THE PALAIS-SMALE CONDITION AND MAÑÉ’S CRITICAL VALUES

GONZALO CONTRERAS, RENATO ITURRIAGA, GABRIEL P. PATERNAIN, AND MIGUEL PATERNAIN

Abstract. Let 𝕃 be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [22]. We define energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of 𝕃 to any covering of N equals the infimum of the values of k such that the energy level t satisfies the Palais-Smale condition for every t > k provided that the Peierls barrier is finite. When the static set is not empty, the Peierls barrier is always finite and thus we obtain a characterization of the critical value of 𝕃 in terms of the Palais-Smale condition.

We also show that if an energy level without conjugate points has energy strictly bigger than c_{u}(𝕃) (the critical value of the lift of 𝕃 to the universal covering of N), then two different points in the universal covering can be joined by a unique solution of the Euler-Lagrange equation that lives in the given energy level. Conversely, if the latter property holds, then the energy of the energy level is greater than or equal to c_{u}(𝕃). In this way, we obtain a characterization of the energy levels where an analogue of the Hadamard theorem holds. We conclude the paper showing other applications such as the existence of minimizing periodic orbits in every homotopy class with energy greater than c_{u}(𝕃) and homologically trivial periodic orbits such that the action of 𝕃 + k is negative if c_{u}(𝕃) < k < c_{a}(𝕃), where c_{a}(𝕃) is the critical value of the lift of 𝕃 the abelian covering of N. We also prove that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange flow in the given energy level.

1. Introduction

In this paper we study geometric and dynamical properties of convex and superlinear Lagrangians, and it can be considered as a continuation of our previous paper [9]. This time we study the action functional from the viewpoint of Morse theory and we show, among other results, that for a compact manifold the critical value as defined by R. Mañé in [22] can be characterized by the Palais-Smale condition. This will follow from more general results to be precisely stated below.

Date: 1999. Gonzalo Contreras was partially supported by a CONACYT-México grant # 28489-E and by CNPq-

Brazil.

Renato Iturriaga was partially supported by a CONACYT-México grant # 28489-E. Miguel Paternain was partially supported by a CSIC-Uruguay grant. Gabriel Paternain is on leave from Centro de Matemática, Facultad de Ciencias, Iguá 4225, 11400

Montevideo, Uruguay.

It is well known that the action functional of the Lagrangian

L(x, v) = \frac{1}{2}|v|_{x}^{2}

arising from a Riemannian metric satisfies the Palais-Smale condition (cf. [28]). This condition ensures that the minimax principle holds and from the latter many standard properties of geodesics easily follow, namely, The Hopf-Rinow theorem, the Hadamard theorem and the existence of closed geodesics in each homotopy class. As we explain below, these and other properties as well as the approach in [28] and [30] hold for energy levels of convex and superlinear autonomous Lagrangians if the energy is greater than the critical value.

In order to describe precisely our results let us recall some preliminaries.

Let N be a closed connected smooth manifold and let 𝕃 : T N → ℝ be a smooth convex superlinear Lagrangian. This means that 𝕃 restricted to each T_{x}N has positive definite Hessian and that for some Riemannian metric we have that

lim_{|v|→∞} \frac{𝕃(x, v)}{|v|} = ∞,

uniformly on x ∈ N.

Since N is compact, the extremals of 𝕃 give rise to a complete flow ϕ_{t} : T N → T N called the Euler-Lagrange flow of the Lagrangian.

Recall that the energy 𝔼_{𝕃} : T N → ℝ is defined by

𝔼_{𝕃}(x, v) = \frac{∂𝕃}{∂v}(x, v).v − 𝕃(x, v).

Since 𝕃 is autonomous, 𝔼_{𝕃} is a first integral of the flow ϕ_{t}.

Recall also that the action of the Lagrangian 𝕃 on an absolutely continuous curve γ : [a, b] → N is defined by

S_{𝕃}(γ) = ∫_{a}^{b} 𝕃(γ(t), γ̇(t)) dt.

Given two points, q_{1} and q_{2} in N and T > 0 denote by C(q_{1}, q_{2}; T) the set of absolutely continuous curves γ : [0, T] → N, with γ(0) = q_{1} and γ(T) = q_{2}. For each k ∈ ℝ we define the action potential Φ_{k} : N × N → ℝ by

Φ_{k}(q_{1}, q_{2}) = inf{S_{𝕃+k}(γ) : γ ∈ ∪_{T >0}C(q_{1}, q_{2}; T)}.

The critical value of 𝕃, which was introduced by Mañé in [22], is the real number c(𝕃) defined as the infimum of k ∈ ℝ such that for some q ∈ N, Φ_{k}(q, q) > −∞. Since 𝕃 is convex and superlinear and N is compact such a number exists and it has various important properties [22, 6]. We briefly mention a few of them since we shall need them below. For any k ≥ c(𝕃), the action potential Φ_{k} is a Lipschitz function that satisfies a triangle inequality. In general the action potential is not symmetric but if we define d_{k} : N × N → ℝ by setting

d_{k}(q_{1}, q_{2}) = Φ_{k}(q_{1}, q_{2}) + Φ_{k}(q_{2}, q_{1}),

then d_{k} is a distance function for all k > c(𝕃) and a pseudo-distance for k = c(𝕃). In [22, 6] the critical value is characterized in other ways relating it to minimizing measures or to the existence of Tonelli minimizers with fixed energy between two points.

We can also consider the critical value of the lift of the Lagrangian 𝕃 to a covering of the compact manifold N. Suppose that p : M → N is a covering space and consider the Lagrangian L : T M → ℝ given by L := 𝕃 ∘ dp. For each k ∈ ℝ we can define an action potential Φ_{k} in M × M just as above and similarly we obtain a critical value c(L) for L. It can be easily checked that if M_{1} and M_{2} are coverings of N such that M_{1} covers M_{2}, then

c(L_{1}) ≤ c(L_{2}), (1)

where L_{1} and L_{2} denote the lifts of the Lagrangian 𝕃 to M_{1} and M_{2} respectively.

Among all possible coverings of N there are two distinguished ones; the universal covering which we shall denote by Ñ, and the abelian covering which we shall denote by \overline{N}. The latter is defined as the covering of N whose fundamental group is the kernel of the Hurewicz homomorphism π_{1}(N) ↦ H_{1}(N, ℝ). When π_{1}(N) is abelian, Ñ is a finite covering of \overline{N}.

