Structural Aspects of Closed Characteristics on S^{1}-invariant Hypersurfaces

by

Frank W. Josellis

Wilhelm Klingenberg

Abstract

Let the standard symplectic space (R^{2n}, ω) be endowed with the S^{1}- structure induced by the representation of the circle group S^{1} → Sp(n) which is given by t ↦ exp (\frac{2π}{τ} tJ). Supposing that Σ ⊂ R^{2n} \ \{0\} is a compact connected S^{1}-hypersurface of contact type with H_{dR}^{1}(Σ) = 0, we show that Σ carries at least n distinct symmetric closed characteristics. Moreover, if the total number of closed characteristics on Σ is finite, then all of them turn out to be symmetric. We also discuss the special case of star-shaped S^{1}- hypersurfaces. In particular, there is a bijection f ↦ Σ_{f} between the functions f : CP^{n−1} → R and the star-shaped S^{1}-hypersurfaces in R^{2n} which allows to reduce the investigation of symmetric closed characteristics on Σ_{f} to the study of the critical points of f.

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

1. Introduction and results The standard symplectic structure on the Euclidean space (ℝ^{2n}, ⟨ · , · ⟩) is given by the 2-form ω(· , ·) = ⟨J· , · ⟩ where J denotes the complex operator

J = ( \begin{matrix}0 & −id_{ℝ^{n}} \\ id_{ℝ^{n}} & 0 \end{matrix} ) ∈ Hom (ℝ^{n}× ℝ^{n}) . (1)

In coordinates corresponding to a symplectic orthonormal basis {e_{1}, …, e_{2n}} of ℝ^{2n} the symplectic form is expressed by ω = ∑_{j=1}^{n} dx_{j} ∧ dx_{n+j}.

For every τ > 0 there is a distinguished parametrization 𝕊_{τ}^{1} := ℝ/τℤ of the circle group 𝕊^{1} ≅ O(2). A symplectic representation of 𝕊^{1} on ℝ^{2n} is then defined by the homomorphism of Lie groups 𝕊_{τ}^{1} → Sp(n), t ↦ exp (\frac{2π}{τ} tJ). We shall write

(t, x) ↦ γ_{t} x := exp (\frac{2π}{τ} tJ) x (2)

for the 𝕊^{1}-action 𝕊_{τ}^{1} × ℝ^{2n} → ℝ^{2n} obtained in this way. Let Orb x := { γ_{t} x | t ∈ 𝕊_{τ}^{1} } stand for the 𝕊^{1}-orbit of x ∈ ℝ^{2n}. We note that Fix (ℝ^{2n}) := { x | γ_{t} x = x ∀ t ∈ 𝕊_{τ}^{1} } consists exactly of the origin 0 ∈ ℝ^{2n} while on ℝ^{2n} \{0} the 𝕊^{1}-action γ is free. Moreover, we observe that γ_{t}^{∗} ω = ω since 𝕊^{1} acts by symplectic transformations.

Let Σ ⊂ ℝ^{2n} \ {0} be a compact connected smooth hypersurface satisfying γ_{t}Σ = Σ for all t ∈ 𝕊_{τ}^{1}. For the sake of brevity, such Σ will henceforward simply be called an 𝕊^{1}-hypersurface. As an additional hypothesis we require the existence of an 𝕊^{1}-invariant contact structure on Σ which is compatible with the symplectic structure of the ambient space ℝ^{2n}. More precisely, let ω_{Σ} denote the pull-back of the symplectic form ω to Σ and assume that there exists a 1-form β on Σ such that

(i) γ_{t}^{∗} β = β for every t ∈ 𝕊_{τ}^{1}, i.e., β is 𝕊^{1}-invariant,

(ii) dβ = ω_{Σ}, and

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

In view of the conditions (ii) and (iii) the hypersurface Σ is of contact type in the sense of A. Weinstein [17]. Taking the 𝕊^{1}-symmetry of Σ into account, we shall call Σ an 𝕊^{1}-hypersurface of contact type provided that all the conditions (i)-(iii) are satisfied.

A given hypersurface Σ having the just stated properties may be viewed as a regular level surface of a suitable 𝕊^{1}-invariant Hamiltonian function H : ℝ^{2n} → ℝ. In this note we study some of the orbit structure of the corresponding Hamiltonian flow on Σ. We are particularly interested in the relation between the 𝕊^{1}-structure of the phase space ℝ^{2n} and the periodic Hamiltonian trajectories on Σ. Formulating the problem with emphasis on the geometric viewpoint, we ask for those closed characteristics on Σ which coincide with orbits of the 𝕊^{1}-action γ.

To make the statements precise, let us briefly recapitulate some basic terminology. Given a smooth hypersurface Σ as above, its characteristic line bundle 𝔏_{Σ} is the 1-dimensional subbundle of T Σ which is defined by the kernel of ω_{Σ}:

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

A characteristic vector field on Σ is a nowhere vanishing section X : Σ → 𝔏_{Σ}. The (unparametrized) integral curves of a characteristic vector field X are called the characteristics on Σ. In particular, a closed characteristic on Σ can be described as an embedding u : 𝕊^{1} → Σ whose tangent map is an embedding u_{∗} : T 𝕊^{1} → 𝔏_{Σ}. Two closed characteristics u, v : 𝕊^{1} → Σ are said to be (geometrically) distinct if u(𝕊^{1}) ∩ v(𝕊^{1}) = ∅. With respect to the 𝕊^{1}-symmetry we now define:

Definition 1 A closed characteristic u : 𝕊^{1} → Σ on an 𝕊^{1}-hypersurface Σ will be called symmetric if it admits a parametrization of the form u(t) = γ_{t} x.

We present a proof of the following multiplicity result for symmetric closed characteristics:

Theorem 1 Let Σ ⊂ ℝ^{2n} \{0} be a compact connected smooth 𝕊^{1}-hypersurface of contact type having trivial first de Rham cohomology, H_{dR}^{1}(Σ) = 0. Then

(4) # { distinct symmetric closed characteristics on Σ } ⩾ n ,

where n = Index_{ℂ} 𝕊^{2n−1} is the cohomological 𝕊^{1}-index of the unit sphere 𝕊^{2n−1} ⊂ ℝ^{2n}.

Hence, under the assumptions of the theorem, the minimal number of symmetric closed characteristics on the 𝕊^{1}-hypersurface Σ in question is determined by the 𝕊^{1}- equivariant homotopy invariants of the sphere 𝕊^{2n−1}. Since the canonical projection x ↦ Orb x defines exactly the Hopf fibration 𝕊^{2n−1} → 𝕊^{2n−1}/𝕊^{1} ≅ ℂP^{n−1}, the 𝕊^{1}-homotopy invariants of 𝕊^{2n−1} are given by the cohomology ring of ℂP^{n−1}.

From our proof of Theorem 1 it will be seen that the assumption H_{dR}^{1}(Σ) = 0 can be replaced as follows. Let λ denote the 1-form on ℝ^{2n} defined by

λ := \frac{1}{2} ∑_{j=1}^{n} x_{j} dx_{n+j} − x_{n+j} dx_{j} , (5)

and let λ_{Σ} be its pull-back to Σ. We observe that γ_{t}^{∗} λ_{Σ} = λ_{Σ} and dλ_{Σ} = ω_{Σ}. Recalling the contact 1-form β on Σ, our proof requires only that the 1-form β − λ_{Σ} is exact. Thus the role of the distinguished 1-form λ here is similar to that of the Liouville form for analogous results in cotangent bundles, cf. [13].

A special situation occurs if β = λ_{Σ}. In this case the 𝕊^{1}-hypersurface Σ is star-shaped, meaning that the map x ↦ x/|x| defines a diffeomorphism Σ → 𝕊^{2n−1}, cf. [17, Example 1]. A star-shaped 𝕊^{1}-hypersurface Σ can be described by an ℝvalued function on ℂP^{n−1} whose critical points correspond to the symmetric closed characteristics on Σ:

Theorem 2 Let 𝕊^{2n−1} ⊂ ℝ^{2n} be the unit sphere and let ℂP^{n−1} be identified with the orbit space 𝕊^{2n−1}/𝕊^{1} = { Orb x | x ∈ 𝕊^{2n−1} }. To a smooth function f : ℂP^{n−1} → ℝ we assign a star-shaped 𝕊^{1}-hypersurface Σ_{f} ⊂ ℝ^{2n} by setting

Σ_{f} := { e^{f(Orb x)} x | x ∈ 𝕊^{2n−1} } . (6)

Then the distinct symmetric closed characteristics on Σ_{f} correspond uniquely to the distinct critical points of f.

Furthermore, for every star-shaped 𝕊^{1}-hypersurface Σ ⊂ ℝ^{2n} there exists a unique function f : ℂP^{n−1} → ℝ such that Σ = Σ_{f}.

Star-shaped hypersurfaces appear in a natural way as regular level surfaces of homogeneous Hamiltonian functions. Conversely, to a given star-shaped 𝕊^{1}-hypersurface Σ_{f} we may associate a distinguished homogeneous Hamiltonian H_{f}. Describing the symmetric closed characteristics on Σ_{f} by periodic integral curves of the Hamiltonian vector field X_{H_{f}} = J grad H_{f}, we find a simple expression for their minimal periods:

Corollary 1 Given a star-shaped 𝕊^{1}-hypersurface Σ_{f} ⊂ ℝ^{2n} \ {0}, we denote by H_{f} : ℝ^{2n}\{0} → ℝ the unique Hamiltonian which is homogeneous of degree 2 and satisfies Σ_{f} = H_{f}^{−1}(1). Suppose that Orb x ∈ ℂP^{n−1} is a critical point of f. Then for every ϱ > 0 the X_{H_{f}} -orbit starting at ϱ x has the minimal period

τ(x) = π e^{2f(Orb x)} . (7)

This holds true, in particular, for the X_{H_{f}} -orbit starting at e^{f(Orb x)} x ∈ Σ_{f}.

