to Σ_{0} then, for |s| < δ, the corresponding orbit u_{s}^{i} is non-degenerate mod 𝕊^{1} relative to Σ_{s}. The periodic X_{H}-orbit u_{s}^{i} is non-degenerate mod 𝕊^{1} relative to U if, in addition, h′(s) ≠ h′′(s), cf. [10, Lemma 4].

5. Hamiltonian functions defined by Riemannian structures As indicated in section 3, we shall extend the Hamiltonian H : U → ℝ from Proposition 1 by the use of a metric on T^{∗}P ⟶^{π^{∗}} P. By this process we generate principal periodic Hamiltonian orbits outside U which we call technical periodic orbits. In this section we study their properties which are important for the actual choice of the extension of H described in the next paragraph.

If g is any Riemannian metric on T P ⟶^{π} P then we denote by i_{g} : T P → T^{∗}P the induced isomorphism i_{g}(ξ_{∗}) = g(ξ_{∗}, ·). The corresponding tangent map is again an isomorphism T i_{g} : T T P → T T^{∗}P.

Let K : T T P → T P be the Levi-Cività connection associated to g. There is an associated Riemannian connection K^{∗} : T T^{∗}P → T^{∗}P such that the following diagram commutes:

T T P ⟶^{K} T P
↓^{T i_{g}}  ↓^{i_{g}}
T T^{∗}P ⟶^{K^{∗}} T^{∗}P

The connection K^{∗} defines a decomposition T T^{∗}P = T_{h}T^{∗}P ⊕ T_{v}T^{∗}P into the horizontal and the vertical subbundle. By K^{∗}|_{T_{v}T^{∗}P} : T_{v}T^{∗}P → T^{∗}P the vertical subbundle can be identified with T^{∗}P. Similarly, by T π^{∗}|_{T_{h}T^{∗}P} : T_{h}T^{∗}P → T P the horizontal subbundle is isomorphic to T P.

Let g^{∗} be the scalar product on T^{∗}P dual to g, i.e., g^{∗}(i_{g}(ξ_{∗}), i_{g}(η_{∗})) = g(ξ_{∗}, η_{∗}). Writing now X ∈ T T^{∗}P as (X_{h}, X_{v}) ∈ T P ⊕T^{∗}P ≅ T_{h}T^{∗}P ⊕T_{v}T^{∗}P, the symplectic form ω can be expressed by

ω(X, Y) = X_{v}(Y_{h}) − Y_{v}(X_{h}) = g^{∗}(X_{v}, i_{g}(Y_{h})) − g^{∗}(Y_{v}, i_{g}(X_{h})) . (34)

This follows readily from a representation ω = ∑ dy_{i} ∧ dx_{i} in symplectic local coordinates (see also [11, Prop. 3.1.3]). Furthermore, applying the Liouville form λ to X ∈ T_{ξ}T^{∗}P, we find λ(X) = ξ(T π^{∗}X) = ξ(X_{h}) . (35)

We observe that the identification T_{h}T^{∗}P ⊕ T_{v}T^{∗}P ≅ T P ⊕ T^{∗}P gives rise to an inner product on the fibres of T T^{∗}P by setting

⟨X, Y ⟩ := g(X_{h}, Y_{h}) + g^{∗}(X_{v}, Y_{v}) (36)

so that T_{h}T^{∗}P and T_{v}T^{∗}P are orthogonal with respect to ⟨·, ·⟩. An adapted complex operator J(ξ) on T_{ξh}T^{∗}P ⊕ T_{ξv}T^{∗}P ≅ T_{π^{∗}(ξ)}P ⊕ T_{π^{∗}(ξ)}^{∗}P is given by

J(ξ)(X_{h}, X_{v}) := (i_{g}^{−1}(X_{v}), −i_{g}(X_{h})) . (37)

The identity ω(X, JY) = ⟨X, Y ⟩ is verified by direct computation.

Lemma 4 Let ϱ > 0 and let Σ^{∗}(ϱ) := { ξ ∈ T^{∗}P | g^{∗}(ξ, ξ) = ϱ^{2} } denote the cosphere bundle over P of radius ϱ. Then Σ^{∗}(ϱ) ⊂ T^{∗}P is a strict Liouville hypersurface satisfying the assumptions of Theorem 1. In particular, the symplectic dilation X_{λ} is normal to Σ^{∗}(ϱ) with respect to ⟨ · , · ⟩.

Proof Because P is compact and Σ^{∗}(ϱ) bounds the compact set { ξ | |ξ|_{g^{∗}} ⩽ ϱ }, it remains to be shown that the symplectic dilation X_{λ} associated to the Liouville form is transverse to Σ^{∗}(ϱ). For ξ ∈ T^{∗}P and Y ∈ T_{ξ}T^{∗}P we infer from

ξ(Y_{h}) = λ(Y) = (X_{λ} ⌟ ω)(Y) = g^{∗}(X_{λv}, i_{g}(Y_{h})) − g^{∗}(Y_{v}, i_{g}(X_{λh})) (38)

the representation X_{λ}(ξ) = (0, ξ) ∈ T_{ξh}T^{∗}P ⊕ T_{ξv}T^{∗}P . (39)

On the other hand, with G(ξ) := \frac{1}{2} g^{∗}(ξ, ξ), we have dG(Y)|_{ξ} = ⟨ grad G, Y ⟩_{ξ} = g^{∗}(ξ, Y_{v}) for Y ∈ T_{ξ}T^{∗}P, so that ⟨X_{λ}, Y ⟩_{ξ} = ⟨ grad G, Y ⟩_{ξ}.

Given ξ ∈ Σ^{∗}(ϱ), a tangent vector Y ∈ T_{ξ}T^{∗}P is contained in T_{ξ}Σ^{∗}(ϱ) if and only if dG(Y)|_{ξ} = 0. This means that X_{λ}(ξ) is normal to Σ^{∗}(ϱ) at ξ with respect to ⟨ · , · ⟩, and the proof is complete. □

The preceeding observations hold true for any metric g on T P ⟶^{π} P. The metric we choose to extend the Hamiltonian in the fibres of T^{∗}P is, however, required to be special because we shall need an explicit knowledge of the principal closed characteristics on a cosphere bundle Σ^{∗}(ϱ).

For this reason let us consider a connection κ on the principal 𝕊^{1}-bundle P. Identifying the Lie algebra T_{0}𝕊_{τ}^{1} of 𝕊_{τ}^{1} with ℝ, we denote the canonical basis vector by \frac{d}{dt}|_{t=0}. Recall that a connection κ on P is given by a 1-form κ(x) : T_{x}P → T_{0}𝕊_{τ}^{1} such that

(i) κ(x) ċ_{0}(x) = \frac{d}{dt}|_{t=0} where ċ_{t}(x) := \frac{d}{dt} c_{t}(x).

(ii) κ(c_{t}(x)) ∘ T c_{t} = κ(x).

The connection κ provides a decomposition T P = T_{h}P ⊕ T_{v}P into the horizontal subbundle T_{h}P = ker κ and the vertical subbundle T_{v}P = ker T p. Hence we may identify T_{h}P ⟶^{T p} T (P/𝕊^{1}) and T_{v}P ⟶^{κ} T 𝕊_{τ}^{1} (see e.g. [3, Ch. 16]).

Let g_{0} denote the flat metric on 𝕊_{τ}^{1} = ℝ/τℤ inherited from the standard metric on ℝ, which is normalized by setting

g_{0}(\frac{d}{dt}|_{t=0}, \frac{d}{dt}|_{t=0}) = 1/τ^{2} for \frac{d}{dt}|_{t=0} ∈ T_{0}𝕊_{τ}^{1} . (40)

Moreover, let g_{1} be a fixed Riemannian metric on P/𝕊^{1}. Then we define the Riemannian metric g on P by setting

g(ξ_{∗}, η_{∗}) := g_{1}(ξ_{_{∗}h}, η_{_{∗}v}) + g_{0}(ξ_{_{∗}v}, η_{_{∗}v}) = g_{1}(T p ξ_{∗}, T p η_{∗}) + g_{0}(κ ξ_{∗}, κ η_{∗}) (41)

where ξ_{∗} = (ξ_{_{∗}h}, ξ_{_{∗}v}) ∈ T_{h}P ⊕ T_{v}P ≅ T (P/𝕊^{1}) ⊕ T 𝕊^{1}. We note that |ċ_{0}(x)|_{g} = 1/τ by the above normalization.