The universal covering of N gives rise to the critical value

c_{u}(𝕃) =^{def} c(lift of 𝕃 to Ñ),

and the abelian covering of N gives rise to the critical value

c_{a}(𝕃) =^{def} c(lift of 𝕃 to \overline{N}).

From inequality (1) it follows that

c_{u}(𝕃) ≤ c_{a}(𝕃),

but in general the inequality may be strict as it was shown in [29].

The critical values have another important feature: they single out those energy levels in which relevant globally minimizing objects (orbits or measures) live [10, 22, 6]. The study of these globally minimizing objects has a long history that goes back to M. Morse [26] and G.A. Hedlund [18]. Recent work on this subject has been done by V. Bangert [3, 4], M.J. Dias Carneiro [10], A. Fathi [13, 14, 15, 16], R. Mañé [22, 23] and J. Mather [24, 25]. We refer to [8, 17] for comprehensive accounts of the theory. Static and semistatic curves are the paradigms of what we mean by globally minimizing orbits and since they will play an important role in our results we give now their definition (cf. [22, 25]).

Set c = c(L). We say that x : [a, b] → M is a semistatic curve if it is absolutely continuous and:

S_{L+c} (x|_{[t_{0},t_{1}]}) = Φ_{c} (x(t_{0}), x(t_{1})) , (2)

for all a < t_{0} ≤ t_{1} < b; and that it is a static curve if

S_{L+c} (x|_{[t_{0},t_{1}]}) = −Φ_{c}(x(t_{1}), x(t_{0})) (3)

for all a < t_{0} ≤ t_{1} < b. Observe that since

Φ_{c}(x(t_{0}), x(t_{1})) + Φ_{c}(x(t_{1}), x(t_{0})) = d_{c}(x(t_{0}), x(t_{1})) ≥ 0

a static curve is a semistatic curve for which d_{c}(x(t_{0}), x(t_{1})) = 0 for all a < t_{0} ≤ t_{1} < b. Let Σ̂(L) be the set of vectors v ∈ T M such that the solutions γ : ℝ → M of the Euler-Lagrange equation satisfying γ̇(0) = v are static. We call Σ̂(L) the static set.

As we mentioned before, one of our aims in this paper is to relate the Morse theory of the action functional to the critical values. Let H^{1}(ℝ^{k}) be the set of absolutely continuous curves x : [0, 1] → ℝ^{k} such that

∫_{0}^{1} |ẋ(t)|^{2} dt < ∞.

It is well known that H^{1}(ℝ^{k}) is a Hilbert space with the inner product defined by

⟨x, y⟩_{1} = ⟨x(0), y(0)⟩_{ℝ^{k}} + ∫_{0}^{1} ⟨ẋ(t), ẏ(t)⟩_{ℝ^{k}} dt,

where ⟨ · , · ⟩_{ℝ^{k}} is the standard inner product of ℝ^{k}.

Given any Riemannian metric of M we may assume, on account of the Nash embedding theorem, that M is isometrically embedded in some ℝ^{k}. Take q_{1} and q_{2} in M and let Ω(q_{1}, q_{2}) be the set of elements of H^{1}(ℝ^{k}) such that x([0, 1]) ⊂ M, x(0) = q_{1} and x(1) = q_{2}. It follows from the arguments in [28] that Ω(q_{1}, q_{2}) inherits a Hilbert manifold structure compatible with the Riemannian metric on M. We now define another action A_{L} closely related to the action S_{L} we defined before. Given the Lagrangian L : T M → ℝ define

A_{L} : ℝ^{+} × Ω(q_{1}, q_{2}) → ℝ

by

A_{L}(b, x) = ∫_{0}^{1} b L(x(t), ẋ(t)/b) dt.

Observe that

A_{L}(b, x) = S_{L}(y),

where y(t) = x(t/b).

We now recall the definition of the Palais-Smale condition. In fact, this is a rather stronger version of the condition in [28] and [30] that we borrow from [19] and [21].

Definition 1. Let f : X → ℝ be a C^{1} map where X is an open set of a Hilbert manifold. We say that f satisfies the Palais-Smale condition if every sequence {x_{n}} such that {f(x_{n})} is bounded and ||d_{x_{n}}f|| → 0 as n → ∞ has a converging subsequence.

We remark that the manifold ℝ^{+} × Ω(q_{1}, q_{2}) is not complete, however if q_{1} ≠ q_{2} then the set {A_{L+k} ≤ b} is complete when k is strictly bigger that the critical value. (see Lemma 15). This is important when we apply the minimax principle (see Proposition 21).

In order to show that the action functional A_{L+k} is C^{2} and satisfies the Palais-Smale condition we need the Lagrangian to be quadratic at infinity:

Definition 2. We say that 𝕃 : T N → ℝ is quadratic at infinity if there exists a Riemannian metric on N and R > 0 such that for each x ∈ N and |v|_{x} > R, 𝕃(x, v) has the form

𝕃(x, v) = \frac{1}{2}|v|_{x}^{2} + θ_{x}(v) − V (x),

where θ is a smooth 1-form on N and V : N → ℝ a smooth function.

A lifted Lagrangian L : T M → ℝ is said to be quadratic at infinity if it is the lift of a Lagrangian quadratic at infinity on N .

In Section 3 (cf. Proposition 18) we shall show that given a Lagrangian L and k ∈ ℝ there is a Lagrangian L_{0} quadratic at infinity such that L(x, v) = L_{0}(x, v) for all (x, v) with E(x, v) ≤ k + 1. We shall also show (cf. Lemma 19) that given two Lagrangians L and L_{0} which agree for any (x, v) with E(x, v) ≤ c(L) + 1, then c(L) = c(L_{0}). These properties motivate the following definition:

Definition 3. We say that the energy level E^{−1}(k) of a convex and superlinear Lagrangian L satisfies the Palais-Smale condition if there is a Lagrangian L_{0} quadratic at infinity such that L and L_{0} agree for any (x, v) with E(x, v) ≤ k + 1 and the action functional A_{L_{0}+k} satisfies the Palais-Smale condition on ℝ^{+} × Ω(q_{1}, q_{2}) for every q_{1} ≠ q_{2}.

In Section 3 we shall prove:

Theorem A. Let M be any covering of the closed manifold N and let L : T M → ℝ be the lifted Lagrangian. If the static set Σ̂(L) is not empty, then

c(L) = inf {t ∈ ℝ : E_{L}^{−1}(k) satisfies the Palais-Smale condition for every k > t} .