Aside from the symmetric closed characteristics, an 𝕊^{1}-hypersurface of contact type as in Theorem 1 may carry additional non-symmetric closed characteristics. These are, however, cohomologically invisible, and Theorem 1 does not yield any hint on their existence or non-existence. Let us point out that the existence of one nonsymmetric closed characteristic on Σ already guarantees the existence of uncountably many distinct non-symmetric closed characteristics:

Theorem 3 Let Σ ⊂ ℝ^{2n} be a compact connected 𝕊^{1}-hypersurface of contact type that carries a non-symmetric closed characteristic u : 𝕊^{1} → Σ. Then the family of closed characteristics { γ_{t} u : 𝕊^{1} → Σ | t ∈ 𝕊_{τ}^{1} } generated by u contains uncountably many geometrically distinct elements.

Consequently, if Σ is known to carry a finite number of distinct closed characteristics in all, then each of its closed characteristics is symmetric.

Example Let us illustrate the above situation by means of the harmonic oscillator with n degrees of freedom, usually given by the system of Lagrangian equations m_{j} q̈_{j} = −k_{j} q_{j} for j = 1, …, n. Introducing the momenta p_{j} = m_{j}q̇_{j}, the total energy of the system is described by the Hamiltonian function

H̃(p, q) = ∑_{j=1}^{n} \frac{1}{2m_{j}} p_{j}^{2} + \frac{k_{j}}{2} q_{j}^{2} . (8)

For fixed c > 0 the level surface H̃^{−1}(c) is an ellipsoid in the (p, q)-space which, in general, fails to be an 𝕊^{1}-hypersurface. Denoting by ω_{j} = \sqrt{k_{j}/m_{j}} the eigenfrequencies, the system is transformed to its so-called normal form by the following symplectic change of coordinates:

x_{j} = \frac{p_{j}}{\sqrt{m_{j} ω_{j}}} and x_{n+j} = q_{j} \sqrt{k_{j}/ω_{j}} , j = 1, …, n. (9)

Then the transformed Hamiltonian function

H(x) = \frac{1}{2} ∑_{j=1}^{n} ω_{j} (x_{j}^{2} + x_{n+j}^{2}) (10)

is invariant under the 𝕊^{1}-action γ and, consequently, the ellipsoid H^{−1}(c) is a starshaped 𝕊^{1}-hypersurface. Now define f : ℂP^{n−1} → ℝ by

(11) f(Orb x) := ln \sqrt{\frac{c}{H(x)}} for x ∈ 𝕊^{2n−1},

so that H^{−1}(c) = Σ_{f}. The Hamiltonian H_{f} is then given by H_{f}(x) := c^{−1}H(x). Setting ω′_{j} := ω_{j}/c, we obtain the representation

H_{f}(x) = \frac{1}{2} ∑_{j=1}^{n} ω′_{j} (x_{j}^{2} + x_{n+j}^{2}) . (12)

By the Lagrange multiplier rule we see that Orb x is a critical point of f if and only if grad H_{f}(x) = μx, in which case μ is equal to some ω′_{j}. In particular, if the ω′_{j} are mutually distinct then Σ_{f} carries exactly n symmetric closed characteristics, corresponding to the so-called normal mode solutions. For the associated minimal periods, Corollary 1 recovers the familiar expression

T_{j} = 2π/ω′_{j} . (13)

Note that in the presence of resonances, i.e., in case ω_{i}/ω_{j} ∈ ℤ for some i ≠ j, the level surface Σ_{f} carries uncountably many additional closed characteristics which are not symmetric unless ω_{i} = ω_{j}. □

Let us finally point out two phenomena which can already be observed in the above simple model where the hypersurface Σ_{f} is an ellipsoid:

Firstly, the appearance of non-symmetric closed characteristics on Σ_{f} is linked to a resonance condition which can be formulated in terms of the minimal periods T_{j} of the symmetric ones. The question arises whether the existence of non-symmetric periodic X_{H_{f}} -orbits on a star-shaped 𝕊^{1}-hypersurface can be excluded by certain non-resonance conditions. This motivates the following

Problem 1 Suppose that f : ℂP^{n−1} → ℝ is a smooth function having finitely many critical points x^{1}, …, x^{ℓ} such that the corresponding minimal periods τ_{j} := τ(x^{j}) according to (7) satisfy τ_{i}/τ_{j} ∉ ℤ for all i ≠ j. Is it possible for the hypersurface Σ_{f} to carry any non-symmetric closed characteristics?

Secondly, the 𝕊^{1}-symmetry is evidently not preserved under symplectic diffeomorphisms of ℝ^{2n}. Hence it is a natural question to ask for conditions which guarantee an 𝕊^{1}-symmetrization of hypersurfaces by such symplectic diffeomorphisms:

Problem 2 Suppose that Σ ⊂ ℝ^{2n} is a compact connected hypersurface of contact type with H_{dR}^{1}(Σ) = 0. Does there exist a symplectic diffeomorphism φ : ℝ^{2n} → ℝ^{2n} such that the hypersurface φ(Σ) is 𝕊^{1}-invariant?

The property of a hypersurface Σ to be of contact type persists under symplectic diffeomorphisms. In case φ(Σ) is 𝕊^{1}-invariant, the contact 1-form on φ(Σ) can also be chosen 𝕊^{1}-invariant by averaging the given contact form over the Lie group 𝕊^{1}.

Consequently, the existence of such a symplectic symmetrization φ already would imply the solution of a more general multiplicity problem for periodic Hamiltonian orbits on hypersurfaces in ℝ^{2n}. For the background, history, and earlier results connected with that we refer to [1, 6, 7, 11, 15].

An analogous problem can be posed for Liouville hypersurfaces in cotangent bundles of free 𝕊^{1}-manifolds. The corresponding 𝕊^{1}-symmetric case has been studied in [13]. Clearly, the above assumption H_{dR}^{1}(Σ) then has to be replaced in a suitable manner. These problems are going to be treated in some forthcoming work.

2. Organization of the proofs Most of the remainder of this work is concerned with the proof of Theorem 1. Here we present an outline of the strategy for this proof which is subdivided into five steps (sections 3-7). The proofs of Theorem 2 and Corollary 1 (section 8) as well as that of Theorem 3 (section 9) are more elementary and independent of the preceding paragraphs, although we shall use some of the notation introduced in section 3.

First of all, we observe that it suffices to carry out the proof of Theorem 1 under the hypothesis that the set of distinct symmetric closed characteristics on the 𝕊^{1}-hypersurface Σ in question is finite, possibly empty. Of course, the existence of a closed characteristic on Σ is guaranteed by C. Viterbo’s result [16], but in the situation in hand it is not a priori obvious whether there is a symmetric one.

The desired symmetric closed characteristics will be found by a variational approach that utilizes the representation of closed characteristics by periodic Hamiltonian orbits. The first ingredient of the proof consists of choosing an appropriate period T > 0 and constructing a suitable Hamiltonian H_{T} : ℝ^{2n} → ℝ such that the symmetric closed characteristics on Σ correspond to certain periodic X_{H_{T}} -orbits whose minimal period is T. Here X_{H_{T}} = J grad H_{T} : ℝ^{2n} → ℝ^{2n} stands for the Hamiltonian vector field associated to H_{T}.

Our goal is to derive a characterization of these particular T -periodic X_{H_{T}} - orbits as the positive critical points of the action integral A_{H_{T}} : Ω → ℝ, considered as a function on a suitable space of loops Ω ⊂ C^{∞}(𝕊_{T}^{1} , ℝ^{2n}). We recall that for a smooth loop u : 𝕊_{T}^{1} → ℝ^{2n} the Hamiltonian action integral is given by

A_{H_{T}} (u) = ∫_{0}^{T} \{\frac{1}{2} ⟨ −Ju̇ , u ⟩ − H_{T} (u)\} dt . (14)

Speaking of a positive critical point, we mean a critical point u ∈ Ω of the action integral such that A_{H_{T}} (u) > 0. A crucial aspect in the variational formulation is that any two geometrically distinct positive critical points u and v (i.e., u(t) and v(t) describe geometrically distinct T -periodic X_{H_{T}} -orbits in ℝ^{2n}) should correspond to geometrically distinct closed characteristics on Σ. If this condition is satisfied, the desired counting of closed characteristics on Σ can be done by counting the geometrically distinct positive critical points of A_{H_{T}} on Ω.

The requirements on A_{H_{T}} have to be taken into account for the designing of the Hamiltonian H_{T}. This task will be carried out in two steps: the local construction near Σ (section 3) and the global extension to ℝ^{2n} (section 4). For a summary of the technical aspects entering in the construction of H_{T} see the comments concluding section 4. Whereas the local part is practically the same as for the corresponding construction in cotangent bundles [13], the extension is considerably simpler in the present situation, and the choice for the asymptotic behavior of H_{T} as |x| → 0 and |x| → ∞ is essentially borrowed from Hofer-Zehnder [12].

The set up for the variational formulation will be completed in section 5. In particular, we adapt the parametrization for the 𝕊^{1}-action to the previously fixed T > 0 by setting γ_{t} = exp (\frac{2π}{T} tJ). Then the space of symmetric loops of minimal period T, defined by Ω_{sym} := { γ_{t} x | x ∈ ℝ^{2n} } ⊂ C^{∞}(𝕊_{T}^{1} , ℝ^{2n}), is the natural choice for the loop space Ω. In view of the isomorphism Ω_{sym} ≅ ℝ^{2n} the variational problem for A_{H_{T}} : Ω_{sym} → ℝ can be solved by estimating the number of positive critical 𝕊^{1}-orbits of some 𝕊^{1}-invariant function F : ℝ^{2n} → ℝ.

The remaining problem is treated by an 𝕊^{1}-equivariant Lyusternik-Schnirelman approach for the −grad F-flow on ℝ^{2n}, using the concept of isolated invariant sets. As a general reference for the basic definitions and notations concerning isolated invariant sets we refer to C. Conley’s and E. Zehnder’s article [3, Sec. 3].

By our construction of H_{T} it turns out that the positive critical 𝕊^{1}-orbits of F constitute a Morse decomposition of a distinguished isolated invariant set 𝔐^{+}, called the positive subcomplex of the free Morse complex of F. The relevant 𝕊^{1}- homotopy invariants of 𝔐^{+} are analyzed in section 6. Our arguments presented here are somewhat simplified and more geometric compared to the corresponding passage in [13]. Note, however, that the proofs given for cotangent bundles can be adapted to the present case, and vice versa.