If ξ_{∗} ∈ T_{x}P then the vertical component has the form ξ_{_{∗}v} = r ċ_{0}(x) for some r ∈ ℝ, and we obtain

T c_{t} ξ_{∗} = T c_{t} (ξ_{_{∗}h}, r ċ_{0}(x)) = (ξ_{_{∗}h}, r ċ_{t}(x)) ∈ T_{t·x}P . (42)

Consequently, it follows readily that T c_{t} ξ_{∗} is parallel along c_{t}(x) with respect to the Levi-Cività connection associated to g.

Lemma 5 Let g be the metric from (41). Then the corresponding dual metric g^{∗} is 𝕊^{1}-invariant, i.e., g^{∗}(γ_{t} ξ, γ_{t} η) = g^{∗}(ξ, η).

Proof We note that p ∘ c_{t} = p implies T p ∘ T c_{t} = T p. Hence, together with κ ∘ T c_{t} = κ, it follows from (41) that g(T c_{t} ξ_{∗}, T c_{t} η_{∗}) = g(ξ_{∗}, η_{∗}). By the definition of γ_{t} we have

γ_{t} ξ = g(i_{g}^{−1}(ξ), T c_{−t} ·) = g(T c_{−t}(T c_{t} i_{g}^{−1}(ξ)), T c_{−t} ·) = g(T c_{t} i_{g}^{−1}(ξ), ·) , (43)

so that γ_{t} = i_{g} ∘ T c_{t} ∘ i_{g}^{−1}. Consequently, we obtain

g^{∗}(γ_{t} ξ, γ_{t} η) = g(T c_{t} i_{g}^{−1}(ξ), T c_{t} i_{g}^{−1}(η)) = g(i_{g}^{−1}(ξ), i_{g}^{−1}(η)) = g^{∗}(ξ, η) . (44)

□

As a consequence, the corresponding cosphere bundle Σ^{∗}(ϱ) is invariant under the 𝕊^{1}-action γ_{t}. Moreover, the principal closed characteristic on Σ^{∗}(ϱ) can be described as follows:

Lemma 6 Let g be the metric from (41) and let g^{∗} be the corresponding dual metric. Given ϱ > 0 and a parametrization of the circle by 𝕊_{τ}^{1} = ℝ/τℤ, we have:

(a) Every 𝕊^{1}-orbit γ_{t} χ_{x} with χ_{x} := i_{g}(ϱ τ ċ_{0}(x)) is a closed characteristic on Σ^{∗}(ϱ).

(b) Consider the Hamiltonian function G(ξ) = \frac{1}{2} g^{∗}(ξ, ξ) on T^{∗}P. Every principal periodic orbit u(t) of X_{G} on Σ^{∗}(ϱ) is of the form u(t) = γ_{ϱτt} χ_{x} and has the minimal period 1/ϱ.

Proof Suppose that ξ ∈ Σ^{∗}(ϱ). Then v(t) = γ_{t} ξ represents a closed characteristic on Σ^{∗}(ϱ) if v̇(t) = μ(t) X_{G}(v(t)) with a scalar function μ(t) > 0.

Let us introduce the co-geodesic spray S^{∗} : T^{∗}P → T T^{∗}P which is defined by S^{∗}(ξ) = (i_{g}^{−1}(ξ), 0) ∈ T P ⊕ T^{∗}P ≅ T_{h}T^{∗}P ⊕ T_{v}T^{∗}P. Then one has S^{∗}(ξ) = X_{G}(ξ), cf. [11, Ch. 3.1]. Hence, if ξ ∈ T_{x}^{∗}P, we may write

X_{G}(γ_{t} ξ) = S^{∗}(γ_{t} ξ) = (i_{g}^{−1}(γ_{t} ξ), 0) ∈ T_{t·x}P ⊕ T_{t·x}^{∗}P . (45)

Associated to ξ ∈ T_{x}^{∗}P there is a section 𝕊^{1} · x → T^{∗}P, defined by t · x ↦ γ_{t} ξ, whose tangent map T γ_{t} ξ maps ċ_{t}(x) to γ̇_{t} ξ := \frac{d}{dt} γ_{t} ξ. Denoting by ∇ the covariant derivative associated to the Levi-Cività connection K, we obtain

\begin{aligned}K^{∗}(γ̇_{t} ξ) & = & (K^{∗}∘ T γ_{t} ξ)(ċ_{t}(x)) \\ & = & (i_{g} ∘ K ∘ T i_{g}^{−1} ∘ T γ_{t} ξ)(ċ_{t}(x)) \\ & = & (i_{g} ∘ K ∘ T (i_{g}^{−1}(γ_{t} ξ)))(ċ_{t}(x)) \\ & = & i_{g}(∇_{ċ_{t}(x)}i_{g}^{−1}(γ_{t} ξ)) \\ & = & i_{g}(∇_{ċ_{t}(x)}(T c_{t} i_{g}^{−1}(ξ))) . \end{aligned} (46)

Recalling that T c_{t} is the parallel displacement, we get ∇_{ċ_{t}(x)}(T c_{t} i_{g}^{−1}(ξ)) = 0 and, therefore, γ̇_{t} ξ ∈ ker K^{∗}, i.e., γ̇_{t} ξ ∈ T_{h}T^{∗}P ⊕ {0}. From π^{∗}(γ_{t} ξ) = c_{t}(x) we finally obtain T π^{∗}(γ̇_{t} ξ) = ċ_{t}(x), implying the representation

γ̇_{t} ξ = (ċ_{t}(x), 0) ∈ T_{t·x}P ⊕ T_{t·x}^{∗}P . (47)

Using (45) and (47), we deduce that the equation γ̇_{t} ξ = μ(t) X_{G}(γ_{t} ξ) is equivalent to μ(t) γ_{t} ξ = i_{g}(ċ_{t}(x)). Because |γ_{t} ξ|_{g^{∗}} = ϱ and |ċ_{t}(x)|_{g} = 1/τ it follows that μ(t) ≡ 1/ϱτ. Hence v(t) = γ_{t} ξ describes a periodic X_{G}-orbit on Σ^{∗}(ϱ) if and only if ξ = i_{g}(ϱτ ċ_{0}(x)) = χ_{x}. The assertion (b) then follows by the reparametrization u(t) := v(ϱτt), and the proof is complete. □

Finally let us point out the following fact: if g is the metric from (41) and g^{∗} its dual metric, then the corresponding scalar product ⟨ · , · ⟩ is 𝕊^{1}-invariant, ⟨T γ_{t} X, T γ_{t} Y ⟩ = ⟨X, Y ⟩. We recall that the map ξ ↦ γ_{t} ξ is a point transformation so that γ_{t}^{∗}ω = ω. Hence it follows that J(γ_{t} ξ) ∘ T γ_{t} = T γ_{t} ∘ J(ξ) for all ξ ∈ T^{∗}P and t ∈ 𝕊^{1}.

6. Global extension of the Hamiltonian From now on let g always be the 𝕊^{1}-invariant metric defined in (41), with the corresponding dual metric g^{∗}. Then there exists ϱ_{1} > 0 such that

\overline{U} ⊂ int \{ ξ ∈ T^{∗}P | |ξ|_{g^{∗}} ⩽ ϱ_{1} \} . (48)

Furthermore, we can choose a > 0 small and ϱ_{2} > 0 large such that

\frac{a ϱ_{1}^{2}}{2} < h(ŝ) < \frac{a ϱ_{2}^{2}}{2} , (49)

where h(ŝ) is the maximal value of the Hamiltonian H on U, see (19).

We shall denote by φ_{s} the flow of the symplectic dilation X_{λ}. From the representation X_{λ}(ξ) = (0, ξ) ∈ T_{ξh}T^{∗}P ⊕ T_{ξv}T^{∗}P it follows that

φ_{s}(ξ) = e^{s}ξ for s ∈ ℝ, ξ ∈ T^{∗}P . (50)

Using the flow φ_{s}, the complement of the 0-section in T^{∗}P can be represented by a parametrized family of hypersurfaces modelled over a fixed cosphere bundle, say Σ^{∗}(ϱ_{2}). It is convenient to label the cosphere bundles by the flow parameter s rather than the radius ϱ. Hence we introduce the notation

Σ_{s}^{∗} := φ_{s}(Σ^{∗}(ϱ_{2})) = Σ^{∗}(ϱ_{2} e^{s}) , s ∈ ℝ . (51)

Now we set s_{1} := ln(ϱ_{1}/ϱ_{2}) < 0 and we choose a smooth function h_{∞} : ℝ → ℝ according to the following conditions:

\{ \begin{matrix}h_{∞}(s) ≡ h(ŝ) & if & s ⩽ s_{1} \\ h_{∞}(s) = \frac{a ϱ_{2}^{2}}{2} e^{2s} & if & s ⩾ 0 \\ 0 < h′_{∞}(s) < h′′_{∞}(s) & if & s > s_{1} . \end{matrix} (52)

Recall that Σ = ∂D with D a relatively compact neighborhood of the 0-section 0_{T^{∗}P}, and the neighborhood U of Σ was defined such that 0_{T^{∗}P} ⊂ D \ U. The Hamiltonian H|_{U} : U → ℝ defined in (20) is now extended to a smooth Hamiltonian H : T^{∗}P → ℝ by setting

(53) H(ξ) := \{ \begin{matrix}0 & if & ξ ∈ D\U \\ H|_{U}(ξ) & if & ξ ∈ U \\ h_{∞}(s) & if & ξ ∈ T^{∗}P \(D ∪ U) with e^{−s}ξ ∈ Σ_{0}^{∗} . \end{matrix}

In particular, we note that H ≡ 0 in a neighborhood of 0_{T^{∗}P}, and

g^{∗}(ξ, ξ) = ϱ_{2}^{2} e^{2s} for e^{−s}ξ ∈ Σ_{0}^{∗} , s ∈ ℝ , (54)

so that H(ξ) = \frac{a}{2} g^{∗}(ξ, ξ) = \frac{a}{2} |ξ|_{g^{∗}}^{2} if |ξ|_{g^{∗}} ⩾ ϱ_{2}.