When M is compact the set Σ̂(L) is not empty [6, 22] and therefore we obtain:

Corollary. Suppose that M is compact. Then

c(L) = inf {t ∈ ℝ : E_{L}^{−1}(k) satisfies the Palais-Smale condition for every k > t} .

S. Bolotin in [5] explores ideas which are similar to the ones we develop here. Using a somewhat different language he also notes that the Palais-Smale condition holds for the action A_{L+k} for large values of k but he does not give a characterization of c(L) as before.

Theorem A will follow from Theorems B and C below. To state these theorems we need two more definitions. For k ∈ ℝ, let

Φ_{k}(q_{1}, q_{2}; T) := inf_{γ∈C(q_{1},q_{2};T)} S_{L+k}(γ)
h_{k}(q_{1}, q_{2}) := lim inf_{T →+∞} Φ_{k}(q_{1}, q_{2}; T).

The function h_{c} is known as the Peierls barrier [14, 25].

Theorem B. Assume that L is quadratic at infinity and h_{k}(q_{1}, q_{2}) = +∞. Then the action functional

A_{L+k} : ℝ^{+} × Ω(q_{1}, q_{2}) → ℝ

satisfies the Palais-Smale condition provided q_{1} ≠ q_{2}.

Theorem C. Assume that L is quadratic at infinity and that for some pair (q_{1}, q_{2}), h_{c}(q_{1}, q_{2}) < +∞. Then the action functional

A_{L+c} : ℝ^{+} × Ω(q_{1}, q_{2}) → ℝ

does not satisfy the Palais-Smale condition.

At the end of Section 3 we explain in detail how to derive Theorem A from Theorems B and C. Two extra ingredients needed for this derivation are given by Corollary 12 in Section 3, which states that the Peierls barrier is finite for every pair (q_{1}, q_{2}) in M when the static set Σ̂(L) is not empty and Lemma 10 which states that h_{k}(q_{1}, q_{2}) = +∞ for every pair (q_{1}, q_{2}) provided that k > c(L). We remark that if h_{c}(q_{1}, q_{2}) is finite for some pair (q_{1}, q_{2}) then it is finite for all pairs (q_{1}, q_{2}).

In the appendix we give an example of a Lagrangian on ℝ^{2} for which the static set is empty and the Peierls barrier h_{c} is infinite (and hence the Palais-Smale condition holds at critical energy). Even though this Lagrangian is not the lift of a Lagrangian on a compact manifold, it shows that most likely Theorems A, B and C are optimal.

It is unknown whether the energy level E_{L}^{−1}(k) satisfies the Palais-Smale condition for k < c(L) (some authors have assumed that the Palais-Smale condition holds at subcritical energies for magnetic Lagrangians and this gap has been pointed out by S. Bolotin, see [32] for a discussion of the problem).

In Section 4 we prove an analogue of the Hadamard theorem on fixed energy levels. A pair of points (x_{1}, v_{1}), (x_{2}, v_{2}) in T M are said to be conjugate if (x_{2}, v_{2}) = ϕ_{t}(x_{1}, v_{1}) for t ≠ 0 and dϕ_{t}(V (x_{1}, v_{1})) intersects V (x_{2}, v_{2}) non-trivially. Here, V (x, v) is the vertical fibre at (x, v) defined as usual as the kernel of dπ_{(x,v)} : T_{(x,v)}T N → T_{x}N where π : T N → N is the canonical projection. We have:

Theorem D. Let 𝕃 : T N → ℝ be a convex and superlinear Lagrangian. Assume that there is q_{0} ∈ N such that q_{0} has no conjugate points in E_{𝕃}^{−1}(k). Let q̃_{0} be a lift of q_{0} to Ñ, the universal covering of N and let L be the lift of 𝕃 to Ñ. Denote the following statement by (H):

(H) For every q̃ in Ñ there is a unique solution of the Euler-Lagrange equation of L

with energy k, joining q̃_{0} to q̃. Then,

k > c_{u}(𝕃) ⟹ (H)
(H) ⟹ k ≥ c_{u}(𝕃).

We note that when k = c_{u}(𝕃) there are examples in [7] where (H) does not hold. Also there are examples in [7] of multivalued Lagrangians, that become honests Lagrangians in the universal covering for which (H) does hold.

In Section 5 we recall some results on Morse theory that we will use in the last section. In Section 6 we give applications such as the existence of minimizing periodic orbits in every non-trivial free homotopy class with energy greater than c_{u}(𝕃) and homologically trivial periodic orbits such that the action of 𝕃 + k is negative if c_{u}(𝕃) < k < c_{a}(𝕃), where c_{a}(𝕃) is the critical value of the lift of 𝕃 to the abelian covering of N. These results should be compared with the work of S.P. Novikov, I. Taimanov and A. Bahri and I. Taimanov on the existence of closed orbits for magnetic Lagrangians. See [2, 27, 32, 33] and the extensive references therein. We also prove in Section 5 that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange flow in the given energy level.

Acknowledgments: M. Paternain thanks the CIMAT, Guanajuato, for hospitality while this work was in progress.

2. First and Second Variations

In this section we calculate the first and second variations of the action functional A_{L+k}. These computations do not need any assumptions on the Lagrangian. However, if we want them to be the first and second derivative of the action functional A_{L+k} we need the Lagrangian to be quadratic at infinity.

Take a curve s → (b_{s}, x_{s}) ∈ ℝ^{+} × Ω(q_{1}, q_{2}) and set b := b_{0} , x := x_{0}, ξ(t) := \frac{∂x_{s}}{∂s} |_{s=0}(t), α := \frac{db_{s}}{ds} |_{s=0} and g(s) := A_{L+k}(b_{s}, x_{s}). A straightforward calculation in local coordinates gives:

Lemma 4.

d_{(b,x)}A_{L+k}(α, ξ) = g′(0) = α ∫_{0}^{1}{k − E_{L}(x(t), ẋ(t)/b)} dt
+ ∫_{0}^{1}{bL_{x}(x(t), ẋ(t)/b)ξ(t) + L_{v}(x(t), ẋ(t)/b)ξ̇(t)} dt.

Remark 5. If (b, x) is a critical point of A_{L+k}, then y : [0, b] → M given by y(t) = x(t/b) is a solution of the Euler-Lagrange equation of L with energy k (see [1] and [6]). Indeed, the second term in the last equation is equal to

∫_{0}^{b}(L_{x}(y, ẏ) − \frac{d}{dt}L_{v}(y, ẏ)) η dt ,

where η(t) := ξ(t/b). Since it is zero for all variations η, then y satisfies the Euler-Lagrange equation. Since the first term is zero for α = 1, then E_{L}(y, ẏ) ≡ k.