The proof of the theorem is finally completed in section 7 by applying the 𝕊^{1}-equivariant Lyusternik-Schnirelman method to 𝔐^{+}, which yields the required estimate for the number of the positive critical 𝕊^{1}-orbits of F.

3. The Hamiltonian near the hypersurface Let Σ ⊂ ℝ^{2n}\{0} be a compact connected smooth 𝕊^{1}-hypersurface of contact type. The hypothesis that Σ be of contact type is alternatively expressed as follows: there exists a symplectic dilation X, defined on some open neighborhood V of Σ, which is transverse to Σ (cf. [17, Lemma 2]). We recall that a vector field X : V → ℝ^{2n} is said to be a symplectic dilation if the Lie derivative of the symplectic form ω along X satisfies L_{X}ω = ω.

The existence of a transverse symplectic dilation follows essentially from the fact that the contact form β can be extended to a 1-form α̃ on some open neighborhood V of Σ such that dα̃ = ω. Since Σ is an 𝕊^{1}-set, every neighborhood of Σ contains an 𝕊^{1}-invariant neighborhood, and we may therefore assume that the neighborhood V in question is 𝕊^{1}-invariant. Then, by averaging α̃ over the group 𝕊^{1}, we obtain an 𝕊^{1}-invariant 1-form α on V that again satisfies dα = ω and whose pull-back α_{Σ} is equal to β. For the vector field X_{α} : V → ℝ^{2n}, defined by the identity α = X_{α} ⌟ ω , (15) we find L_{X_{α}}ω = X_{α} ⌟ dω + d(X_{α} ⌟ ω) = dα = ω by Cartan’s formula. Moreover, since α and ω are 𝕊^{1}-invariant, it follows readily that X_{α} is 𝕊^{1}-equivariant.

To see that X_{α} is transverse, we recall that the compact hypersurface Σ has a canonical positive orientation given by the outward normal vector field N(x), x ∈ Σ. Then 0 ≠ JN(x) ∈ T_{x}Σ, and because of ω(X, JN) = ⟨JX, JN⟩ = ⟨X, N⟩ = 0 for every X ∈ T_{x}Σ it follows that JN(x) indeed spans 𝔏_{Σ}|_{x}. Since α_{Σ} ∧ ω_{Σ}^{n−1} is a (positive) volume form, we get 0 < α(JN) = ω(X_{α}, JN) = ⟨X_{α}, N⟩ and, therefore, X_{α} is transverse to Σ.

Because the vector field X_{α} is 𝕊^{1}-equivariant, so is its flow which will be denoted by (s, x) ↦ ψ_{s}(x), i.e., we have ψ_{s}(γ_{t} x) = γ_{t} ψ_{s}(x). In view of the compactness of Σ there exists a constant ŝ > 0 such that for every x ∈ Σ the trajectory s ↦ ψ_{s}(x) is defined for all |s| < ŝ. Hence { Σ_{s} := ψ_{s}(Σ) | |s| < ŝ} constitutes a parametrized family of 𝕊^{1}-hypersurfaces modelled on Σ. Subsequently let us assume that ŝ is chosen sufficiently small such that, in addition, ⟨ N(ψ_{s}(x)) , X_{α}(ψ_{s}(x)) ⟩ > 0 for all x ∈ Σ, |s| < ŝ. Clearly, this latter assumption guarantees that each Σ_{s} is of contact type. For later convenience we also introduce the 𝕊^{1}-invariant neighborhood U of Σ by setting U := { ψ_{s}(x) | x ∈ Σ , |s| < ŝ} ⊂ V . (16)

Example For a star-shaped 𝕊^{1}-hypersurface Σ we may choose V := ℝ^{2n} and α := λ with the 1-form λ given by (5). The corresponding transverse symplectic dilation X_{λ} : ℝ^{2n} → ℝ^{2n} and its flow ψ_{s}(x) then are easily seen to be determined by

X_{λ}(x) = \frac{1}{2} x and ψ_{s}(x) = e^{s/2}x ∀ x ∈ ℝ^{2n}, s ∈ ℝ . (17)

There are no restrictions on the choice of ŝ > 0 since X_{λ} is transverse to each of the hypersurfaces Σ_{s} := e^{s/2} Σ.

Having completed these preparations, we now introduce a particular 𝕊^{1}-invar-iant Hamiltonian function H : U → ℝ. For this purpose let 0 < δ < ŝ be fixed and let the smooth function g : (−ŝ, ŝ) → ℝ_{0}^{+} be defined by

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

Defining h : (−ŝ, ŝ) → ℝ_{0}^{+} by

h(s) := ∫_{−ŝ}^{s} g(t) dt , (19)

the desired Hamiltonian H : U → ℝ is obtained by setting

H(ψ_{s}(x)) := h(s) for x ∈ Σ , |s| < ŝ . (20)

The gradient vector field grad H : U → ℝ^{2n}, defined according to dH = ⟨ grad H, · ⟩, is 𝕊^{1}-equivariant since H is 𝕊^{1}-invariant. The associated Hamiltonian vector field X_{H} : U → ℝ^{2n} is defined by the identity

− dH = X_{H} ⌟ ω , (21) and satisfies X_{H}(x) = J grad H(x). Consequently, X_{H} is 𝕊^{1}-equivariant. We shall write the Hamiltonian equations as

u̇ = X_{H}(u) . (22)

The following lemma states some properties of the non-constant periodic solutions of the Hamiltonian system (22):

Lemma 1 For every |s| < δ the restriction X_{H}|_{Σ_{s}} defines a characteristic vector field on Σ_{s} = ψ_{s}(Σ). If u_{0} is a periodic solution of (22) on Σ having the minimal period τ_{0} > 0, then there exists a corresponding periodic solution u_{s} on Σ_{s} which is given by u_{s}(t) = (ψ_{s} ∘ u_{0}) (\frac{t}{σ(s)}) (23)

and has the minimal period τ(s) = σ(s)τ_{0}, where

σ(s) = e^{s} \frac{h′(0)}{h′(s)} , |s| < δ . (24)

Furthermore, every non-constant periodic solution of u̇ = X_{H}(u) has a representation of the form (23).

Proof From dH(X_{H}) = ω(X_{H}, X_{H}) = 0 it follows that H remains constant along integral curves of X_{H} so that X_{H}(ψ_{s}(x)) ∈ T_{ψ_{s}(x)}Σ_{s} for all |s| < ŝ. Moreover, from

ω(X, X_{H}) = dH(X) = ⟨ grad H, X ⟩ = 0 ∀ X ∈ T_{ψ_{s}(x)}Σ_{s} (25)

we see that X_{H}(ψ_{s}(x)) ∈ 𝔏_{Σ_{s}}|_{ψ_{s}(x)}. Hence X_{H}|_{Σ_{s}} is a characteristic vector field if X_{H}(ψ_{s}(x)) ≠ 0 everywhere on Σ_{s}. Observing that

\begin{aligned}α(X_{H})|_{ψ_{s}(x)} & = & dH(X_{α})|_{ψ_{s}(x)} = L_{X_{α}}H|_{ψ_{s}(x)} = \frac{d}{dt}|_{t=0}ψ_{t}^{∗}H(ψ_{s}(x)) \\ & = & \frac{d}{dt}|_{t=s}h(t) = h′(s) , \end{aligned} (26)

we conclude by the definition of h that X_{H}(ψ_{s}(x)) ≠ 0 if and only if |s| < δ. This proves the first assertion of the lemma.

The gradient of H is orthogonal to the level surfaces Σ_{s} so that there is a smooth function μ : U → ℝ with grad H(ψ_{s}(x)) = μ(ψ_{s}(x))N(ψ_{s}(x)), where N again denotes the outward normal vector field on Σ_{s}. Consequently, in view of (26), we have

h′(s) = L_{X_{α}}H|_{ψ_{s}(x)} = ⟨ grad H, X_{α} ⟩|_{ψ_{s}(x)} = μ ⟨ N, X_{α} ⟩|_{ψ_{s}(x)} . (27)

Recalling that ⟨ N(ψ_{s}(x)) , X_{α}(ψ_{s}(x)) ⟩ > 0 for all ψ_{s}(x) ∈ U, the gradient of H can be expressed by

grad H(ψ_{s}(x)) = \frac{h′(s)}{⟨ N(ψ_{s}(x)) , X_{α}(ψ_{s}(x)) ⟩} N(ψ_{s}(x)) . (28)

Now let s ∈ (−δ, δ) be fixed. The map ψ_{s} is thus defined for all x′ = ψ_{s′}(x) ∈ U that satisfy |s+s′| < ŝ. Since L_{X_{α}}ω = ω implies ψ_{s}^{∗}ω = e^{s}ω, we obtain the following identity for the Jacobian Dψ_{s}(x′):

Dψ_{s}(x′)^{T} J Dψ_{s}(x′) = e^{s} J . (29)

Consequently, the transformation ψ_{s} behaves like a symplectic transformation up to the additional dilation factor e^{s}. Hence, if u̇_{0}(t) = X_{H}(u_{0}(t)) is a (periodic) solution on Σ, we infer

\begin{aligned}\frac{d}{dt} (ψ_{s} ∘ u_{0})(t) & = & e^{s} X_{H∘ψ_{−s}}((ψ_{s} ∘ u_{0})(t)) \\ & = & e^{s} J grad(H ∘ ψ_{−s})((ψ_{s} ∘ u_{0})(t)) . \end{aligned} (30)

To reparametrize ψ_{s} ∘ u_{0} as an integral curve of X_{H}, we observe that similar to (28) the gradient of H ∘ ψ_{−s} at ψ_{s}(x) ∈ Σ_{s} is represented by

grad(H ∘ ψ_{−s})(ψ_{s}(x)) = \frac{h′(0)}{⟨ N(ψ_{s}(x)) , X_{α}(ψ_{s}(x)) ⟩} N(ψ_{s}(x)) . (31)

Consequently, for all |s| < δ and x ∈ Σ we find

X_{H∘ψ_{−s}}(ψ_{s}(x)) = \frac{h′(0)}{h′(s)} X_{H}(ψ_{s}(x)) . (32)

Combining (30) and (32), the reparametrization u_{s}(t) := (ψ_{s} ∘ u_{0})(t/σ(s)) with σ : (−δ, δ) → ℝ^{+} given by (24) indeed yields a (periodic) integral curve of X_{H} on Σ_{s}. If u_{0} is periodic, the transformation of the minimal period according to τ_{0} ↦ σ(s)τ_{0} follows in an obvious way.