Lemma 7 The principal periodic orbits of X_{H} on Σ_{0}^{∗} are given by

v_{0}(t; x) := γ_{aϱ_{2}τt}χ_{x} for x ∈ P , (55)

where χ_{x} = i_{g}(ϱ_{2}τ ċ_{0}(x)) ∈ Σ_{0}^{∗}, and where the circle is parametrized by 𝕊_{τ}^{1} = ℝ/τℤ. Each orbit v_{0}(t; x) has the minimal period 1/aϱ_{2}.

Proof We point out that X_{H}(ξ) = a X_{G}(ξ) if ξ ∈ Σ_{0}^{∗}, where G is the Hamiltonian from Lemma 6. The assertion thus is an immediate consequence of Lemma 6. □

The flow φ_{s} provides a transfer of the orbits v_{0}(t; x) from Σ_{0}^{∗} to Σ_{s}^{∗}. The transferred orbits can be parametrized as integral curves of X_{H} by a procedure analogous to Lemma 1 in section 4:

Lemma 8 Let v_{0}(t) stand for any of the principal periodic orbits v_{0}(t; x) of X_{H} on Σ_{0}^{∗}, having the minimal period τ_{0}^{∗} := 1/aϱ_{2}. If s > s_{1} then an associated principal periodic orbit v_{s} of X_{H} on Σ_{s}^{∗} is defined by

v_{s}(t) := ( φ_{s} ∘ v_{0} ) (\frac{t}{σ^{∗}(s)}) , (56)

where

σ^{∗}(s) := e^{s} \frac{h′_{∞}(0)}{h′_{∞}(s)} , s > s_{1} . (57)

The orbit v_{s} has the minimal period

τ^{∗}(s) := σ^{∗}(s) τ_{0}^{∗} . (58)

For every T > 0 the equation T = τ^{∗}(s) has exactly one solution s(T ) ∈ (s_{1}, +∞) which depends smoothly on T and satisfies s(T ) → s_{1} as T →∞.

Proof It follows immediately from our definitions that (φ_{s} ∘ v_{0})(t) is a principal closed characteristic on Σ_{s}^{∗}. Using that φ_{s}^{∗}ω = e^{s}ω, we readily obtain for s > s_{1}

\frac{d}{dt} (φ_{s}∘v_{0})(t) = e^{s} J((φ_{s}∘v_{0})(t)) grad(H ∘φ_{−s})((φ_{s}∘v_{0})(t)) , (59)

where J(ξ) ∈ Hom(T_{ξ}T^{∗}P) is the complex operator according to (37). Furthermore, let ⟨ · , · ⟩ = ω( · , J ·) denote the associated scalar product on T_{ξ}T^{∗}P, having the corresponding norm ∥ · ∥.

For ξ ∈ Σ_{0}^{∗} and s ∈ ℝ let N^{∗}(φ_{s}(ξ)) := X_{λ}(φ_{s}(ξ))/∥X_{λ}(φ_{s}(ξ))∥ denote the outward unit normal vector on Σ_{s}^{∗} at φ_{s}(ξ). If, in addition, s > s_{1} then we can represent the gradient of H at φ_{s}(ξ) by

grad H(φ_{s}(ξ)) = \frac{h′_{∞}(s)}{⟨ N^{∗}(φ_{s}(ξ)) , X_{λ}(φ_{s}(ξ)) ⟩} N^{∗}(φ_{s}(ξ)) . (60)

This expression is derived analogously to (24) in the proof of Proposition 1. Using a similar representation for the gradient of H ∘φ_{−s}, it is readily seen that

(61) grad(H ∘φ_{−s})(φ_{s}(ξ)) = \frac{h′_{∞}(0)}{h′_{∞}(s)} grad H(φ_{s}(ξ)) .

Combined with (59) this yields v̇_{s}(t) = X_{H}(v_{s}(t)).

Evidently, τ^{∗}(s) is the minimal period of v_{s} provided that τ_{0}^{∗} is the minimal period of v_{0}. The condition h′_{∞}(s) < h′′_{∞}(s) for s > s_{1} in (52) implies that τ^{∗}′(s) < 0. Hence τ^{∗}(s) is strictly decreasing. Moreover, we have τ^{∗}(s) → +∞ as s → s_{1} because of h′_{∞}(s) → 0, and τ^{∗}(s) → 0 as s → +∞ because of σ^{∗}(s) = e^{−s} for s > 0. Consequently, τ^{∗} : (s_{1}, +∞) → (0, +∞) is invertible, and the lemma follows. □

In the next section it will turn out to be a crucial observation that the Hamiltonian action of the orbit v_{s(T )} is negative if the minimal period T is sufficiently large:

Lemma 9 Given T > 0, let s = s(T ) and v_{s} be defined according to Lemma 8. The Hamiltonian action of v_{s} is then given by

A_{H}(v_{s}) = T (h′_{∞}(s) − h_{∞}(s)) . (62)

In particular, there exists T_{2} > 0 such that A_{H}(v_{s(T )}) < 0 for all T ⩾ T_{2}.

Proof In respect of v̇_{s} = X_{H}(v_{s}) we have

v_{s}^{∗}λ = ω(X_{λ}, X_{H})|_{v_{s}(t)} dt = λ(X_{H})|_{v_{s}(t)} dt = h′_{∞}(s) dt . (63)

Since H(v_{s}(t)) = h_{∞}(s) we conclude

A_{H}(v_{s}) = ∫_{𝕊_{T}^{1}} v_{s}^{∗}λ − ∫_{0}^{T} H(v_{s}(t)) dt = T (h′_{∞}(s) − h_{∞}(s)) . (64)

Furthermore, we have s(T ) → s_{1} as T → +∞, which implies h′_{∞}(s(T )) → 0 and h_{∞}(s(T )) → h(ŝ) > 0. Hence there exists T_{2} > 0 such that h′_{∞}(s(T )) < h_{∞}(s(T )) if T ⩾ T_{2}, so that A_{H}(v_{s(T )}) < 0. □

Let us summarize the aspects of the Hamiltonian description which are meaningful for the subsequent variational setup:

Proposition 2 Let Σ ⊂ T^{∗}P be a Liouville hypersurface satisfying the hypotheses of Theorem 1 and carrying at most finitely many principal closed characteristics. Let the Hamiltonian H : T^{∗}P → ℝ be defined by (53), and let T_{1}, T_{2} > 0 be chosen in accordance with Lemma 3 and Lemma 9, respectively. If T ⩾ max{T_{1}, T_{2}} then the set of principal periodic X_{H}-orbits of minimal period T is described by

{ u_{s_{i}^{±}}^{i}(t) | i = 1, …, ℓ } ∪ { v_{s}(t; x) | s=s(T ), x∈P } . (65)

Furthermore, if u is any principal periodic X_{H}-orbit of minimal period T with A_{H}(u) > 0, then there exist i ∈ {1, …, ℓ} and ϑ ∈ 𝕊_{T}^{1} such that u(t) = u_{s_{i}^{+}}^{i}(t + ϑ).