Remark 6.

\frac{∂A_{L+k}}{∂b} (α) = α ∫_{0}^{1}\{k − E_{L}(x(t), ẋ(t)/b)\} dt,

= \frac{α}{b} ∫_{0}^{b}\{k − E_{L}(y(s), ẏ(s))\} ds,

where y(s) = x(s/b).

Lemma 7.

\frac{∂^{2}A_{L+k}}{∂^{2}x} (ξ, ξ) = ∫_{0}^{b} \{ηL_{x}(y, ẏ)η + ηL_{xv}(y, ẏ)η̇ + η̇L_{vx}(y, ẏ)η + η̇L_{vv}(y, ẏ)η̇\} dt,

where η(t) = ξ(t/b) and y(t) = x(t/b).

Proof . Calculate g′′(0) where g(s) = A_{L+k}(b, x_{s}).

The following lemma is an immediate consequence of the Morse Index Theorem for convex Lagrangians (cf. [11]).

Lemma 8. Let y, (b, x) be as in the previous lemma. If in addition y is a solution of the Euler-Lagrange equation with no conjugate points, then \frac{∂^{2}A_{L+k}}{∂^{2}x} is positive definite.

The following theorem is a particular case of a theorem due to Smale [30].

Theorem 9. If a Lagrangian L is quadratic at infinity then the corresponding action functional A_{L+k} : ℝ^{+} × Ω(q_{1}, q_{2}) → ℝ is a C^{2} function. Moreover, the first and second variations described above are the first and second derivatives of A_{L+k}.

Lemma 8 motivates the following:

Question: Is it true that (b, x) is a local minimum of A_{L+k} provided that x(t/b) is a solution of the Euler-Lagrange equation with no conjugate points?

The next example shows that the answer to this question is negative. This example was motivated by the referee who pointed out a mistake in a previous version of the mansucript. We thank him or her for this observation.

Let L be the Lagrangian on ℝ^{2} given by:

L(x, y, ẋ, ẏ) = \frac{1}{2}(ẋ^{2} + ẏ^{2}) − xẏ.

We take k = 1/2, q_{1} = (1, 0) and q_{2} = (−1, 0). The orbits of L with energy 1/2 are circles of radius one oriented counterclockwise. Let c_{0} be the half circle of radius one connecting q_{1} to q_{2}. Let (−ε, ε) ∋ s ↦ c_{s} be a small variation of c_{0} by circles with center at (0, s) and radius \sqrt{1 + s^{2}}. We parametrize these circles by arc length, hence c_{s} connects q_{1} to q_{2} in time b(s) = \sqrt{1 + s^{2}}(π + 2 a(s)), where a(s) is the angle in (−π/2, π/2) whose tangent is s. By observing that

(L + \frac{1}{2})(x, y, ẋ, ẏ) = 1 − x ẏ,

when ẋ^{2} + ẏ^{2} = 1 one obtains:

A(s) := A_{L+1/2}(b_{s}, x_{s}) = S_{L+1/2}(c_{s}) = b(s) − (1 + s^{2}) (\frac{π}{2} + a(s) + \frac{1}{2} sin(2a(s))) .

A somewhat tedious computation shows that:

A′(0) = 0,  A′′(0) = 0,  A′′′(0) = −2.

Hence s ↦ A(s) does not have a local minumum at s = 0. On the other hand the piece of orbit [0, π] ∋ s ↦ c_{0}(s) does not have conjugate points [8, Example A.3]. Finally we observe that if we consider a ball in ℝ^{2} with radius, let us say, four, then L restricted to this ball can be embedded into a convex superlinear Lagrangian on a closed surface.

3. The Palais-Smale condition for Lagrangians quadratic at infinity

3.1. The Peierls barrier. For k ∈ ℝ, let

Φ_{k}(q_{1}, q_{2}; T) := inf_{γ∈C(q_{1},q_{2},T)} S_{L+k}(γ)
h_{k}(q_{1}, q_{2}) := lim inf_{T →+∞} Φ_{k}(q_{1}, q_{2}; T).

The function h_{c} is called the Peierls barrier [14].

Lemma 10. Set c := c(L).

If k > c then h_{k}(q_{1}, q_{2}) = +∞ for all q_{1}, q_{2} ∈ M.

If k < c then h_{k}(q_{1}, q_{2}) = −∞ for all q_{1}, q_{2} ∈ M.

Proof. If k > c, we have that

Φ_{k}(q_{1}, q_{2}; T) ≥ Φ_{c}(q_{1}, q_{2}) + (k − c) T.

Hence h_{k}(q_{1}, q_{2}) = +∞.

If k < c, since Φ_{k}(q_{2}, q_{2}) = −∞, there is a curve γ ∈ C(q_{2}, q_{2}; T) with T > 0 and S_{L+k}(γ) < 0. Then

Φ_{k}(q_{1}, q_{2}; 1 + n T) ≤ Φ_{k}(q_{1}, q_{2}; 1) + n S_{L+k}(γ) ⟶^{n} −∞.

Thus h_{k}(q_{1}, q_{2}) = −∞.

Proposition 11. h_{c}(p, p) = 0 iff p ∈ πΣ̂(L).

Proof. First take p ∈ πΣ̂(L) and let γ : ℝ → M be a static curve with γ(0) = p. Let ε > 0 be given. For any t > 0 there is a curve γ_{t} : [0, T_{t}] → M such that γ_{t}(0) = γ(t), γ_{t}(T_{t}) = γ(0) and

S_{L+c}(γ_{t}) ≤ Φ_{c}(γ(t), p) + ε.

Then

0 ≤ h_{c}(p, p) ≤ lim inf_{t→+∞} S_{L+c}(γ|_{[0,t]} ∗ γ_{t})

≤ lim inf_{t→+∞} (S_{L+c}(γ|_{[0,t]}) + S_{L+c}(γ_{t}))

≤ lim inf_{t→∞} (Φ_{c}(p, γ(t)) + Φ_{c}(γ(t), p)) + ε

= ε,

where the last equality holds because γ is a static curve.