Given a non-constant periodic integral curve u(t) of X_{H}, then u lies entirely on some level surface Σ_{s} of H for some |s| < δ. Thus, by interchanging the role of Σ and Σ_{s} in the preceding arguments, u can be transformed into a periodic Hamiltonian orbit u_{0} on Σ. The inverse transformation then shows that u(t) = u_{s}(t), and the proof of the lemma is complete. □

The lemma shows that the closed characteristics on Σ can be described as the periodic orbits of X_{H}|_{Σ} or, more generally, as the non-constant periodic solutions of the Hamiltonian system u̇ = X_{H}(u). In the sequel, we shall speak of Hamiltonian orbits (trajectories, integral curves) rather than solutions, meaning that a Hamiltonian orbit is an equivalence class of solutions where u ∼ v if u(t) = v(t+t′) for some t′ ∈ ℝ. Consequently, two periodic orbits of X_{H}|_{Σ} are considered to be distinct if and only if the corresponding closed characteristics are geometrically distinct. Moreover, a periodic X_{H}-orbit on Σ will be called symmetric if the corresponding closed characteristic is symmetric.

We point out that the 𝕊^{1}-symmetry of Σ and H has so far not been used, and Lemma 1 remains valid if the 𝕊^{1}-invariance is not assumed. Even for an 𝕊^{1}- hypersurface Σ the statements refer to arbitrary periodic Hamiltonian orbits, not just symmetric ones. Note, however, that the 𝕊^{1}-equivariance of X_{α} implies that a symmetric X_{H}-orbit u_{0} on Σ is transformed into a symmetric X_{H}-orbit u_{s} on Σ_{s}.

The next lemma serves as a preparation for the variational formulation of Theorem 1:

Lemma 2 Let the 𝕊^{1}-hypersurface Σ satisfy the hypotheses of Theorem 1 and let the Hamiltonian H : U → ℝ be given by (20). In addition, suppose that Σ carries only a finite number of symmetric periodic X_{H}-orbits. Then there exists T_{0} > 0 such that every T ⩾ T_{0} has the following property:

(∗) If u_{0} is a symmetric periodic X_{H}-orbit on Σ then there exists a uniquely determined s ∈ (−δ, δ), depending on u_{0} and T, such that the corresponding X_{H}-orbit u_{s} on Σ_{s} satisfies the two conditions

(a) The X_{H}-orbit u_{s} has the minimal period T.

(b) The Hamiltonian action integral A_{H}(u_{s}) is positive, i.e.,

A_{H}(u_{s}) = ∫_{0}^{T} \frac{1}{2} ⟨ −Ju̇_{s} , u_{s} ⟩ dt − ∫_{0}^{T}(H ∘u_{s}) dt > 0 .

Proof Leet us assume that there are ℓ > 0 symmetric periodic X_{H}-orbits on Σ, denoted by u_{0}^{1}, …, u_{0}^{ℓ}, and let τ_{0,i} > 0 be the minimal period of u_{0}^{i}. For |s| < δ let the corresponding symmetric periodic X_{H}-orbits u_{s}^{i} on Σ_{s} be defined according to Lemma 1. The minimal period of u_{s}^{i} then is given by τ_{i}(s) = σ(s)τ_{0,i} with σ : (−δ, δ) → ℝ^{+} defined in (24).

In view of the choice (19) of h, the derivative σ′(s) has exactly one zero at s_{0} which is the unique solution of the equation h′(s) = h′′(s) in (−δ, δ). We note that σ′(s) < 0 for −δ < s < s_{0} and σ′(s) > 0 for s_{0} < s < δ. Moreover, we observe that σ(s) → +∞ as |s| → δ.

As a consequence, for every T > σ(s_{0})τ_{0,i} = τ_{i}(s_{0}) the equation T = τ_{i}(s) has exactly two solutions s_{i}^{+} ∈ (−δ, s_{0}) and s_{i}^{−} ∈ (s_{0}, δ). These T -dependent solutions satisfy s_{i}^{+} → −δ and s_{i}^{−} → δ monotonically as T → +∞. We note that the associated X_{H}-orbits u_{s_{i}^{±}}^{i} lie on Σ_{s_{i}^{±}} and have the minimal period T.

Now let us fix a period T > max { τ_{i}(s_{0}) | i = 1, …, ℓ } and let us consider a T -periodic smooth loop u : 𝕊_{T}^{1} ≅ ℝ/T ℤ → U. Recalling the 1-form λ from (5), the pull-back of λ by u can be written as

\begin{aligned}u^{∗}λ & = & \frac{1}{2} ∑_{j=1}^{n} \{ u_{j} du_{n+j} − u_{n+j} du_{j} \} \\ & = & \frac{1}{2} ∑_{j=1}^{n} \{ u_{j} u̇_{n+j} − u_{n+j} u̇_{j} \} dt \\ & = & \frac{1}{2} ⟨ −Ju̇ , u ⟩ dt . \end{aligned} (33)

Similarly, the pull-back of the 1-form α = X_{α} ⌟ ω by u has the representation

u^{∗}α = ω(X_{α}(u), u̇) dt = ⟨ −Ju̇ , X_{α}(u)⟩ dt . (34)

We observe that the homotopy [ 0, 1 ] × U → U defined by (r, ψ_{s}(x)) ↦ ψ_{(1−r)s}(x) retracts U to Σ so that H_{dR}^{1}(U) ≅ H_{dR}^{1}(Σ) = 0. Consequently, since dα = dλ = ω, the 1-form α−λ on U is exact and we have α = λ+dF for some function F : U → ℝ. By integration over the circle 𝕊_{T}^{1} we obtain

∫_{𝕊_{T}^{1}} u^{∗}λ = ∫_{𝕊_{T}^{1}} u^{∗}α − ∫_{𝕊_{T}^{1}} d(u^{∗}F) = ∫_{0}^{T} ⟨ −Ju̇ , X_{α}(u)⟩ dt . (35)

Now suppose that u = u_{s} where u_{s} stands for any of the above symmetric T -periodic X_{H}-orbits u_{s_{i}^{±}}^{i}, i = 1, …, ℓ. Then we have −Ju̇_{s} = grad H(u_{s}), and, in view of the representation (28) of grad H, the right hand integral becomes

∫_{0}^{T}⟨ grad H(u_{s}), X_{α}(u_{s})⟩ dt = ∫_{0}^{T} \frac{h′(s)}{⟨ N(u_{s}) , X_{α}(u_{s}) ⟩} ⟨N(u_{s}), X_{α}(u_{s})⟩ dt . (36)

Thus from (33), (35), and (36) we deduce

∫_{0}^{T} \frac{1}{2} ⟨ −Ju̇_{s} , u_{s} ⟩ dt = T h′(s) . (37)

In combination with ∫_{0}^{T} H(u_{s}(t)) dt = T h(s) , (38)

this yields the identity A_{H}(u_{s}) = T (h′(s) − h(s)) , (39)

where u_{s} = u_{s_{i}^{±}}^{i} with s_{i}^{±} being the solutions of T = τ_{i}(s) for i = 1, …, ℓ.

We claim that A_{H}(u_{s_{i}^{+}}^{i}) > 0 and A_{H}(u_{s_{i}^{−}}^{i}) < 0 for i = 1, …, ℓ in case the prescribed period T is sufficiently large. For this purpose let us recall that s_{i}^{+} → −δ and s_{i}^{−} → δ as T → +∞.

With regard to the fact that h(δ) < 0 and h′(δ) = 0 we can find some ε_{1} > 0 such that h′(s) − h(s) < 0 if s ∈ (δ − ε_{1}, δ). Moreover, choosing 0 < ε_{2} < min {δ, 1}, we see that h′ is strictly increasing on (−δ, −δ + ε_{2}). Hence, by the mean value theorem, it follows for −δ < s < −δ + ε_{2} that

h(s) = ∫_{−δ}^{s} h′(r) dr = h′(r̄) ∫_{−δ}^{s} dr < h′(s) ε_{2} < h′(s) (40)

with some r̄ ∈ (−δ, s). Consequently, for ε := min{ε_{1}, ε_{2}} we have

\begin{matrix}h′(s) − h(s) > 0 & if & s ∈ (−δ, −δ + ε) \\ h′(s) − h(s) < 0 & if & s ∈ (δ − ε, δ) . \end{matrix} (41)

Now we can choose T_{0} > min { τ_{i}(s_{0}) | i = 1, …, ℓ } so large that for every T ⩾ T_{0} the corresponding solutions s_{i}^{±} satisfy s_{i}^{+} ∈ (−δ, −δ +ε) and s_{i}^{−} ∈ (δ −ε, δ) for every i = 1, …, ℓ, and the claim follows.

This proves the assertion of the lemma if the assumed number of symmetric periodic X_{H}-orbits on Σ is ℓ > 0. In the case ℓ = 0 the condition (∗) is trivially satisfied for every choice of T_{0} > 0, and the proof is hereby complete. □

4. The global extension of the Hamiltonian Observing that the neighborhood U of Σ introduced in (16) is relatively compact, we can find R_{1} > 0 sufficiently large such that

U ⊂ { x ∈ ℝ^{2n} | |x| ⩽ R_{1} } . (42)

If T_{0} is again the constant from Lemma 2, then we can choose T > T_{0} and R_{2} > R_{1} so large that \frac{3π}{2T} R_{1}^{2} < h(ŝ) < \frac{3π}{2T} R_{2}^{2} , (43)

where h(ŝ) is the maximal value of the Hamiltonian H defined in (20). Now we pick a smooth function h_{T} : ℝ_{0}^{+} → ℝ^{+} that satisfies

h_{T} (r) ≡ h(ŝ)  if r ⩽ R_{1}
h_{T} (r) = \frac{3π}{2T} r^{2}  if r ⩾ R_{2}
h_{T} (r) > \frac{3π}{2T} r^{2}  if R_{1} < r < R_{2}
0 < h′_{T} (r) < \frac{3π}{T} r if R_{1} < r < R_{2} .