7. The variational formulation and the associated Morse complex Let T > 0 be fixed such that Proposition 2 holds true. Correspondingly we fix the parametrization of the circle by 𝕊_{T}^{1} = ℝ/T ℤ. To every ξ ∈ T^{∗}P we assign the principal loop u_{ξ} : 𝕊_{T}^{1} → T^{∗}P, defined by u_{ξ}(t) := γ_{t} ξ, which has the minimal period T . The principal loop u_{ξ}(t) is a principal periodic X_{H}-orbit if it solves the equation u̇_{ξ}(t) = X_{H}(u_{ξ}(t)) , ξ ∈ T^{∗}P . (66)

We have to derive lower bounds for the number of geometrically distinct solutions of (66) which, moreover, have a positive Hamiltonian action. Recall that the Hamiltonian action of u_{ξ} is given by

A_{H}(u_{ξ}) = ∫_{𝕊_{T}^{1}} u_{ξ}^{∗}λ − ∫_{0}^{T} H(u_{ξ}(t)) dt . (67)

Let Ω_{prin} := { u_{ξ} | ξ ∈ T^{∗}P } denote the manifold of T -periodic principal loops, which can be seen as a finite dimensional submanifold of C^{∞}(𝕊_{T}^{1} , T^{∗}P). Indeed, by ξ ↦ u_{ξ} we have a diffeomorphism T^{∗}P → Ω_{prin}. Considering A_{H} as a function Ω_{prin} → ℝ, then the critical points of A_{H} on Ω_{prin} are precisely the principal periodic Hamiltonian orbits from (66).

Instead of formulating the variational set up for A_{H} on Ω_{prin} we study the corresponding problem on T^{∗}P:

Lemma 10 (a) The Hamiltonian action integral A_{H} : Ω_{prin} → ℝ gives rise to a smooth 𝕊^{1}-invariant function F : T^{∗}P → ℝ by setting

F(ξ) := \frac{1}{T} A_{H}(u_{ξ}) . (68)

(b) The gradient of F is given by

(69) grad F(ξ) = −J(ξ) γ̇_{0} ξ − grad H(ξ) .

In particular, grad F(ξ) = 0 if and only if u_{ξ}(t) satisfies the equation (66).

(c) Let ˜∇ denote the Levi-Cività covariant derivative associated to the metric ⟨ · , · ⟩ on T^{∗}P so that the Hessean form of F is represented by

D^{2}F(ξ)(X, Y) = ⟨ ˜∇_{X} grad F(ξ) , Y ⟩ for X, Y ∈ T_{ξ}T^{∗}P . (70)

Then { γ_{t} ξ | t ∈ 𝕊_{T}^{1} } is a non-degenerate critical manifold of F if and only if u_{ξ}(t) is non-degenerate mod 𝕊^{1} as a periodic orbit of X_{H}, i.e., u_{ξ}(t) has only one simple Floquet multiplier equal to 1.

Proof (a) Let ξ ∈ T^{∗}P with π^{∗}(ξ) = x ∈ P. We observe that

\begin{aligned}u_{ξ}^{∗}λ & = & λ(u̇_{ξ})|_{u_{ξ}(t)} dt = ω(X_{λ}, u̇_{ξ})|_{u_{ξ}(t)} dt \\ & = & g^{∗}(γ_{t} ξ, i_{g}(ċ_{t}(x))) dt = g^{∗}(γ_{t} ξ, γ_{t} i_{g}(ċ_{0}(x))) dt \\ & = & g^{∗}(ξ, i_{g}(ċ_{0}(x))) dt . \end{aligned} (71)

Since, moreover, H(u_{ξ}(t)) = H(γ_{t} ξ) = H(ξ), we deduce

F(ξ) = g^{∗}(ξ, i_{g}(ċ_{0}(x))) − H(ξ) . (72)

From this expression it is seen that F is indeed a smooth function.

(b) To derive a representation of grad F(ξ) let us proceed as follows. Consider an 𝕊^{1}-equivariant vector field X on T^{∗}P and let χ_{s} denote its flow. The corresponding flow χ̂_{s} on Ω_{prin} is given by (χ̂_{s}(u_{ξ}))(t) := χ_{s}(γ_{t} ξ) = γ_{t} χ_{s}(ξ). Then we have F(χ_{s}(ξ)) = \frac{1}{T} A_{H}(χ̂_{s}(u_{ξ})), and we conclude

⟨ grad F(ξ) , X(ξ) ⟩ = \frac{1}{T} \frac{d}{ds}|_{s=0} A_{H}(χ̂_{s}(u_{ξ})) . (73)

The directional derivative on the right hand side is readily determined: in view of Cartan’s formula L_{X}λ = X ⌟ dλ + d(X ⌟ λ) and dλ = ω we obtain

\frac{d}{ds} ∫_{𝕊_{T}^{1}} (χ_{s} ∘ u_{ξ})^{∗}λ = ∫_{𝕊_{T}^{1}} \frac{d}{ds} u_{ξ}^{∗}(χ_{s}^{∗}λ) = ∫_{𝕊_{T}^{1}} u_{ξ}^{∗}(L_{X}λ) = ∫_{𝕊_{T}^{1}} u_{ξ}^{∗}(X ⌟ ω) . (74)

Consequently,

\frac{d}{ds}|_{s=0} A_{H}(χ̂_{s}(u_{ξ})) = ∫_{𝕊_{T}^{1}} u_{ξ}^{∗}(X ⌟ ω) − ∫_{0}^{T} dH(X)|_{γ_{t}ξ} dt . (75)

Decomposing X = (X_{h}, X_{v}) ∈ T_{h}T^{∗}P ⊕T_{v}T^{∗}P ≅ T P ⊕T^{∗}P into the horizontal and the vertical subbundle according to section 5, we find

\begin{aligned}u_{ξ}^{∗}(X ⌟ ω) & = & ω(X, u̇_{ξ})|_{γ_{t}ξ} dt = g^{∗}(X_{v}(γ_{t} ξ), i_{g}(ċ_{t}(x))) dt \\ & = & g^{∗}(γ_{t} X_{v}(ξ), γ_{t} i_{g}(ċ_{0}(x))) dt = g^{∗}(X_{v}(ξ), i_{g}(ċ_{0}(x))) dt \\ & = & ⟨ X(ξ), −J(ξ) γ̇_{0} ξ ⟩ dt . \end{aligned} (76)

Hence it follows that

(77) ⟨ grad F(ξ), X(ξ) ⟩ = ⟨ −J(ξ) γ̇_{0} ξ − grad H(ξ) , X(ξ) ⟩ ,

Let grad F(ξ) = 0. Then we find T γ_{t} grad F(ξ) = −J(γ_{t} ξ) T γ_{t} γ̇_{0} ξ − grad H(γ_{t} ξ) = −J(u_{ξ}(t)) u̇_{ξ}(t) − grad H(u_{ξ}(t)) = 0, so that the critical points of F correspond uniquely to the solutions of (66).

(c) The 𝕊^{1}-orbit 𝕊^{1}·ξ = { γ_{t} ξ | t ∈ 𝕊_{T}^{1} } is a non-degenerate critical manifold of F if ξ is a critical point and the kernel of ˜∇grad F(γ_{t} ξ) is spanned by γ̇_{t} ξ. We can introduce a symplectic orthonormal basis e_{1}, …, e_{2n} in T_{ξ}T^{∗}P by choosing e_{1}, …, e_{n} ∈ T_{ξh}T^{∗}P ⊕ {0} and e_{n+i} := −J(ξ)e_{i} ∈ {0} ⊕ T_{ξv}T^{∗}P. An orthonormal basis at γ_{t} ξ then is given by e_{i}(t) := T γ_{t} e_{i}.

Let a_{i,j} = ⟨ ˜∇_{e_{i}(t)}grad H(γ_{t} ξ), e_{j}(t) ⟩ denote the components of the Hessean matrix A := (a_{i,j}) of H expressed in the above basis. Note that a_{i,j} = a_{j,i} is indeed independent of t in view of the 𝕊^{1}-invariance of H and the choice of the basis.

Using the coordinate representation, the linearized Hamiltonian equation along the T -periodic critical orbit u_{ξ} takes the form \frac{d}{dt} X = J(ξ)A X of an autonomous linear differential equation in T_{ξ}T^{∗}P ≅ ℝ^{2n}. Hence it follows that u_{ξ}(t) is nondegenerate mod 𝕊^{1} if the non trivial T -periodic solutions of this equation are given by X(t) ≡ μ γ̇_{0}ξ for μ ∈ ℝ \ {0}, and the proof is complete. □

Now we are ready to reformulate the statements of Theorems 1 and 2 in terms of critical 𝕊^{1}-orbits of F:

Proposition 3 Suppose that the hypotheses of Proposition 2 are satisfied. Then there exist ℓ ⩾ CL(P/𝕊^{1}) critical points ξ^{1}, …, ξ^{ℓ} of F such that

(i) 𝕊^{1}·ξ^{i} ∩ 𝕊^{1}·ξ^{j} = ∅ if i ≠ j.

(ii) F(ξ^{i}) > 0 for i = 1, …, ℓ.