Now assume that h_{c}(p, p) = 0. Then there is T_{n} → ∞ and Tonelli minimizers γ_{n} : [0, T_{n}] → M with γ_{n}(0) = γ_{n}(T_{n}) = p and S_{L+c}(γ_{n}) → 0. By Lemma 16 the speed of the Tonelli minimizers γ_{n} is bounded and hence there is a subsequence such that γ̇_{n}(0) → v. Let γ be the solution of the Euler-Lagrange equation with initial conditions (p, v). We are going to show that γ is static. Take 0 < t_{1} < t_{2} and ε > 0. Take n so big that T_{n} > t_{2},

S_{L+c}(γ_{n}) < ε,
Φ_{c}(γ(t_{1}), γ(t_{2})) < Φ_{c}(γ_{n}(t_{1}), γ_{n}(t_{2})) + ε

and

Φ_{c}(γ(t_{2}), γ(t_{1})) < Φ_{c}(γ_{n}(t_{2}), γ_{n}(t_{1})) + ε.

Next observe that

Φ_{c}(γ_{n}(t_{1}), γ_{n}(t_{2})) ≤ S_{L+c}(γ_{n}|_{[t_{1},t_{2}]})

and

Φ_{c}(γ_{n}(t_{2}), γ_{n}(t_{1})) ≤ S_{L+c}(γ_{n}|_{[t_{2},T_{n}]}) + S_{L+c}(γ_{n}|_{[0,t_{1}]}).

Hence

Φ_{c}(γ(t_{1}), γ(t_{2})) + Φ_{c}(γ(t_{2}), γ(t_{1})) ≤ 3ε,

and since ε was arbitrary we deduce that γ|_{(0,∞)} is static. By Proposition 3.5.5 in [8], γ : ℝ → M is a static curve as desired.

Corollary 12. If the static set Σ̂(L) is not empty, then for any pair (q_{1}, q_{2}) we have

h_{c}(q_{1}, q_{2}) ≤ Φ_{c}(q_{1}, p) + Φ_{c}(p, q_{2}),

for any p ∈ πΣ̂(L).

Proof. From the definition of the Peierls barrier h_{c} we have:

h_{c}(q_{1}, q_{2}) ≤ Φ_{c}(q_{1}, p) + h_{c}(p, q) + Φ_{c}(q, q_{2}).

Set p = q ∈ πΣ̂(L) and use Proposition 11.

Even though we do not need the next result for the proof of the theorems in the introduction, we include it here because it gives an interesting characterization of the Peierls barrier in terms of the action potential. This result should be compared with Fathi’s results in [14], in which he characterizes the Peierls barrier in terms of conjugate weak KAM solutions.

Proposition 13. If M is compact then

h_{c}(x, y) = inf_{p∈π(Σ̂(L))} {Φ_{c}(x, p) + Φ_{c}(p, y)} .

Proof. Using Corollary 12 we get that

h_{c}(x, y) ≤ inf_{p∈πΣ̂(L)}[ Φ_{c}(x, p) + Φ_{c}(p, y) ].

Now let γ_{n} ∈ C(x, y; T_{n}) be Tonelli minimizers with T_{n} → +∞ and S_{L+c}(γ_{n}) → h_{c}(x, y) < +∞. Then \frac{1}{T_{n}} S_{L+c}(γ_{n}) → 0. Observe that by Lemma 16 the speed of the Tonelli minimizers γ_{n} is bounded. Let μ be a weak limit of a subsequence of the probability measures μ_{γ_{n}} supported on the piece of orbits

[0, T_{n}] ∋ t ↦ (γ_{n}(t), γ̇_{n}(t)).

Then μ is a minimizing measure (see [6, 22]). Let q ∈ π(supp(μ)) and q_{n} ∈ γ_{n}([0, T_{n}]) be such that lim_{n} q_{n} = q. Then,

Φ_{c}(x, q) + Φ_{c}(q, y) ≤ Φ_{c}(x, q_{n}) + Φ_{c}(q_{n}, y) + Φ_{c}(q_{n}, q) + Φ_{c}(q, q_{n})
≤ S_{L+c}(γ_{n}) + Φ_{c}(q_{n}, q) + Φ_{c}(q, q_{n}).

Letting n → ∞, we get that

Φ_{c}(x, q) + Φ_{c}(q, y) ≤ h_{c}(x, y).

3.2. Proof of Theorem B. We begin with the following lemma:

Lemma 14. Suppose that L is quadratic at infinity. Given any coordinate chart there are positive numbers δ, B, C and D such that in the given chart we have:

C|v − w|^{2} ≤ (L_{v}(x, v) − L_{v}(y, w)) · (v − w) + [B (|v| + |w|) + D] |v − w| d(x, y)

provided that d(x, y) < δ.

Proof . Take δ such that if d(x, y) < δ then x, y belong to the same coordinate chart of M. Next we work in local coordinates as if L were defined in ℝ^{2n}.

∫_{0}^{1}(v − w) · L_{vv}(t (x, v) + (1 − t) (y, w)) · (v − w) dt =

= (L_{v}(x, v) − L_{v}(y, w)) · (v − w)

− ∫_{0}^{1}(v − w) · L_{xv}((x, v)t + (y, w)(1 − t)) · (x − y) dt.

Now, since L is quadratic at infinity |L_{xv}(x, v)| ≤ B_{1} |v|_{x}+D_{1} for some positive constants B_{1} and D_{1}. On the other hand, since L_{vv} is positive definite, there is C > 0 such that

u L_{vv} u ≥ C |u|^{2}

for every u. If we now take into account the equivalence between d and the euclidean metric in the given coordinate chart we can easily obtain the statement of the lemma for appropriate constants B and D.

The next lemma will ensure that if q_{1} ≠ q_{2} then the set {A_{L+k} ≤ b} is complete when h_{k}(q_{1}, q_{2}) = ∞. This observation is needed when we apply the minimax principle given by Proposition 21.

Lemma 15. If q_{1} ≠ q_{2} and A_{L+k}(b_{n}, x_{n}) ≤ D then b_{n} is bounded away from zero.

Proof. Since our Lagrangian is quadratic at infinity, there are positive constants D_{1} and D_{2} such that

L(x, v) ≥ D_{1} |v|^{2} − D_{2},

consequently

D ≥ A_{L+k}(b_{n}, x_{n}) ≥ \frac{D_{1}}{b_{n}} ∫_{0}^{1} |ẋ_{n}|^{2} dt + (k − D_{2}) b_{n},

and thus there is D_{3} > 0 such that for all n,

∫_{0}^{1} |ẋ_{n}|^{2} dt ≤ D_{3} b_{n}. (4)

Observe that if b_{n} → 0 then ||x_{n}||_{1} → 0 and then the length of x_{n} goes to 0 which is absurd provided q_{1} ≠ q_{2}.