(44)

Moreover, we shall assume that h_{T} is chosen in such a way that there exists only one solution r_{crit} ∈ (R_{1}, R_{2}) of the equation

h′_{T} (r) = \frac{2π}{T} r . (45)

Note that, by the Jordan-Brouwer separation theorem, ℝ^{2n}\Σ consists of a bounded connected component U_{0} and an unbounded connected component U_{∞}. Now we extend the Hamiltonian H : U → ℝ defined in (20) to a smooth function on ℝ^{2n} as follows:

H_{T} (x) := \{ \begin{matrix}0 & if & x ∈ U_{0} \\ H(x) & if & x ∈ U \\ h_{T} (|x|) & if & x ∈ U_{∞}\U . \end{matrix} (46)

If x ∈ U_{∞}\U we have: grad H_{T} (x) = \frac{h′_{T} (|x|)}{|x|} x. Consequently, for the Hamiltonian system u̇ = X_{H_{T}} (u) on ℝ^{2n} we find two types of symmetric periodic X_{H_{T}} -orbits that have the minimal period T:

(i) The significant periodic X_{H_{T}} -orbits u_{s} in U which have been described in the previous section.

(ii) The technical periodic X_{H_{T}} -orbits generated by the above extension. These are of the form u(t) = exp (\frac{2π}{T} tJ) x where |x| = r_{crit} solves the equation (45).

The choice of the function h_{T} implies that the corresponding Hamiltonian action A_{H_{T}} (u) of a technical periodic X_{H_{T}} -orbit is negative:

Lemma 3 Suppose that u is a symmetric periodic X_{H_{T}} -orbit that has the minimal period T and satisfies A_{H_{T}} (u) > 0. Then u is a significant periodic X_{H_{T}} -orbit.

Proof For a technical X_{H_{T}} -orbit u(t) = exp (\frac{2π}{T} tJ) x with |x| = r_{crit} we obtain

\frac{1}{T} A_{H_{T}} (u) = \frac{1}{T} ∫_{0}^{T}\{\frac{1}{2} ⟨ −Ju̇ , u ⟩ − H_{T} (u)\} dt
= \frac{1}{2} h′_{T} (|x|) |x| − h_{T} (|x|)
< \frac{3π}{2T} r_{crit}^{2} − h_{T} (r_{crit}) < 0

(47)

since r_{crit} ∈ (R_{1}, R_{2}), and the statement thus follows. □

To summarize the contents of the last two sections, we note that Lemma 1 allows to associate to each symmetric closed characteristics on Σ a so-called ‘tower’ { u_{s} | |s| < δ } of symmetric periodic X_{H}-orbits where u_{s} lies on the hypersurface Σ_{s} and has the minimal period τ(s). If, in addition, the number of these towers is assumed to be finite, then Lemma 2 guarantees that for every sufficiently large T > 0 each of these towers contains a unique X_{H}-orbit u_{s} that has the prescribed minimal period T and satisfies A_{H}(u_{s}) > 0. Consequently, the counting of the geometrically distinct closed characteristics on Σ can be done by counting these particular T -periodic X_{H}-orbits.

Fixing T so large that the condition (43) is satisfied, we have extended the Hamiltonian H : U → ℝ to H_{T} : ℝ^{2n} → ℝ such that the additional symmetric periodic X_{H_{T}} -orbits of minimal period T outside U have a negative Hamiltonian action and do not affect the above orbit counting. Therefore, to prove Theorem 1 it suffices to show:

Proposition 1 Let Σ ⊂ ℝ^{2n} be an 𝕊^{1}-hypersurface satisfying the hypotheses of Theorem 1 such that, in addition, the number of symmetric closed characteristics on Σ is finite. Suppose T > 0 is fixed in accordance with (43) so that the distinct symmetric periodic X_{H_{T}} -orbits u of minimal period T and A_{H_{T}} (u) > 0 correspond uniquely to the distinct symmetric closed characteristics on Σ. Then the number of these X_{H_{T}} -orbits u is at least n.

5. The variational approach and the free Morse complex The choice of the period T in the hypotheses of Proposition 1 provides a distinguished parametrization 𝕊_{T}^{1} = ℝ/ℤ of the circle group. To every x ∈ ℝ^{2n} we may now assign the parametrized 𝕊^{1}-orbit u_{x} : 𝕊_{T}^{1} → ℝ^{2n} by setting u_{x}(t) := γ_{t} x. The 𝕊^{1}-orbit u_{x} describes a symmetric T -periodic X_{H_{T}} -orbit if it solves the equation

u̇_{x} = X_{H_{T}} (u_{x}) , x ∈ ℝ^{2n} , (48)

and T is the minimal period if x ≠ 0. Hence, setting Ω_{sym} := { u_{x} | x ∈ ℝ^{2n} }, the desired periodic X_{H_{T}} -orbits of Proposition 1 can be found as those critical points u_{x} of the restricted action integral A_{H_{T}} |_{Ω_{sym}} : Ω_{sym} → ℝ for which

A_{H_{T}} (u_{x}) = ∫_{0}^{T} \{\frac{1}{2} ⟨ −Ju̇_{x} , u_{x} ⟩ − H_{T} (u_{x})\} dt (49) attains a positive value. However, the bijection ℝ^{2n} → Ω_{sym}, x ↦ u_{x}, suggests to study the corresponding variational problem on ℝ^{2n}:

Lemma 4 The Hamiltonian action integral A_{H_{T}} |_{Ω_{sym}} : Ω_{sym} → ℝ gives rise to a smooth 𝕊^{1}-invariant function F : ℝ^{2n} → ℝ by setting

F(x) := \frac{1}{T} A_{H_{T}} (u_{x}) . (50)

In particular, x ∈ ℝ^{2n} is a critical point of F if and only if u_{x} ∈ Ω_{sym} solves the Hamiltonian equation u̇_{x} = X_{H_{T}} (u_{x}).

Proof Evaluating A_{H_{T}} (u_{x}), we get the expression F(x) = \frac{π}{T} |x|^{2} − H_{T} (x). Then, clearly, grad F(x) = 0 is equivalent to \frac{2π}{T} γ_{t} x = grad H_{T} (γ_{t} x). On the other hand, −Ju̇_{x}(t) = \frac{2π}{T} u_{x}(t) = \frac{2π}{T} γ_{t} x, and the statement follows. □

The assertion of Proposition 1 can now be restated as follows:

Proposition 2 Suppose that the hypotheses of Proposition 1 are satisfied. Then there exist ℓ ⩾ n critical points x^{1}, …, x^{ℓ} of F such that

(i) Orb x^{i} ∩ Orb x^{j} = ∅ if i ≠ j.

(ii) F(x^{i}) > 0 for i = 1, …, ℓ.

The proof will be based on 𝕊^{1}-equivariant Lyusternik-Schnirelman theory for flows, involving C. Conley’s concept of isolated invariant sets. The flow in question is the −grad F-flow on ℝ^{2n}, i.e., the map ℝ^{2n} × ℝ → ℝ^{2n}, (x, τ) ↦ x·τ, solving the initial value problem

\{ \begin{matrix}\frac{d}{dτ} (x·τ) & = & −grad F(x·τ) \\ x·0 & = & x ∀ x ∈ ℝ^{2n} . \end{matrix} (51)

Note that the trajectory τ ↦ x·τ of each x ∈ ℝ^{2n} is indeed defined for all τ ∈ ℝ because grad F is linearly bounded. The 𝕊^{1}-equivariance of grad F implies that (γ_{t} x)·τ = γ_{t} (x·τ) for all (x, τ) ∈ ℝ^{2n}× ℝ.

Given r_{2} > r_{1} > 0, we define the compact 𝕊^{1}-set Q_{r_{1},r_{2}} ⊂ ℝ^{2n}\{0} by setting

Q_{r_{1},r_{2}} := { x ∈ ℝ^{2n} | r_{1} ⩽ |x| ⩽ r_{2} } , (52)

which has the boundary ∂Q_{r_{1},r_{2}} = 𝕊_{r_{1}}^{2n−1} ∪ 𝕊_{r_{2}}^{2n−1} , (53)

where 𝕊_{r}^{2n−1} denotes the Euclidean (2n−1)-sphere of radius r.

Lemma 5 There exists a choice of r_{2} > ϱ > r_{1} > 0 such that:

(i) Q := Q_{r_{1},r_{2}} is an isolating block for the −grad F-flow, containing all critical points of F except for the origin 0 ∈ ℝ^{2n} and having the exit set Q^{−} = ∂Q_{r_{1},r_{2}}.

(ii) F(x) = \frac{π}{T} ϱ^{2} if x ∈ 𝕊_{ϱ}^{2n−1}. In particular, max_{x∈ℝ^{2n}} F(x) > 0 is attained in Q.

The isolated invariant set 𝔐 ⊂ int Q will be called the free Morse complex of F.

Proof Let B_{r}(0) denote the closed 2n-ball of radius r centered at the origin. From the definition (16) of U we infer B_{ϱ}(0) ∩ \overline{U} = ∅ if ϱ > 0 is chosen small enough. Then H_{T} |_{B_{ϱ}(0)} = 0 and, consequently, F|_{B_{ϱ}(0)} = \frac{π}{T} ϱ^{2}.