In case that all the critical 𝕊^{1}-orbits 𝕊^{1}·ξ^{i} satisfying (i) and (ii) are non-degenerate critical manifolds of F then it follows that ℓ ⩾ SB(P/𝕊^{1}).

For the proof of Proposition 3 we have to study the structure of the Morse complex consisting of the critical points of F together with their connecting orbits under the gradient flow. For this purpose we observe

Lemma 11 The function F : T^{∗}P → ℝ is bounded above, and for every c ∈ ℝ the set F_{c} := { ξ ∈ T^{∗}P | F(ξ) ⩾ c } (78)

is compact. As a consequence, the critical set Crit(F) := { ξ | grad F(ξ) = 0 } is compact.

Proof Let ξ ∈ Σ_{0}^{∗} and consider φ_{s}(ξ) = e^{s}ξ for s ⩾ 0. Then, by the definition of H, we have H(φ_{s}(ξ)) = h_{∞}(s), and consequently, with x = π^{∗}(ξ),

F(e^{s} ξ) = e^{s} g^{∗}(ξ, i_{g}(ċ_{0}(x))) − \frac{a ϱ_{2}^{2}}{2} e^{2s} , (79)

Since g^{∗}(ξ, i_{g}(ċ_{0}(x))) ⩽ |ξ|_{g^{∗}} |ċ_{0}(x)|_{g} = \frac{ϱ_{2}}{T}, we infer F(e^{s} ξ) ⩽ \frac{ϱ_{2}}{T} e^{s} − \frac{a ϱ_{2}^{2}}{2} e^{2s} → −∞ as s → +∞, and the first statement of the lemma follows.

In view of Proposition 2 the structure of the critical set of A_{H} on Ω_{prin} is known by the choice of H and T . It follows immediately that the set of critical values of F is bounded so that Crit(F) ⊂ F_{c} for some suitable c. This implies the second statement since F_{c} is compact and Crit(F) is closed. □

Let us now introduce the flow of −grad F, i.e., \frac{d}{dτ} (ξ · τ) = −grad F(ξ · τ). Because the gradient vector field of F is linearly bounded in the fibres of T^{∗}P, its flow is globally defined as a map T^{∗}P × ℝ → T^{∗}P. By the semi-boundedness of F its Morse complex is given by

𝔐 := { ξ ∈ T^{∗}P | F(ξ·τ) > −∞ ∀ τ ∈ ℝ } . (80)

We recall that a Morse decomposition of 𝔐 consists of a finite collection of compact non-empty subsets M_{i} ⊂ 𝔐 which can be ordered, say (M_{1}, …, M_{k}). This ordering is said to be amissible if for every ξ ∈ 𝔐 \ (M_{1} ∪ … ∪ M_{k}) there exist unique subscripts i < j with ξ·τ → M_{i} as τ → +∞ and ξ·τ → M_{j} as τ → −∞.

From Lemma 9 and Proposition 2 it follows that 0 is a regular value of F. Consequently, 𝔐 admits a Morse decomposition (𝔐^{−}, 𝔐^{+}) where

𝔐^{+} := { ξ ∈ 𝔐 | F(ξ·τ) > 0 ∀ τ ∈ ℝ } , (81)

and 𝔐^{−} := { ξ ∈ 𝔐 | F(ξ·τ) < 0 ∀ τ ∈ ℝ } . (82)

We shall refer to 𝔐^{+} and 𝔐^{−} as the positive and the negative Morse complex of F, respectively. Note that 𝔐^{−} ≠ ∅ because by Proposition 2 it contains the technical critical 𝕊^{1}-orbits v_{s}. Moreover, 𝔐^{+} ≠ ∅ since F_{0} ⊃ 0_{T^{∗}P} ≠ ∅ so that in view of Lemma 11 there exists at least one critical 𝕊^{1}-orbit on a positive level.

Let us fix a regular level c < 0 of F such that the Morse complex 𝔐 is contained in F_{c}. Using Lemma 11, we can choose c such that F^{−1}({c}) is a cosphere bundle.

Associated to the Morse decomposition (𝔐^{−}, 𝔐^{+}) there is a filtration of index pairs N_{0} ⊂ N_{1} ⊂ N_{2} for the −grad F-flow in the sense of Conley, cf. [2]. For this purpose we set

N_{0} := F^{−1}(\{c\}) ,  N_{1} := \overline{F_{c}\F_{0}} ,  N_{2} := F_{c} . (83)

The pair (N_{2}, N_{1}) is an index pair for 𝔐^{+}, i.e.,

(a) N := \overline{N_{2}\N_{1}} = F_{0} is an isolating neighborhood for 𝔐^{+} having the exit set ∂N = F^{−1}({0}).

(b) N_{1} is positively invariant relative to N_{2}.

(c) If ξ ∈ N_{2} and ξ ·[ 0, ∞) ⊄ N_{2} then there exists τ ⩾ 0 such that ξ ·[ 0, τ ] ⊂ N_{2} and ξ·τ ∈N_{1}.

Similarly, F_{c} is an isolating neighborhood for the total Morse complex 𝔐, having the exit set F^{−1}({c}) = ∂F_{c}.

It is a crucial observation that the interior of N contains an 𝕊^{1}-invariant subset which carries the 𝕊^{1}-cohomology of the base space P:

Lemma 12 There exists r > 0 such that

Q := { r i_{g}(ċ_{0}(x)) | x ∈ P } ⊂ F_{0} . (84)

Moreover, Q is 𝕊^{1}-equivariantly homotopic to the zero section 0_{T^{∗}P}.

Proof If r > 0 is small enough then we have H|_{Q} = 0 by the definition of the Hamiltonian. From (72) it follows that F|_{Q} = r |ċ_{0}(x)|_{g}^{2} = r/T > 0 so that Q ⊂ F_{0}. Clearly, the map 0_{T^{∗}P} × [ 0, 1 ] → T^{∗}P, defined by (x, μ) ↦ μ r i_{g}(ċ_{0}(x)), deforms 0_{T^{∗}p} equivariantly into Q. □

8. Equivariant index maps and cuplength estimates for 𝔐^{+} If (X, A) is a paracompact 𝕊^{1}-pair then let H∗_{𝕊^{1}}(X, A) denote the 𝕊^{1}-equivariant Alexander-Spanier cohomology with real coefficients, cf. e.g. [6]. Since we deal with a free 𝕊^{1}-action, the equivariant cohomology is isomorphic to the ordinary cohomology of (X/𝕊^{1}, A/𝕊^{1}), i.e., we have H∗_{𝕊^{1}}(X, A) ≅ H^{∗}(X/𝕊^{1}, A/𝕊^{1}). Likewise, we denote by H∗^{𝕊^{1}}(X, A) := H_{∗}(X/𝕊^{1}, A/𝕊^{1}) singular 𝕊^{1}-homology of (X, A) with real coefficients.

Given an 𝕊^{1}-space X, the 𝕊^{1}-equivariant cuplength of X is defined by

𝕊^{1}-cuplength(X) := sup { k ∈ ℕ_{0} | ∃ ω_{i} ∈Hq_{i}_{𝕊^{1}}(X) , q_{i} ⩾1 , i=1, …, k ,
such that ω_{1} ⌣ … ⌣ ω_{k} ≠ 0 } .

(85)

If (X, A) is an 𝕊^{1}-pair, then A is called 𝕊^{1}-categorical in X if A can be deformed equivariantly through X into a single 𝕊^{1}-orbit. The 𝕊^{1}-category of a non empty 𝕊^{1}-invariant subset A⊂X in X is defined by

𝕊^{1}-cat_{X}(A) := inf { k ∈ ℕ | there exist closed 𝕊^{1}-categorical subsets

A_{1}, …, A_{k} ⊂ X with A ⊂ A_{1} ∪ … ∪ A_{k} } , (86)

whereas 𝕊^{1}-cat_{X}(∅) := 0.

The following proposition relates the 𝕊^{1}-homotopy invariants of the base space P to those of the positive Morse complex 𝔐^{+}:

Proposition 4 Let 𝔐^{+} be the positive Morse complex of F introduced in (81), and let F_{c} be chosen as in (83). Then

(87) 𝕊^{1}-cat_{F_{c}}(𝔐^{+}) ⩾ 1 + 𝕊^{1}-cuplength(P) .

Proof Since F_{c} can equivariantly be retracted to 0_{T^{∗}P} and since the zero section can equivariantly be deformed into Q ⊂ F_{0}, there exists an 𝕊^{1}-equivariant strong deformation retraction ḡ : F_{c} × [ 0, 1 ] → F_{c} deforming F_{c} into Q. The retraction map g := ḡ(·, 1) : F_{c} → Q induces an isomorphism g^{∗} : H∗_{𝕊^{1}}(Q) → H∗_{𝕊^{1}}(F_{c}). In particular, H∗_{𝕊^{1}}(Q) ≅ H∗_{𝕊^{1}}(P) implies 𝕊^{1}-cuplength(F_{c}) = 𝕊^{1}-cuplength(P).