Let us begin now with the proof of Theorem B.

Take {(b_{n}, x_{n})} such that {A_{L+k}(b_{n}, x_{n})} is bounded and ||d_{(b_{n},x_{n})}A_{L+k}||_{1} → 0. Let y_{n}(t) = x_{n}(t/b_{n}). Then {b_{n}} is bounded, for if not, we may assume that b_{n} → +∞ and then

A_{L+k}(b_{n}, x_{n}) = S_{L+k}(y_{n}) ≥ Φ_{k}(q_{1}, q_{2}; b_{n}) ⟶^{n} +∞.

So, we can assume that {b_{n}} converges. Let b = lim_{n} b_{n}. Let

w_{n}(s) = \begin{cases}y_{n}(s) & if s ≤ b_{n}, \\ q_{2} & if b_{n} ≤ s ≤ b + 1. \end{cases}

Then w_{n} ∈ C(q_{1}, q_{2}; b + 1) and

S_{L}(w_{n}) ≤ S_{L}(y_{n}) + L(q_{2}, 0) (b + 1 − b_{n})
≤ A_{L+k}(b_{n}, x_{n}) − k b + 2 L(q_{2}, 0),

if b_{n} ≥ b − 1. By the same arguments in the proof of Tonelli’s Theorem and by the Arzela-Ascoli Theorem there is a convergent subsequence of w_{n} in the C^{0} topology (cf.

[8, 24]). This implies that also {y_{n}} and {x_{n}} have convergent subsequences in the C^{0} topology. For the sequel we work with a convergent subsequence of {x_{n}}. We shall assume without loss of generality that the limit point of the sequence {x_{n}} is contained in a coordinate chart, for if not we can cover it with a finite number of charts and we do the argument below on each chart. Hence for n large enough x_{n} has its image contained in the same local chart as the limit point.

By Lemma 15 we may assume that b_{n} → b ≠ 0. Write z_{n} = x_{n}/b_{n}. Also by Lemma 15 (see inequality (4)) there is K > 0 such that

||x_{n}||_{1} ≤ K and ||z_{n}||_{1} ≤ K

(recall that || . ||_{1} is the norm in H^{1}(ℝ^{k})). Now we follow the lines of Lemma 7.1 in [30]. Since ||d_{(b_{n},x_{n})}A_{L+k}||_{1} → 0 , given ε there is N such that

||d_{(b_{n},x_{n})}A_{L+k}(η) − d_{(b_{m},x_{m})}A_{L+k}(η)||_{1} < ε

for every n, m ≥ N and ||η||_{1} ≤ 2K. We can take in particular η = x_{n} − x_{m} and therefore,

| ∫_{0}^{1}{b_{n}L_{x}(x_{n}, ż_{n}) − b_{m}L_{x}(x_{m}, ż_{m})} (x_{n} − x_{m}) dt +
+ ∫_{0}^{1}{L_{v}(x_{n}, ż_{n}) − L_{v}(x_{m}, ż_{m})} (ẋ_{n} − ẋ_{m}) dt| < ε

for m, n > N. Since our Lagrangian is quadratic at infinity, then there are positive constants a and c such that ||L_{x}|| < a |v|_{x}^{2} + c. Using (4) in the proof of Lemma 15 we have that the first term is bounded by (2 a D_{3} + (b_{n} + b_{m})c)||x_{n} − x_{m}||_{∞}. Consequently the second integral is small for big m and n.

Now we apply Lemma 14 to obtain

C ∫_{0}^{1} |ż_{n} − ż_{m}|^{2} dt ≤ ∫_{0}^{1}{ L_{v}(x_{n}, ż_{n}) − L_{v}(x_{m}, ż_{m}) } · (ż_{n} − ż_{m}) dt +
+ B ∫_{0}^{1} (|ż_{n}| + |ż_{m}|) |ż_{n} − ż_{m}| |x_{n} − x_{m}| dt
+ D ∫_{0}^{1} |ż_{n} − ż_{m}| |x_{n} − x_{m}| dt.

Now

∫_{0}^{1} (|ż_{n}| + |ż_{m}|) |ż_{n} − ż_{m}| dt ≤ (||z_{n}||_{1} + ||z_{m}||_{1}) ∥z_{n} − z_{m}∥_{1}

by the Cauchy-Shwartz inequality. Therefore

C ∥ż_{n} − ż_{m}∥_{L^{2}}^{2} ≤ ∫_{0}^{1}{ L_{v}(x_{n}, ż_{n}) − L_{v}(x_{m}, ż_{m}) } · (ż_{n} − ż_{m}) dt +
+ (4 K^{2} B + 2KD) ||x_{n} − x_{m}||_{∞}.

Since x_{n} converges in the C^{0} topology and the integral is small, we conclude that z_{n} converges in the H^{1}-norm. Again, since b ≠ 0, x_{n} also converges in the H^{1}-norm, finishing the proof of the theorem.

3.3. Proof of Theorem C.

Lemma 16 ([24]). For B > 0 there exists C = C(B) > 0 such that if x, y ∈ M and γ ∈ C_{T} (x, y) is a solution of the Euler-Lagrange equation with A_{L}(γ) ≤ B T, then |γ̇(t)| < C for all t ∈ [0, T].

Proof. By the superlinearity there is D > 0 such that L(x, v) ≥ |v| − D for all (x, v) ∈ T M. Since S_{L}(γ) ≤ B T, the mean value theorem implies that there is t_{0} ∈ (0, T) such that

|γ̇(t_{0})| ≤ D + B.

The conservation of the energy implies that there is C = C(B) > 0 such that |γ̇| ≤ B.

Lemma 17. For all x, y ∈ M and ε > 0, the function t ↦ Φ_{k}(x, y; t) is Lipschitz on ε < t < +∞.

Proof. Fix ε > 0. If T > ε and γ ∈ C(x, y; T) is a Tonelli minimizer, from Lemma 16 there exists C = C(ε) > 0 such that |E(γ, γ̇)−k| ≤ C(ε)+|k|. Denote h(s) := Φ_{k}(x, y; s). If γ_{s}(t) := γ(T t/s) with t ∈ [0, s], then h(s) ≤ A_{L+k}(γ_{s}) =: B(s). Using Remark 6 we have that

f(T) : = lim sup_{δ→0} \frac{h(T + δ) − h(T)}{δ}

≤ B′(T) = \frac{1}{T} ∫_{0}^{T} [ E(γ, γ̇) − k ] dt

≤ C(ε) + |k|.