Let us now fix r_{1} < ϱ and r_{2} > R_{2} with the constant R_{2} introduced in (43). Observing that F(x) = \frac{π}{T} |x|^{2} if |x| ⩽ ϱ and F(x) = \frac{π}{T} |x|^{2} − \frac{3π}{2T} |x|^{2} = −\frac{π}{2T} |x|^{2} if |x| ⩾ R_{2}, we infer that F attains its global maximum inside Q on a positive level and, moreover,

(54) − grad F(x) = \{ \begin{matrix}−\frac{2π}{T} x & if & x ∈ 𝕊_{r_{1}}^{2n−1} \\ \frac{π}{T} x & if & x ∈ 𝕊_{r_{2}}^{2n−1} . \end{matrix}

Hence Q is an isolating block for the −grad F-flow that has the exit set Q^{−} = ∂Q (cf. [2, p. 5]). In addition, we see that grad F(x) = 0 and x ∈ ℝ^{2n}\Q imply x = 0, which completes the proof. □

An admissibly ordered Morse decomposition of 𝔐 is a tupel (M_{1}, …, M_{k}) of compact non empty subsets of 𝔐 such that for every x ∈ 𝔐\(M_{1} ∪ … ∪ M_{k}) there exist unique indices i < j with x·τ → M_{i} as τ → +∞ and x·τ → M_{j} as τ → −∞. Let us consider the subsets

𝔐^{+} := { x ∈ 𝔐 | F(x·τ) > 0 ∀τ ∈ ℝ } (55)

and 𝔐^{−} := { x ∈ 𝔐 | F(x·τ) < 0 ∀τ ∈ ℝ } , (56)

which we call the positive and the negative subcomplex of 𝔐, respectively. Note that 𝔐^{+} contains the critical points x of F that correspond to the significant periodic X_{H_{T}} -orbits u_{x} with A_{H_{T}} (u_{x}) > 0, whereas 𝔐^{−} contains their negative counterparts together with the sphere 𝕊_{r_{crit}}^{2n−1} which gives rise to the technical periodic X_{H_{T}} -orbits.

As a consequence of Lemma 5 we have 𝔐^{+} ≠ ∅, and it follows readily that (𝔐^{−}, 𝔐^{+}) is indeed an admissibly ordered Morse decomposition of 𝔐. Setting Q^{0} := { x ∈ Q | F(x) ⩽ 0 }, we define a filtration of index pairs N_{0} ⊂ N_{1} ⊂ N_{2} associated to (𝔐^{−}, 𝔐^{+}) by

N_{0} := Q^{−} , N_{1} := Q^{−}∪ Q^{0} , N_{2} := Q . (57)

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

(a) N := \overline{N_{2}\N_{1}} = \{ x ∈ Q | F(x) ⩾ 0 \} is an isolating neighborhood for 𝔐^{+} having the exit set N^{−} = ∂N = 𝕊_{r_{1}}^{2n−1} ∪ { x ∈ Q | F(x) = 0 }.

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

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

The proof of Poposition 2 requires an estimate for the number ℓ of distinct critical 𝕊^{1}-orbits Orb x^{i}, i = 1, …, ℓ, contained in 𝔐^{+}. So far we only know that ℓ < +∞ by our assumptions on the hypersurface Σ, and that ℓ ⩾ 1 because F attains its global maximum along some Orb x^{i} ⊂ 𝔐^{+}. To derive the claimed inequality ℓ ⩾ n we have to investigate the 𝕊^{1}-equivariant topology of 𝔐^{+} which will be carried out in the subsequent section.

6. 𝕊^{1}-index maps related to the positive subcomplex 𝔐^{+} The observation that the sphere 𝕊_{ϱ}^{2n−1} introduced in Lemma 5 is contained in the isolating block N of 𝔐^{+} turns out to be crucial. This fact causes a topological linking which allows to estimate the 𝕊^{1}-category of 𝔐^{+} in N by the cohomological 𝕊^{1}-index of 𝕊_{ϱ}^{2n−1}. We merely recall some relevant facts on the involved 𝕊^{1}-index maps and refer to [8, 9, 10] for further details.

Given a compact 𝕊^{1}-pair (X, A), we denote by H∗_{𝕊^{1}}(X, A) the 𝕊^{1}-equivariant Alexander-Spanier cohomology with real coefficients, cf. e.g. [9]. Since the 𝕊^{1}-action under consideration is free, 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 shall denote by H∗^{𝕊^{1}}(X, A) := H_{∗}(X/𝕊^{1}, A/𝕊^{1}) singular 𝕊^{1}-homology of (X, A) with real coefficients.

The cohomological 𝕊^{1}-index is defined in terms of universal characteristic classes. The universal principal 𝕊^{1}-bundle E𝕊^{1} →^{p} B𝕊^{1} is obtained as the inductive limit of the Hopf fibration

𝕊^{3} ⊂ 𝕊^{5} ⊂ … ⊂ 𝕊^{2n−1} ⊂ …
↓  ↓  ↓
𝕊^{2} ⊂ ℂP^{2} ⊂ … ⊂ ℂP^{n−1} ⊂ …,

i.e., E𝕊^{1} = 𝕊^{∞} and B𝕊^{1} = ℂP^{∞}. The base space ℂP^{∞} then is called the classifying space for 𝕊^{1}. The cohomology ring H^{∗}(ℂP^{∞}; ℤ) is the polynomial ring generated by the first Chern class c_{1} ∈ H^{2}(ℂP^{∞}; ℤ), cf. [14, Thm. 14.5]. Introducing real coefficients, let c ∈ H^{2}(ℂP^{∞}) denote the cohomology class corresponding to c_{1}.

Now let X be a paracompact free 𝕊^{1}-space and let f : X/𝕊^{1} → ℂP^{∞} denote a classifying map for the principal 𝕊^{1}-bundle X →^{p} X/𝕊^{1}. The corresponding induced map f^{∗} : H^{∗}(ℂP^{∞}) → H^{∗}(X/𝕊^{1}) is a ring homomorphism, and the cohomological 𝕊^{1}-index of X ≠ ∅ is defined by

(58) Index_{ℂ} X := 1 + sup { k ∈ ℕ_{0} | f^{∗}(c^{k}) ≠ 0 } .

Moreover, we set Index_{ℂ} ∅ := 0.

If X = 𝕊^{2n−1} ⊂ ℝ^{2n} is the Euclidean (2n − 1)-sphere (of any radius ϱ > 0), then the 𝕊^{1}-action γ gives rise to the Hopf map 𝕊^{2n−1} →^{p} ℂP^{n−1}. The corresponding classifying map f is the inclusion ℂP^{n−1} → ℂP^{∞}, and the cohomology ring H^{∗}(ℂP^{n−1}) is the truncated polynomial ring, terminating in dimension 2(n − 1), which is generated by f^{∗}(c) ∈ H^{2}(ℂP^{n−1}), cf. [14, Thm. 14.4]. Hence we infer

Index_{ℂ} 𝕊^{2n−1} = n . (59)

Given a free 𝕊^{1}-pair (X, A), then A is called 𝕊^{1}-categorical in X if A can be deformed 𝕊^{1}-equivariantly through X into a single 𝕊^{1}-orbit. The 𝕊^{1}-category of a non empty 𝕊^{1}-set 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} } , (60)

whereas 𝕊^{1}-cat_{X}(∅) := 0. Furthermore, we write 𝕊^{1}-cat(X) := 𝕊^{1}-cat_{X}(X).

Since the 𝕊^{1}-action on (X, A) is supposed to be free, the 𝕊^{1}-category is related to the ordinary Lyusternik-Schnirelman category (see e.g. [4, Ch. 27.3]) by

𝕊^{1}-cat_{X}(A) = cat_{X/𝕊^{1}}(A/𝕊^{1}) . (61)

The above definition of 𝕊^{1}-cat_{X}(A) differs from E. Fadell’s [8] in that we require the covering sets A_{1}, …, A_{k} to be closed. However, being concerned with free 𝕊^{1}-pairs (X, A) where X/𝕊^{1} is an absolute neighborhood retract (ANR) and the subset A/𝕊^{1} is closed, this difference does not affect the statements below.

In terms of the 𝕊^{1}-category and the cohomological 𝕊^{1}-index, the desired estimate can now be expressed as follows:

Proposition 3 Let 𝔐^{+} be the positive subcomplex defined in (55), and let N denote the isolating neighborhhood obtained from the filtration of index pairs (57). Then

(62) 𝕊^{1}-cat_{N}(𝔐^{+}) ⩾ Index_{ℂ} 𝕊^{2n−1} .

Proof First we note that N is a smooth manifold with boundary ∂N equipped with a proper free Lie group action of the circle group 𝕊^{1}. Hence the orbit space N/𝕊^{1} has the unique structure of a smooth manifold with boundary ∂(N/𝕊^{1}) = (∂N)/𝕊^{1} such that the projection N → N/𝕊^{1} is a submersion, cf. [5, Prop. 5.2]. In particular, N/𝕊^{1} is an ANR so that 𝕊^{1}-cat_{N} has the continuity property.

Let us recall that 𝕊_{ϱ}^{2n−1} ⊂ int N where ϱ > 0 is the constant introduced in Lemma 5. The inequality (62) will now be derived in three steps:

Claim 1: 𝕊^{1}-cat_{N}(𝔐^{+}) ⩾ 𝕊^{1}-cat_{N}(𝕊_{ϱ}^{2n−1}) . (63)

Proof: Let U_{ε} denote the open ε-neighborhood of 𝔐^{+}. Clearly, U_{ε} is 𝕊^{1}-invariant and, for small ε, we have U_{ε} ⊂ N. Moreover, by the continuity property of 𝕊^{1}-cat_{N} we may assume that ε > 0 is chosen such that 𝕊^{1}-cat_{N}(U_{ε}) = 𝕊^{1}-cat_{N}(𝔐^{+}).

Since 𝔐^{+} is compact and non empty, we find 0 < k := 𝕊^{1}-cat_{N}(𝔐^{+}) < +∞. Consequently, there exist k closed 𝕊^{1}-categorical subsets A_{1}, …, A_{k} ⊂ N such that U_{ε} ⊂ A := A_{1} ∪ … ∪ A_{k}. For every x ∈ \overline{N \A} there exists a unique τ(x) ⩾ 0 such that x·τ(x) ∈ ∂N, i.e., the integral curve τ ↦ x·τ of the −grad F-flow exits N at time τ(x). The compactness of \overline{N \A} then implies

τ^{−} := sup \{ τ(x) | x ∈ \overline{N \A} \} < +∞ . (64)

If τ ⩾ τ^{−} is fixed, it follows that N ⊂ A_{1}·τ ∪ … A_{k}·τ. Setting B_{ν} := (A_{ν}·τ) ∩ 𝕊_{ϱ}^{2n−1} for ν = 1, …, k, we thus have found k closed 𝕊^{1}-categorical subsets of N such that 𝕊_{ϱ}^{2n−1} ⊂ B_{1} ∪ … ∪ B_{k}. As a consequence, 𝕊^{1}-cat_{N}(𝕊_{ϱ}^{2n−1}) ⩽ k, which proves claim 1.