Assuming that ℓ := 𝕊^{1}-cat_{F_{c}}(𝔐^{+}) ⩽ 𝕊^{1}-cuplength(F_{c}), we argue by contradiction. We have to distinguish between the two cases ℓ = 0 and ℓ > 0.

In the latter case we can find a non-vanishing cup product ω_{1} ⌣ … ⌣ ω_{ℓ} ≠ 0 where ω_{ν} ∈ Hk_{ν}_{𝕊^{1}}(F_{c}) with k_{ν} ⩾ 1 for ν = 1, …, ℓ. Let ⟨ · , · ⟩ denote the duality pairing between cohomology and homology. Then there exists a homology class a_{0} ∈ Hk^{𝕊^{1}}(F_{c}) with k = k_{1} + … + k_{ℓ}, such that

⟨ ω_{1} ⌣ … ⌣ ω_{ℓ} , a_{0} ⟩ ≠ 0 . (88)

We emphasize that T^{∗}P is a proper free smooth 𝕊^{1}-manifold with the 𝕊^{1}-structure induced from P. Hence the orbit space carries the unique structure of a smooth manifold such that T p : T^{∗}P → (T^{∗}P)/𝕊^{1} is a submersion, cf. [4, Prop. 5.2]. In particular, F_{c}/𝕊^{1} is a compact manifold with boundary ∂ (F_{c}/𝕊^{1}) = (∂F_{c})/𝕊^{1}.

Set d := dim F_{c}/𝕊^{1} and let ζ_{F_{c}} ∈Hd^{𝕊^{1}}(F_{c}, ∂F_{c}) be the corresponding fundamental class. By Lefschetz duality there is an isomorphism given by the cap product,

Hd−k_{𝕊^{1}}(F_{c}, ∂F_{c}) ⟶^{⌢ζ_{Fc}} Hk^{𝕊^{1}}(F_{c}) . (89)

Let ω_{0} ∈ Hd−k_{𝕊^{1}}(F_{c}, ∂F_{c}) be the unique class with a_{0} = ω_{0} ⌢ ζ_{F_{c}}. Consequently, we have

⟨ ω_{1} ⌣ … ⌣ ω_{ℓ} , ω_{0} ⌢ ζ_{F_{c}} ⟩ = ⟨ ω_{1} ⌣ … ⌣ ω_{ℓ} ⌣ ω_{0} , ζ_{F_{c}} ⟩ ≠ 0 . (90)

The latter cup product needs to be related to the 𝕊^{1}-cohomology of the isolating neighborhood N = F_{0} of 𝔐^{+}.

We have a sequence F_{c} →^{g} Q →^{i} F_{0} →^{j} F_{c} with j ∘ i ∘ g ≃ id, where i and j are the inclusions. From the induced sequence in 𝕊^{1}-cohomology,

H∗_{𝕊^{1}}(F_{c}) ⟶^{j^{∗}} H∗_{𝕊^{1}}(F_{0}) ⟶^{(i∘g)^{∗}} H∗_{𝕊^{1}}(F_{c}) , (91)

it follows that (i ∘ g)^{∗} ∘ j^{∗} =id^{∗}. Hence (i ∘ g)^{∗} is epimorphic and j^{∗} is monomorphic. Note that j^{∗}(ω_{1} ⌣ … ⌣ ω_{ℓ}) = j^{∗}(ω_{1}) ⌣ … ⌣ j^{∗}(ω_{ℓ}), i.e., j^{∗} commutes with the cup product. We shall abbreviate ω̂_{ν} := j^{∗}(ω_{ν}) ∈ Hkν_{𝕊^{1}}(F_{0}). Consequently we get

ω̂_{1} ⌣ … ⌣ ω̂_{ℓ} ≠ 0 . (92)

To take the relative class ω_{0} into account, we observe that there exists a homotopy h̄ : F_{c} \Q × [ 0, 1 ] → F_{c} \Q which deforms F_{c} \Q equivariantly to ∂F_{c}. Let this retraction be denoted by h := h̄(·, 1) : F_{c} \Q → ∂F_{c}. Thus we obtain a sequence

∂F_{c} →^{j_{0}} F_{c} \ int F_{0} →^{i_{0}} F_{c} \ Q →^{h} ∂F_{c} with h ∘ i_{0} ∘ j_{0} = id, where i_{0} and j_{0} are

the inclusions. Then j_{0}^{∗} ∘ (h ∘ i_{0})^{∗} = id^{∗} for the induced homomorphisms in 𝕊^{1}- cohomology. Now let us consider the following commutative diagram:

[figure: H∗_{𝕊^{1}}(F_{c}, F_{c}\int F_{0}); H∗_{𝕊^{1}}(F_{c}, ∂F_{c}); H∗_{𝕊^{1}}(F_{c}\int F_{0}, ∂F_{c}); H∗−1_{𝕊^{1}}(∂F_{c}); H∗_{𝕊^{1}}(F_{c}); H∗_{𝕊^{1}}(∂F_{c}); H∗−1_{𝕊^{1}}(F_{c}\int F_{0}); H∗−1_{𝕊^{1}}(∂F_{c}); H∗_{𝕊^{1}}(F_{c}\int F_{0}); H∗_{𝕊^{1}}(∂F_{c}); ĵ^{∗}; î^{∗}; ι_{1}^{∗}; ι_{2}^{∗}; ι_{3}^{∗}; δ^{∗}; (h∘i_{0})^{∗}; (h∘i_{0})^{∗}; j_{0}^{∗}; j_{0}^{∗}; id^{∗}; id^{∗}]

Note that the middle row is given by the exact 𝕊^{1}-cohomology sequence of the triple (F_{c}, F_{c} \ int F_{0}, ∂F_{c}) while the columns are obtained by the exact 𝕊^{1}-cohomology sequences of the pairs (F_{c}, ∂F_{c}) and (F_{c}\int F_{0}, ∂F_{c}), respectively.

We claim that ĵ^{∗} is epimorphic. In view of the exactness of the cohomology sequence this claim follows if î^{∗} = 0. Indeed, since j_{0}^{∗} is epimorphic the exactness of the right column implies δ^{∗} = 0 and, therefore, ι_{3}^{∗} is monomorphic. By the exactness of the left column we deduce

ι_{3}^{∗} ∘ î^{∗} = (h ∘ i_{0})^{∗} ∘ ι_{2}^{∗} ∘ ι_{1}^{∗} = 0 , (93)

so that î^{∗} = 0 in view of the injectivity of ι_{3}^{∗}.

Since ĵ^{∗} is an epimorphism it has a right inverse, i.e., there is a homomorphism ĝ such that for the following sequence in relative 𝕊^{1}-cohomology

H∗_{𝕊^{1}}(F_{c}, ∂F_{c}) ⟶^{ĝ} H∗_{𝕊^{1}}(F_{c}, F_{c}\int F_{0}) ⟶^{ĵ^{∗}} H∗_{𝕊^{1}}(F_{c}, ∂F_{c}) , (94)

we have ĵ^{∗} ∘ĝ = id^{∗}. We note that the cohomology groups are finite-dimensional real vector spaces: for H∗_{𝕊^{1}}(F_{c}, ∂F_{c}) this fact follows because the corresponding singular homology groups are finitely generated (cf. [12, p. 298]). For H∗_{𝕊^{1}}(F_{c}, F_{c}\int F_{0}) it follows similarly, using the fact that by Čech-excision there is the following isomorphism which is induced by inclusion (cf. [5, Ch.VIII, 6.17]):

ι_{F_{0}}^{∗} : H∗_{𝕊^{1}}(F_{c}, F_{c}\int F_{0}) ⟶^{≅} H∗_{𝕊^{1}}(F_{0}, ∂F_{0}) . (95)

Let ω̂_{0} := (ι_{F_{0}}^{∗} ∘ ĝ)(ω_{0}) ∈ Hd−k_{𝕊^{1}}(F_{0}, ∂F_{0}). Thus we have a commutative diagram

Hk_{𝕊^{1}}(F_{c}) ⊗ Hd−k_{𝕊^{1}}(F_{c}, ∂F_{c}) ⟶^{⌣} Hd_{𝕊^{1}}(F_{c}, ∂F_{c})
↓^{j^{∗}}  ↓^{ι_{F_{0}}^{∗}∘ĝ}  ↓^{ι_{F_{0}}^{∗}∘ĝ}
Hk_{𝕊^{1}}(F_{0}) ⊗ Hd−k_{𝕊^{1}}(F_{0}, ∂F_{0}) ⟶^{⌣} Hd_{𝕊^{1}}(F_{0}, ∂F_{0})

where the vertical maps are monomorphisms. Hence, setting k = k_{1} + … +k_{ℓ}, we deduce from (90) and (92) the following cup product:

ω̂_{1} ⌣ … ⌣ ω̂_{ℓ} ⌣ ω̂_{0} ≠ 0 . (96)

If ℓ = 𝕊^{1}-cat_{F_{c}}(𝔐^{+}) > 0 then there exist ℓ closed 𝕊^{1}-categorical subsets A_{ν} ⊂ F_{c} with 𝔐^{+} ⊂ A_{1} ∪ … ∪ A_{ℓ}. Since 𝔐^{+} ⊂ int F_{0} we may assume A_{ν} ⊂ int F_{0} and, in addition, (A_{ν}, A_{ν′}) is an excisive pair for any choice of ν, ν′ ∈ {1, …, ℓ}. By ι_{ν} : A_{ν} → F_{c} and j_{ν} : A_{ν} → F_{0} we denote the inclusions. Since A_{ν} is 𝕊^{1}-categorical in F_{c} the inclusion ι_{ν} is equivariantly homotopic to a map ι_{ν}^{0} : A_{ν} → F_{c} where the image ι_{ν}^{0}(A_{ν}) = 𝕊^{1}·ξ_{0} is a single 𝕊^{1}-orbit. Thus we have an 𝕊^{1}-map ι̂_{ν} : A_{ν} → 𝕊^{1}·ξ_{0} and an inclusion κ : 𝕊^{1} · ξ_{0} → F_{c} with ι_{ν}^{0} = κ ∘ ι̂_{ν}. These maps induce the following commutative diagram in 𝕊^{1}-cohomology:

[figure: H∗_{𝕊^{1}}(F_{c}); H∗_{𝕊^{1}}(𝕊^{1}·ξ_{0}); H∗_{𝕊^{1}}(F_{0}); H∗_{𝕊^{1}}(A_{ν}); κ^{∗}; j^{∗}; ι̂_{ν}^{∗}; j_{ν}^{∗}; ι_{ν}^{∗}]

Then ker ι_{ν}^{∗} = H∗_{𝕊^{1}}(F_{c}) for ∗ ⩾ 1, and therefore im j^{∗} ⊂ ker j_{ν}^{∗}. In particular, we infer ω̂_{ν} = j^{∗}(ω_{ν}) ∈ ker j_{ν}^{∗} ⊂ Hk_{ν}_{𝕊^{1}}(F_{0}). For ∗ ⩾ 1 we consider the exact cohomology sequence of the 𝕊^{1}-pair (F_{0}, A_{ν}),

⟶ H∗−1_{𝕊^{1}}(A_{ν}) ⟶^{δ^{∗}} H∗_{𝕊^{1}}(F_{0}, A_{ν}) ⟶^{i_{ν}^{∗}} H∗_{𝕊^{1}}(F_{0}) ⟶^{j_{ν}^{∗}} H∗_{𝕊^{1}}(A_{ν}) ⟶ (97)

where i_{ν} : (F_{0}, ∅) → (F_{0}, A_{ν}) is the inclusion. In view of the exactness of the sequence, for ν = 1, …, ℓ there exists a class ω̃_{ν} ∈ Hk_{ν}_{𝕊^{1}}(F_{0}, A_{ν}) with i_{ν}^{∗}(ω̃_{ν}) = ω̂_{ν}. Setting A:=A_{1} ∪ … ∪ A_{ℓ}, we thus obtain the following commutative diagram

\begin{matrix}Hk_{1}_{𝕊^{1}}(F_{0}, A_{1}) & ⊗ & ⋯ & ⊗ & Hk_{ℓ}_{𝕊^{1}}(F_{0}, A_{ℓ}) & ⟶^{⌣} & Hk_{𝕊^{1}}(F_{0}, A) \\ ↓^{i_{1}^{∗}} & & & & ↓^{i_{ℓ}^{∗}} & & ↓^{i_{A}^{∗}} \\ Hk_{1}_{𝕊^{1}}(F_{0}) & ⊗ & ⋯ & ⊗ & Hk_{ℓ}_{𝕊^{1}}(F_{0}) & ⟶^{⌣} & Hk_{𝕊^{1}}(F_{0}) \end{matrix} (98)

where i_{A} : (F_{0}, ∅) → (F_{0}, A) is the inclusion. We note that ω̂_{1} ⌣ … ⌣ ω̂_{ℓ} = i_{A}^{∗}(ω̃_{1} ⌣ … ⌣ ω̃_{ℓ}) ≠ 0. To derive the desired contradiction to our assumption ℓ ⩽ 𝕊^{1}-cuplength(Q) we use Wazewski’s principle for equivariant flows:

Lemma 13 The exit set ∂F_{0} is a strong 𝕊^{1}-deformation retract of F_{0} \ 𝔐^{+}.

The proof is analogous to [1, Lemma 3]. Now we choose a closed 𝕊^{1}-invariant subset A_{0} ⊂ F_{0} with ∂F_{0} ⊂ A_{0} ⊂ F_{0} \𝔐^{+} such that \overline{F_{0}\(A_{1} ∪ … ∪ A_{k})} ⊂ int A_{0}. By Lemma 13 it follows that ∂F_{0} is an 𝕊^{1}-retract of A_{0}. Hence the inclusion of 𝕊^{1}-pairs i_{0} : (A_{0}, ∂F_{0})→(F_{0}, ∂F_{0}) induces the trivial homomorphism i_{0}^{∗} =0 in 𝕊^{1}-cohomology. In view of the exactness cohomology sequence of the triple (F_{0}, A_{0}, ∂F_{0}),

⟶^{δ^{∗}} H∗_{𝕊^{1}}(F_{0}, A_{0}) ⟶^{j_{0}^{∗}} H∗_{𝕊^{1}}(F_{0}, ∂F_{0}) ⟶^{i_{0}^{∗}} H∗_{𝕊^{1}}(A_{0}, ∂F_{0}) ⟶^{δ^{∗}} (99)

we infer j_{0}^{∗} is onto. Hence we can find a relative class ω̃_{0} ∈ Hd−k_{𝕊^{1}}(F_{0}, A_{0}) with j_{0}^{∗}(ω̃_{0}) = ω̂_{0} ∈ Hd−k_{𝕊^{1}}(F_{0}, ∂F_{0}). Thus we obtain another commutative diagram

Hk_{𝕊^{1}}(F_{0}, A) ⊗ Hd−k_{𝕊^{1}}(F_{0}, A_{0}) ⟶^{⌣} Hd_{𝕊^{1}}(F_{0}, F_{0})
↓^{i_{A}^{∗}}  ↓^{j_{0}^{∗}}  ↓^{i_{F_{0}}^{∗}}
Hk_{𝕊^{1}}(F_{0}) ⊗ Hd−k_{𝕊^{1}}(F_{0}, ∂F_{0}) ⟶^{⌣} Hd_{𝕊^{1}}(F_{0}, ∂F_{0})

where i_{F_{0}} : (F_{0}, ∂F_{0})→(F_{0}, F_{0}) is the inclusion. Now we are led to

\begin{aligned}ω̂_{1} ⌣…⌣ ω̂_{ℓ} ⌣ ω̂_{0} & = & i_{A}^{∗}(ω̃_{1} ⌣…⌣ ω̃_{ℓ}) ⌣ j_{0}^{∗}(ω̃_{0}) \\ & = & i_{F_{0}}^{∗}(ω̃_{1} ⌣…⌣ ω̃_{ℓ} ⌣ ω̃_{0}) = 0 , \end{aligned} (100)

in contradiction to (96).