If S, T > ε we have that

Φ_{k}(x, y; S) ≤ Φ_{k}(x, y; T) + ∫_{T}^{S} f(t) dt
≤ Φ_{k}(x, y; T) + C(ε) + |k| |T − S|.

Since we can reverse the roles of S and T, this implies the Lipschitz condition for T ↦ Φ_{k}(x, y; T).

We begin now with the proof of Theorem C. Let f(t) := Φ_{c}(x, y; t).

Case 1: Suppose first that there is T_{0} > 0 such that f is monotonous on [T_{0}, +∞). Since by Lemma 17 f is Lipschitz on [T_{0}, +∞) by Rademacher’s theorem [12], f is differentiable almost everywhere and

f(t) − f(T_{0}) = ∫_{T_{0}}^{t} f′(s) ds. (5)

Since f is monotonous f′ ≥ 0 or f′ ≤ 0 for all t ≥ T_{0} and lim_{t→+∞} f(t) = lim inf_{t→+∞} f(t) =

h_{c}(x, y) < +∞. This implies that there is a sequence of differentiability points t_{n} → +∞ such that f′(t_{n}) → 0, for otherwise there would exist K > 0 and R > 0 such that f′(s) ≥ K > 0 for s ≥ R or f′(s) ≤ −K for s ≥ R. Consequently equation (5) would imply that lim_{t→+∞} f(t) is infinite.

Let γ_{n} be a Tonelli minimizer in C(x, y; t_{n}) and η_{s}(t) = γ_{n}(\frac{t_{n}}{s} t). Then S_{L+k}(η_{s}) ≥ f(s) for s in an open interval contaning t_{n}. This implies that

f′(t_{n}) = \frac{d}{ds} |_{t_{n}}S_{L+k}(η_{s}) = \frac{1}{t_{n}} ∫_{0}^{t_{n}}[ k − E(γ_{n}, γ̇_{n}) ] dt,

where the second equality follows from Remark 6. Since γ_{n} is a solution of the Euler-Lagrange equation, by Remark 5 and Remark 6, if x_{n}(s) = γ_{n}(s t_{n})

dA_{L+k}(t_{n}, x_{n})(ξ, α) = α f′(t_{n}) → 0.

Observe also that A_{L+k}(t_{n}, x_{n}) → h_{c}(x, y) < +∞. On the other hand (t_{n}, x_{n}) does not converge.

Case 2: Suppose that the set of local minima of f is unbounded. We claim that there exists a sequence t_{n} → +∞ of local minima of f such that lim_{n} f(t_{n}) = lim inf_{t→+∞} f(t). For let s_{n} be an increasing sequence of local minima of f such that s_{n} → +∞. There exists a sequence r_{n} > s_{n} such that lim_{n} f(r_{n}) = lim inf_{t→+∞} f(t). Excluding some s_{n}’s if necessary we can assume that s_{n} < r_{n} < s_{n+1}. Minimizing f on the interval [s_{n}, s_{n+1}], we obtain a local minimum t_{n} ∈ [s_{n}, s_{n+1}] such that lim_{n} t_{n} = lim_{n} s_{n} = +∞ and f(t_{n}) ≤ f(r_{n}) so that lim_{n} f(t_{n}) = lim inf_{t→+∞} f(t).

Let γ_{n} be a Tonelli minimizer in C(x, y; t_{n}) and η_{s}(t) = γ_{n}(\frac{t_{n}}{s} t). Then S_{L+k}(η_{s}) ≥ f(t_{n}) for s in a neighbourhood of t_{n}. In particular, t_{n} is also a local minimum of s ↦ S_{L+k}(η_{s}). Since s ↦ S_{L+k}(η_{s}) is differentiable, \frac{d}{d_{s}} |_{t_{n}}A_{L+k}(η_{s}) = 0. By Remark 5 and Remark 6, if x_{n}(s) = γ_{n}(s t_{n})

dA_{L+k}(t_{n}, x_{n})(ξ, α) = α \frac{d}{d_{s}} |_{t_{n}}A_{L+k}(η_{s}) = 0.

Observe also that A_{L+k}(t_{n}, x_{n}) → h_{c}(x, y) < +∞. On the other hand (t_{n}, x_{n}) does not converge.

3.4. Proof of Theorem A.

Proposition 18. Given a convex and superlinear Lagrangian 𝕃 : T N → ℝ and k ∈ ℝ there is a Lagrangian 𝕃_{0}, convex and quadratic at infinity such that 𝕃_{0}(x, v) = 𝕃(x, v) for every (x, v) such that 𝔼_{𝕃}(x, v) ≤ k + 1.

Proof. Choose R > 0 such that

𝔼_{𝕃}(x, v) ≤ k + 1 implies |v|_{x} ≤ R.

Take C a regular value of 𝕃 so that

|v|_{x} ≤ 2R implies 𝕃(x, v) ≤ C.

Now take R_{1} > 0 such that

𝕃(x, v) ≤ C implies |v|_{x} ≤ R_{1}.

Define ψ : ℝ → ℝ by

ψ(s) = sup_{x,|v|=1} 𝕃(x, sv).

This function is convex because it is the supremum of a family of convex functions. Let φ : ℝ → ℝ be φ(s) = a s^{2} + b where a > 0 and b > 0 are such that φ > ψ on s ≤ R_{1}. Define 𝕃_{1} : T M → ℝ by 𝕃_{1}(x, v) = 𝕃(x, v) when 𝕃(x, v) ≤ C, 𝕃_{1}(x, v) = φ(|v|_{x}) if |v|_{x} ≥ R_{1} and 𝕃_{1}(x, sv) being affine on s for all s, |v|_{x} on the interval { s > 0 | |sv|_{x} < R_{1}, 𝕃(x, sv) ≥ C }. By choosing a suitably large we can make 𝕃_{1} convex. Hence 𝕃_{1} is convex, continuous, quadratic at infinity and coincides with 𝕃 on |v|_{x} < 2R. Now take a convex C^{∞} approximation 𝕃_{0} of 𝕃_{1} coinciding with 𝕃_{1} on |v|_{x} ≤ R and |v|_{x} ≥ R_{1} + 1.

Lemma 19. Assume that L and L_{0} agree on the set of those (x, v) satisfying E_{L}(x, v) ≤ c(L) + 1. Then c(L_{0}) = c(L).