Claim 2: 𝕊^{1}-cat_{N}(𝕊_{ϱ}^{2n−1}) = 𝕊^{1}-cat(𝕊_{ϱ}^{2n−1}) . (65)

Proof: Clearly, 𝕊^{1}-cat_{N}(𝕊_{ϱ}^{2n−1}) ⩽ 𝕊^{1}-cat(𝕊_{ϱ}^{2n−1}), and it remains to show the opposite inequality. For this purpose we observe that the homotopy ḡ : Q × [ 0, 1 ] → Q, defined by ḡ(x, s) := (1−s)x + \frac{sϱ}{|x|} x, deforms Q equivariantly into 𝕊_{ϱ}^{2n−1}. Hence the 𝕊^{1}-map g := ḡ(·, 1)|_{N} : N → 𝕊_{ϱ}^{2n−1} is an 𝕊^{1}-retraction of N to 𝕊_{ϱ}^{2n−1}.

Now suppose that k = 𝕊^{1}-cat_{N}(𝕊_{ϱ}^{2n−1}) and let A_{1}, …, A_{k} be closed 𝕊^{1}-categor-ical subsets of N with 𝕊_{ϱ}^{2n−1} ⊂ A_{1} ∪ … ∪ A_{k}. Moreover, let h_{ν} : A_{ν} × [ 0, 1 ] → N denote the corresponding 𝕊^{1}-homotopy that deforms A_{ν} into a single 𝕊^{1}-orbit. We observe that B_{ν} := A_{ν} ∩ 𝕊_{ϱ}^{2n−1} is 𝕊^{1}-invariant and closed in 𝕊_{ϱ}^{2n−1} for ν = 1, …, k. Furthermore, by the 𝕊^{1}-homotopy g ∘ h_{ν}|_{B_{ν}} : B_{ν} × [ 0, 1 ] → 𝕊_{ϱ}^{2n−1} we see that B_{ν} is 𝕊^{1}-categorical in 𝕊_{ϱ}^{2n−1}. Hence we get 𝕊^{1}-cat(𝕊_{ϱ}^{2n−1}) ⩽ k, and claim 2 follows.

Claim 3:

(66) 𝕊^{1}-cat(𝕊^{2n−1}) ⩾ Index_{ℂ} 𝕊^{2n−1} .

Proof: Arguing by contradiction, we assume that k := 𝕊^{1}-cat(𝕊^{2n−1}) < Index_{ℂ} 𝕊^{2n−1}. As a consequence, there are k closed 𝕊^{1}-categorical subsets A_{ν} ⊂ 𝕊^{2n−1} such that A_{1} ∪ … ∪ A_{k} = 𝕊^{2n−1}. Without loss of generality we may assume that {A_{ν}, A_{ν′}} is an excisive couple for all ν, ν′, i.e., A_{ν} ∪A_{ν′} = int A_{ν} ∪ int A_{ν′} in the relative topology of A_{ν} ∪ A_{ν′}.

Denote by i_{ν} : A_{ν} → 𝕊^{2n−1} the inclusion. Then i_{ν} is 𝕊^{1}-homotopic to a map i_{ν}^{0} : A_{ν} → 𝕊^{2n−1} such that i_{ν}^{0}(A_{ν}) = Orb x^{ν} is a single 𝕊^{1}-orbit. Hence we have an 𝕊^{1}-map ι_{ν} : A_{ν} → Orb x^{ν} and an inclusion κ_{ν} : Orb x^{ν} → 𝕊^{2n−1} such that i_{ν}^{0} = κ_{ν} ∘ι_{ν}. These maps induce the following commutative diagram in 𝕊^{1}-cohomology:

[figure: H∗_{𝕊^{1}}(N); H∗_{𝕊^{1}}(Orb x^{ν}); H∗_{𝕊^{1}}(A_{ν}); κ_{ν}^{∗}; ι_{ν}^{∗}; i_{ν}^{∗}]

Consequently, i_{ν}^{∗} = 0 for ∗ ⩾ 1. Let us now consider the exact 𝕊^{1}-cohomology sequence of the pair (𝕊^{2n−1}, A_{ν}):

⟶ H∗−1_{𝕊^{1}}(A_{ν}) ⟶^{δ^{∗}} H∗_{𝕊^{1}}(𝕊^{2n−1}, A_{ν}) ⟶^{j_{ν}^{∗}} H∗_{𝕊^{1}}(𝕊^{2n−1}) ⟶^{i_{ν}^{∗}} H∗_{𝕊^{1}}(A_{ν}) ⟶ (67)

where j_{ν} : (𝕊^{2n−1}, ∅)→(𝕊^{2n−1}, A_{ν}) is the inclusion. By the exactness of the sequence, i_{ν}^{∗} = 0 implies that j_{ν}^{∗} is epimorphic.

Using the isomorphsm H∗_{𝕊^{1}}(𝕊^{2n−1}) ≅ H^{∗}(𝕊^{2n−1}/𝕊^{1}) ≅ H^{∗}(ℂP^{n−1}), let us denote by α ∈ H2_{𝕊^{1}}(𝕊^{2n−1}) the generator corresponding to the class f^{∗}(c) ∈ H^{2}(ℂP^{n−1}). For

ν = 1, …, k we thus can find α_{ν} ∈ H2_{𝕊^{1}}(𝕊^{2n−1}, A_{ν}) with j_{ν}^{∗}(α_{ν}) = α.

Observing that A_{1}∪…∪A_{k} = 𝕊^{2n−1} and denoting by j : (𝕊^{2n−1}, ∅)→(𝕊^{2n−1}, A) the inclusion, we obtain the following commutative diagram

\begin{matrix}H2_{𝕊^{1}}(𝕊^{2n−1}, A_{1}) & ⊗ & ⋯ & ⊗ & H2_{𝕊^{1}}(𝕊^{2n−1}, A_{k}) & ⟶^{⌣} & H2k_{𝕊^{1}}(𝕊^{2n−1}, 𝕊^{2n−1}) \\ ↓^{j_{1}^{∗}} & & & & ↓^{j_{k}^{∗}} & & ↓^{j^{∗}} \\ H2_{𝕊^{1}}(𝕊^{2n−1}) & ⊗ & ⋯ & ⊗ & H2_{𝕊^{1}}(𝕊^{2n−1}) & ⟶^{⌣} & H2k_{𝕊^{1}}(𝕊^{2n−1}) \end{matrix} (68)

so that j^{∗} = 0. Thus, we find α^{k} = j^{∗}(α_{1} ⌣ … ⌣ α_{k}) = 0 in contradiction to the choice of α which implies α^{k} ≠ 0 for k = 0, …, n−1. Thus the proof of claim 3 is complete.

As a side remark, we point out that indeed 𝕊^{1}-cat(𝕊^{2n−1}) = Index_{ℂ} 𝕊^{2n−1} = n, cf. [10, Cor. 7.8]. The estimate (62) now follows from (63), (65), and (66). The proof of the proposition is with this complete. □

7. Lyusternik-Schnirelman theory for 𝔐^{+} and the proof of Theorem 1 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} := { x ∈ 𝔐^{+} | grad F(x) = 0 and F(x) = c_{i} } . (69)

Then (M_{1}, …, M_{k}) is an admissibly ordered Morse decomposition of the positive subcomplex 𝔐^{+}. Consequently, we obtain

Proposition 4 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_{N}(𝔐^{+}) ⩽ ∑_{j=1}^{k} 𝕊^{1}-cat_{N}(M_{j}) . (70)

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

Proof Let us first consider an admissibly ordered Morse decomposition (𝔐_{1}, 𝔐_{2}) of 𝔐^{+} consisting of only two Morse sets. If 𝕊^{1}-cat_{N}(𝔐_{j}) = ℓ_{j}, j = 1, 2, then there exist ℓ_{1}+ℓ_{2} closed 𝕊^{1}-categorical subsets A_{j,i} ⊂ N with 𝔐_{j} ⊂ int (A_{j,1}∪ … ∪A_{j,ℓ_{j}}). In view of the compactness of 𝔐^{+} there exists τ_{0} ∈ [ 0, +∞) such that for τ ⩾ τ_{0} we have

𝔐^{+} ⊂ ⋃_{i=1}^{ℓ_{1}} A_{1,i} ∪ ⋃_{i=1}^{ℓ_{2}} ((A_{2,i}·τ) ∩ N) . (71)

Thus 𝔐^{+} can be covered by ℓ_{1} +ℓ_{2} closed 𝕊^{1}-categorical subsets of N and, therefore, 𝕊^{1}-cat_{N}(𝔐^{+}) ⩽ 𝕊^{1}-cat_{N}(𝔐_{1}) + 𝕊^{1}-cat_{N}(𝔐_{2}).

Because the Morse sets 𝔐_{1} and 𝔐_{2} are again isolated invariant sets, the same argument can now be applied to Morse decompositions of 𝔐_{1} and 𝔐_{2}. Continuing this process, the desired Morse decomposition (M_{1}, …, M_{k}) of 𝔐^{+} is reached after a finite number of repetitions. This proves the inequality (70).