It remains to consider the case ℓ = 𝕊^{1}-cat_{F_{0}}(𝔐^{+}) = 0, i.e., we assume 𝔐^{+} = ∅. Again by Lefschetz duality we find ω_{0} ∈ Hd_{𝕊^{1}}(F_{c}, ∂F_{c}) having the property that ⟨ 1 , ω_{0} ⌢ ζ_{F_{c}} ⟩ = ⟨ ω_{0} , ζ_{F_{c}} ⟩ ≠ 0, where 1 ∈ H0_{𝕊^{1}}(F_{c}) is the unit element of the 𝕊^{1}-cohomology ring of F_{c}. Then 0 ≠ ω̂_{0} = (ι_{F_{0}}^{∗} ∘ ĝ)(ω_{0}) ∈ Hd_{𝕊^{1}}(F_{0}, ∂F_{0}). On the other hand, since 𝔐^{+} = ∅ we may choose A_{0} := F_{0}. Similar as above, we conclude

ω̂_{0} = j_{0}^{∗}(ω̃_{0}) = 0 , (101)

which yields a contradiction. Thus the proof of the proposition is complete. □

9. Lyusternik-Schnirelman theory and the proof of Theorem 1 We are now prepared to apply Lyusternik-Schnirelman theory for flows to the positive Morse complex 𝔐^{+}. By our assumptions, 𝔐^{+} contains only a finite number of critical 𝕊^{1}-orbits. Let the corresponding critical values c_{j} be ordered according to 0 < c_{1} < … < c_{k}. For j =1, …, k, we define the Morse sets M_{j} by

M_{j} := { ξ ∈ 𝔐^{+} | grad F(ξ) = 0 and F(ξ) = c_{i} } . (102)

Then (M_{1}, …, M_{k}) is an admissible ordering of the Morse decomposition of 𝔐^{+}. Consequently, we obtain

Proposition 5 Let (M_{1}, …, M_{k}) denote the Morse decomposition of 𝔐^{+} induced by the ordering of the positive critical values c_{1} <…<c_{k} of F. Then

𝕊^{1}-cat_{F_{c}}(𝔐^{+}) ⩽ ∑_{j=1}^{k} 𝕊^{1}-cat_{F_{c}}(M_{j}) . (103)

In particular, the number of the distinct critical 𝕊^{1}-orbits of F in 𝔐^{+} is greater or equal 𝕊^{1}-cat_{F_{c}}(𝔐^{+}).

We omit the proof which can be taken over from [10, Prop. 5].

Combining this result with Proposition 4, we have shown the existence of ℓ ⩾ 1 + 𝕊^{1}-cuplength(P) distinct critical 𝕊^{1}-orbits 𝕊^{1} ·ξ^{1}, …, 𝕊^{1} ·ξ^{ℓ} of F such that F(ξ^{i}) > 0. Thus we have verified the first part of Proposition 3 in section 7, which is the variational reformulation of Theorem 1. The proof of Theorem 1 is therefore complete.

10. Morse theory and the proof in the non-degenerate case Suppose now that all the critical 𝕊^{1}-orbits contained in 𝔐^{+} are non-degenerate critical manifolds of F. In this case their number can be estimated by the sum of the equivariant Betti numbers of P.

Let 𝕊^{1} ·ξ^{1}, …, 𝕊^{1} ·ξ^{ℓ} ∈ T^{∗}P denote the distinct critical orbits in 𝔐^{+} and let c_{i} > 0 be the critical level corresponding to M_{i} := 𝕊^{1}·ξ^{i}. Here we assume that the critical sets M_{i} are indexed according to the ordering c_{1} ⩽ … ⩽ c_{ℓ} of the critical values. Then (M_{1}, …, M_{ℓ}) constitutes an admissibly ordered Morse decomposition of 𝔐^{+}.

In the case of free 𝕊^{1}-spaces the equivariant Conley index h_{𝕊^{1}}(A) of an isolated invariant set A is defined to be the ordinary Conley index of the orbit set A/𝕊^{1}:

h_{𝕊^{1}}(A) := h(A/𝕊^{1}) . (104)

Now consider the Morse equation (cf. [2, Thm. 3.3]),

∑_{i=1}^{ℓ} P(t, h_{𝕊^{1}}(𝕊^{1}·ξ^{i})) = P(t, h_{𝕊^{1}}(𝔐^{+})) + (1+t) Q(t) , (105)

where P(t, ·) denotes the Poincaré polynomial and Q(t) is a polynomial with nonnegative integer coefficients. From (105) we deduce the following result:

Proposition 6 Suppose that the critical 𝕊^{1}-orbits 𝕊^{1} · ξ^{i} in 𝔐^{+} are non-degenerate mod 𝕊^{1} and let (𝕊^{1}·ξ^{1}, …, 𝕊^{1}·ξ^{ℓ}) constitute an admissibly ordered Morse decomposition of 𝔐^{+}. Then we have ℓ ⩾ SB (P/𝕊^{1}) , (106)

where SB(P/𝕊^{1}) denotes the sum of the Betti numbers of the orbit space P/𝕊^{1}.

Proof Consider the orbit space (T^{∗}P)/𝕊^{1} endowed with its unique differentiable structure such that the projection T p : T^{∗}P → (T^{∗}P)/𝕊^{1} is a submersion. Then T p(𝕊^{1} · ξ^{i}) is a non-degenerate critical point of the function F^{∗} : (T^{∗}P)/𝕊^{1} → ℝ which is defined by the identity F = F^{∗} ∘ T p.

The Conley index of T p(𝕊^{1} ·ξ^{i}) is the homotopy type of a pointed sphere of dimension ν_{i}, where ν_{i} is equal to the Morse index of T p(𝕊^{1}·ξ^{i}):

h_{𝕊^{1}}(𝕊^{1}·ξ^{i}) = h(T p(𝕊^{1}·ξ^{i})) = [ 𝕊̇^{ν_{i}}] . (107)

Clearly, ν_{i} is the number of the negative eigenvalues of D^{2}F(ξ) ∈ Hom(T_{ξ}T^{∗}P) at every point ξ ∈ 𝕊^{1}·ξ^{i}. From the Poincaré polynomial P(t, [ 𝕊̇^{ν_{i}}]) = t^{ν_{i}} we get

∑_{i=1}^{ℓ} P(t, h_{𝕊^{1}}(𝕊^{1}·ξ^{i})) = ∑_{i=1}^{ℓ} t^{ν_{i}} . (108)

Considering now the right hand side of (105), we observe that the equivariant Conley index of 𝔐^{+} is given by

h_{𝕊^{1}}(𝔐^{+}) = [ ((F_{0}/𝕊^{1})/(∂F_{0}/𝕊^{1}), {∂F_{0}/𝕊^{1}}) ] . (109)

In Alexander-Spanier cohomology (with real coefficients) we have the following isomorphisms (see also [5, Ch.VIII, 6.20]):

\begin{aligned}H^{∗}( (F_{0}/𝕊^{1})/(∂F_{0}/𝕊^{1}), \{∂F_{0}/𝕊^{1}\}) & ≅ & H^{∗}(F_{0}/𝕊^{1}, ∂F_{0}/𝕊^{1}) \\ & ≅ & H∗_{𝕊^{1}}(F_{0}, ∂F_{0}) . \end{aligned} (110)

Consequently, the Poincaré polynomial of h_{𝕊^{1}}(𝔐^{+}) is given by

P(t, h_{𝕊^{1}}(𝔐^{+})) = ∑_{j⩾0} t^{j} dim Hj_{𝕊^{1}}(F_{0}, ∂F_{0}) . (111)

In the proof of Proposition 4 we have already used the monomorphism

ι_{F_{0}}^{∗} ∘ ĝ : H∗_{𝕊^{1}}(F_{c}, ∂F_{c}) → H∗_{𝕊^{1}}(F_{0}, ∂F_{0}) , (112)

which implies the inequality

dim H∗_{𝕊^{1}}(F_{0}, ∂F_{0}) ⩾ dim H∗_{𝕊^{1}}(F_{c}, ∂F_{c}) . (113)

Using Lefschetz duality and recalling that the set Q from Lemma 12 is a strong 𝕊^{1}-deformation retract of F_{c}, we get

Hd−k_{𝕊^{1}}(F_{c}, ∂F_{c}) ≅ Hk^{𝕊^{1}}(F_{c}) ≅ Hk^{𝕊^{1}}(Q) ≅ H_{k}(Q/𝕊^{1}) , (114)

where d = dim F_{c}/𝕊^{1}. Hence we conclude dim Hd−k_{𝕊^{1}}(F_{c}, ∂F_{c}) ⩾ dim H_{k}(P/𝕊^{1}). Summing up the terms on the right hand of (111) which are labelled by j = d−k, we derive for t ⩾ 0 the following estimate from the Morse equation (105):

∑_{i=1}^{ℓ} t^{ν_{i}} ⩾ ∑_{k=0}^{d} t^{d−k} dim H_{k}(P/𝕊^{1}) . (115)

Setting now t=1, we find ℓ ⩾ SB (P/𝕊^{1}) as claimed. □

This proves the second part of Proposition 3 in section 7, and the proof of Theorem 2 is complete.

Acknowledgement The first author gratefully acknowledges the hospitality and the support of the Mathematische Fakultät at the Universität Tübingen.

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Frank W. Josellis Schützenstr. 7 D-45768 Marl Germany

e-mail: josellis@moebius.mathematik.uni-tuebingen.de

Wilhelm Klingenberg Mathematische Fakultät Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen Germany

e-mail: wilhelm@moebius.mathematik.uni-tuebingen.de