Suppose that 𝕊^{1}-cat_{N}(M_{j}) = ℓ and that A_{1}, …, A_{ℓ} ⊂ N are closed 𝕊^{1}-categor-ical subsets covering M_{j}. Then M_{j} contains at least ℓ critical 𝕊^{1}-orbits Orb x_{i} ⊂ A_{i}, i = 1, …, ℓ, which belong to mutually distinct path components of the orbit space N/𝕊^{1}. This follows immediately from the observation that a finite set of 𝕊^{1}-orbits which belongs to the same path component of N/𝕊^{1} must have the 𝕊^{1}-category 1. The assertion of the proposition is thereby proven. □

Proposition 4 combined with Proposition 3 shows that 𝔐^{+} contains at least n distinct 𝕊^{1}-orbits Orb x^{i} such that grad F(x^{i}) = 0, which proves Proposition 2. Consequently, the proof of Theorem 1 is complete. □

8. The proofs of Theorem 2 and Corollary 1 Let f : ℂP^{n−1} → ℝ be a smooth function and let f̄: 𝕊^{2n−1} → ℝ denote a lift of f, according to f̄(x) := f(Orb x). The associated 𝕊^{1}-hypersurface Σ_{f} is then given by

Σ_{f} = { e^{f̄(x)} x | x ∈ 𝕊^{2n−1} } ⊂ ℝ^{2n} , (72)

where 𝕊^{2n−1} stands for the Euclidean unit sphere in ℝ^{2n}. Then Σ_{f} is star-shaped since every radius { μ x | μ ⩾ 0, x ∈ 𝕊^{2n−1} } intersects Σ_{f} transversally. The transversality is seen as follows: if x ∈ 𝕊^{2n−1} and b_{1}, …, b_{2n−1} ∈ T_{x}𝕊^{2n−1} is a basis, then a corresponding basis of T_{e^{f̄(x)}x}Σ_{f} is obtained via tangent map of the transformation 𝕊^{2n−1} → Σ_{f}, i.e., by the map

b_{i} ↦ e^{f̄(x)} (b_{i} + ⟨ grad f̄(x), b_{i} ⟩ x) . (73)

Thus the flow (x, s) ↦ e^{s/s} x of the symplectic dilation X_{λ}(x) = \frac{1}{2} x can be used to introduce the smooth Hamiltonian function H_{f} : ℝ^{2n}\{0} → ℝ by setting

H_{f}(e^{s/2} x) := e^{s} ∀ x ∈ Σ_{f}, s ∈ ℝ , (74)

Note that H_{f} is positively homogeneous of degree 2 and its gradient is positively homogeneous of degree 1, i.e., H_{f}(μx) = μ^{2}H_{f}(x) and grad H_{f}(μx) = μ grad H_{f}(x) for all x ∈ ℝ^{2n−1}\{0} and μ > 0. In particular, we deduce

H_{f}(e^{f̄(x)} x) = e^{2f̄(x)} H_{f}(x) = 1 ∀ x ∈ 𝕊^{2n−1} , (75)

so that f̄ can be expressed by

f̄(x) = −\frac{1}{2} ln H_{f}(x) , x ∈ 𝕊^{2n−1} . (76)

Then the gradient of f̄ satisfies the identity

(77) ⟨ grad f̄(x), X ⟩ = \frac{−1}{2H_{f}(x)} ⟨ grad H_{f}(x), X ⟩ ∀ X ∈ T_{x}𝕊^{2n−1} .

Hence it follows that grad f̄(x) = 0 if and only if x and grad H_{f}(x) are collinear. In this case we have γ̇_{t} x = \frac{2π}{τ} Jγ_{t} x = μ X_{H_{f}} (γ_{t} x) for some μ > 0 and, consequently, the X_{H_{f}} -orbit starting at x can be reparametrized as an 𝕊_{τ}^{1}-orbit. Finally, by the homogeneity of grad H_{f}, it follows that the map 𝕊_{τ}^{1} → Σ_{f} given by t ↦ e^{f(Orb x)} γ_{t} x is a symmetric closed characteristic on Σ_{f}.

Thus the distinct symmetric closed characteristics on Σ_{f} correspond uniquely to the distinct 𝕊^{1}-orbits Orb e^{f̄(x)} x such that grad f̄(x) = 0. On the other hand, the associated distinct 𝕊^{1}-orbits Orb x on 𝕊^{2n−1} represent precisely the distinct critical points of the function f : ℂP^{n−1} → ℝ.

Let now Σ ⊂ ℝ^{2n} be any star-shaped 𝕊^{1}-hypersurface. Setting H(e^{s/2} x) := e^{s} for all (x, s) ∈ Σ × ℝ, we define f : ℂP^{n−1} → ℝ by f(Orb x) := −\frac{1}{2} ln H(x) for all x ∈ 𝕊^{2n−1}. Then we have Σ_{f} = Σ, and the proof of Theorem 1 is complete. □

For the proof of the corollary we use the above notation and we assume that Orb x is a critical point of f, i.e., x ∈ 𝕊^{2n−1} is a critical point of f̄. Then for every s ∈ ℝ the outward normal vector field at e^{s/2}e^{f̄(x)} x ∈ H_{f}^{−1}(e^{s}) satisfies

N(e^{s/2}e^{f̄(x)} x) = x . (78)

Using the symplectic dilation X_{λ}, the gradient of H_{f} at this point is determined by the formula (28). Thus we obtain the expression

grad H_{f}(e^{s/2}e^{f̄(x)} x) = \frac{2 e^{s/2}}{e^{f̄(x)}} x . (79)

Now let the corresponding 𝕊^{1}-orbit u(t) := e^{s/2}e^{f̄(x)} exp (\frac{2π}{τ(x)} tJ) x be parametrized as an integral curve of X_{H_{f}}, i.e., let −Ju̇ = grad H_{f}(u). This is equivalent to

\frac{2π}{τ(x)} e^{s/2}e^{f̄(x)}x = \frac{2 e^{s/2}}{e^{f̄(x)}} x . (80)

Hence the minimal period is determined by

τ(x) = π e^{2f̄(x)} , (81)

such that the corollary follows with ϱ = e^{s/2}. □

9. Proof of Theorem 3 Suppose that Σ ⊂ ℝ^{2n} \{0} is a compact connected 𝕊^{1}-hypersurface which carries a non-symmetric closed characteristic embedding u : 𝕊_{τ}^{1} → Σ. We may consider u as a periodic X_{H}-orbit having the minimal period τ > 0 where H is again the Hamiltonian function from (20). Then the period τ yields a natural choice for the parametrization of the 𝕊^{1}-action γ, i.e., we shall assume γ_{t} x = exp (\frac{2π}{τ} tJ) x.

Denoting the flow of X_{H} by φ_{t} and setting x_{0} = u(0), we obtain the representation u(t) = φ_{t}(x_{0}). Moreover, since X_{H} is 𝕊^{1}-equivariant, it follows that φ_{t}(γ_{s} x_{0}) = γ_{s} φ_{t}(x_{0}) for all s, t ∈ 𝕊_{τ}^{1}. Thus the τ-periodic integral curve u(t) of X_{H} gives rise to an entire family { γ_{s} φ_{t}(x_{0}) | s ∈ 𝕊_{τ}^{1} } of periodic integral curves having the minimal period τ.

Two of these integral curves will be called equivalent if they describe the same geometric X_{H}-orbit on Σ, i.e., if they are related to each other by reparametrization. More precisely, we define φ_{t}(γ_{s} x_{0}) ∼ φ_{t}(γ_{s′} x_{0}) in case there exists t′ ∈ 𝕊_{τ}^{1} such that φ_{t}(γ_{s′} x_{0}) = φ_{t+t′}(γ_{s} x_{0}). We show that, for u non-symmetric, an equivalence class consists of finitely many elements only. Hence there exist uncountably many distinct equivalence classes and, therefore, uncountably many geometrically distinct periodic X_{H}-orbits of the form γ_{s} φ_{t}(x_{0}) as asserted.

Since u is supposed to be non-symmetric, the trajectory t ↦ φ_{t}(γ_{s} x_{0}) intersects the 𝕊^{1}-orbit Orb x_{0} tranversally at γ_{s} x_{0}. Hence we infer

t_{1} := min { t ∈ (0, τ ] | φ_{t}(γ_{s} x_{0}) ∈ Orb x_{0} } > 0 . (82)

The 𝕊^{1}-equivariance of the flow φ implies that t_{1} is independent of the particular choice of s and hence we may fix s = 0 for the moment. Consequently, there exists ϑ ∈ (0, τ] such that φ_{t_{1}}(x_{0}) = γ_{ϑ} x_{0}. By induction it follows now that φ_{mt_{1}}(x_{0}) = γ_{mϑ} x_{0} for every m ∈ ℕ.

We claim that τ = mt_{1} for some m ∈ ℕ. Else there exists a natural number m such that (m−1) t_{1} < τ < mt_{1} < τ + t_{1}. Setting t′_{1} := mt_{1} − τ, we get 0 < t′_{1} < t_{1} and φ_{t′_{1}}(x_{0}) = φ_{mt_{1}}(φ_{−τ}(x_{0})) = φ_{mt_{1}}(x_{0}) = γ_{mt_{1}}x_{0} ∈ Orb x_{0} , (83)

in contradiction to the minimality of t_{1}. Thus we have mt_{1} = τ as claimed. Now we conclude γ_{mϑ} x_{0} = φ_{mt_{1}}(x_{0}) = φ_{τ}(x_{0}) = x_{0} , (84)

which implies ϑ = t_{1} = τ/m. Recalling that x_{0} may be replaced by γ_{s} x_{0} for s ∈ 𝕊_{τ}^{1}, we finally infer that φ_{t+t′}(γ_{s} x_{0}) = φ_{t}(γ_{s′} x_{0}) (85)

holds true for t′, s′−s in the fundamental domain [ 0, τ) of 𝕊_{τ}^{1} = ℝ/τℤ if and only if t′ = s−s′ = kt_{0} for some k = 0, …, m−1. In other words, the equivalence class of the integral curve φ_{t}(γ_{s} x_{0}) is given by { φ_{t}(γ_{s+kt_{0}}x_{0}) | k = 0, …, m−1 }. The proof is herewith complete. □

Acknowledgement The first author expresses his gratitude to the Mathematische Institut at the Universität Tübingen for its hospitality and co-operation. The second author acknowledges the hospitality and support of the Laboratoire de Topologie et Géométrie of the Université Paul Sabatier, Toulouse, and the inspiring discussions with Jean-Claude Sikorav.

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[15] Rabinowitz, P.H. : The prescribed energy problem for periodic solutions of Hamiltonian systems. Contemporary Mathematics 81 (1988) 183-191

[16] Viterbo, C. : A proof of Weinstein’s conjecture in ℝ^{2n}. Ann. Inst. H. Poincaré, Anal. Non Linéaire 4 (1987) 337-356

[17] Weinstein, A. : On the hypotheses of Rabinowitz’ periodic orbit theorems. J. Differential Equations 33 (1979) 353-358

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

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