Proof. We use the following characterization of the critical value taken from [9]

c(L) = inf_{u∈C^{∞}(M,ℝ)} sup_{x∈M} H(x, d_{x}u), (6)

where H : T^{∗}M → ℝ is the Hamiltonian associated to L. Let 0 < ε < 1. Observe that H(x, p) < c(L) + ε implies H_{0}(x, p) = H(x, p) < c(L) + ε, where H_{0} is the Hamiltonian associated to L_{0}. By (6) there is u ∈ C^{∞}(M, ℝ) such that H(x, d_{x}u) < c(L) + ε and hence H_{0}(x, d_{x}u) = H(x, d_{x}u) < c(L) + ε. Since ε is arbitrary, we obtain c(L_{0}) ≤ c(L). Changing the roles of H_{0}, H, c(L_{0}), c(L) we obtain c(L) ≤ c(L_{0}), concluding the proof.

Corollary 20. Given a convex superlinear Lagrangian L : T M → ℝ and k ≥ c(L) there is a Lagrangian L_{0} convex and quadratic at infinty, such that L and L_{0} agree on the set of those (x, v) satisfying E_{L}(x, v) ≤ k + 1 and c(L_{0}) = c(L).

Proof. It follows from Proposition 18 and Lemma 19.

Let us prove Theorem A. Take L_{0} such that c(L_{0}) = c(L) according to the preceding corollary. Since L_{0} and L agree on a neighbourhood of E^{−1}(c(L)), then L and L_{0} have the same static set. Now if k > c(L_{0}) = c(L), the barrier h_{k} of L_{0} is +∞, and then A_{L_{0}+k} satisfies the Palais-Smale condition by Theorem B. This means that E_{L}^{−1}(k) is a Palais-Smale level. On the other hand, since the static set Σ̂(L) is not empty, h_{c} < +∞, and then Theorem C completes the proof.

4. Proof of Theorem D

The statement k > c_{u}(𝕃) ⇒ (H) was proved in [7] and could also be obtained using the corollary of Theorem A in [9]: we reparametrize the energy level in the universal covering to obtain a complete Finsler metric to which we apply Morse theory which is known to hold for Finsler geometry.

Now we prove that (H) ⟹ k ≥ c_{u}(𝕃). Let L be the lift of 𝕃 to T Ñ. Recall that the Hamiltonian H associated to L is given by

H(x, p) = sup {p(v) − L(x, v) | v ∈ T_{x}Ñ }, (7)

and the supremum is achieved at v such that p = L_{v}(x, v). Thus H(x, L_{v}(x, v)) = E(x, v).

Given q̃ ∈ Ñ, let (b_{q}, x_{q}) be the unique critical point of A_{L+k} in ℝ^{+} × Ω(q̃_{0}, q̃). Write yq̃(t) := x_{q}(t/bq̃) and define f : Ñ → ℝ by f(q̃) = A_{L+k}(bq̃, xq̃) = S_{L+k}(yq̃). The uniqueness of yq̃ implies that f is of class C^{1} because it is a composition of the action functional with an analogue of the exponential map expq̃ on E_{L}^{−1}(k). Let α : (−ε, ε) → Ñ be a smooth curve such that α(0) = q̃ and α̇(0) = w ∈ Tq̃Ñ. If we differentiate f(α(s)) at s = 0 using the first variation given by Lemma 4 and integration by parts we obtain:

dq̃f(w) = L_{v}(q̃, ẏq̃(bq̃)) · w.

This implies that

H(x, dq̃f) = E(q̃, ẏq̃(bq̃))) = k.

On [9] we showed that

c_{u}(𝕃) = inf_{f∈C^{1}(Ñ,ℝ)} sup_{x∈Ñ} H(x, d_{x}f) .

We see right away that k ≥ c_{u}(𝕃̃).

5. Some results on Morse theory

In this section we recall some results on Morse theory (cf. [28, 30]) that we shall use in the next section.

Let X be an open set in a Hilbert manifold and f : X → ℝ is a C^{2} map. The following version of the minimax principle (Proposition 21 below) is a modification of that of [19] (see also [31]). The only (minor) difference with the usual minimax principle is that our manifold X is not necessarily complete, but instead each set [f ≤ b] ⊆ X is complete.

Observe that if the vectorfield Y = −∇f is not globally Lipschitz, the gradient flow ψ_{t} of f is a priori only a local flow. Given p ∈ X, t > 0 define

α(p) := sup{ a > 0 | s ↦ ψ_{s}(p) is defined on s ∈ [0, a] }.

We say that a function τ : X → [0, +∞) is an admisible time if τ is differentiable and 0 ≤ τ(x) < α(x) for all x ∈ X. Given and admisible time τ, and a subset F ⊂ X. define

F_{τ} := { ψ_{τ(p)}(p) | p ∈ F }.

Let F be a family of subsets F ⊂ X. We say that F is forward invariant if F_{τ} ∈ F for all F ∈ F and any admisible time τ. Define

c(f, F) = inf_{F ∈F} sup_{p∈F} f(p).

Proposition 21. Let f be a C^{1} function satisfying the Palais-Smale condition. Assume also that F is forward invariant under the gradient flow and there are −∞ < a < c(f, F) < b < +∞ such that the subset [f ≤ b] ⊆ X is complete. Then c(f, F) is a critical value of f.

Proof. We borrow the following lemma from [30],

Lemma 22. Suppose that f : X → ℝ is C^{1}, ψ_{t} is the gradient flow of f and the subset [a ≤ f ≤ b] ⊂ X is complete. Then the flow ψ_{t} is relatively complete on [a ≤ f ≤ b], that is, if a ≤ f(p) ≤ b, then either α(p) = +∞ or f(ψ_{β}(p)) ≤ a for some 0 ≤ β < α(p).

Proof. Let ψ_{t} be the flow of Y = −∇f. Then

f(ψ_{t_{1}}(p)) − f(ψ_{t_{2}}(p)) = − ∫_{t_{1}}^{t_{2}} ∇f(ψ_{s}(p)) · Y (ψ_{s}(p)) ds = ∫_{t_{1}}^{t_{2}} ∥Y (ψ_{s}(p))∥^{2} ds. (8)

Moreover, using the Cauchy-Schwartz inequality, we have that

d(ψ_{t_{1}}(p), ψ_{t_{2}}(p))^{2} ≤ [ ∫_{t_{1}}^{t_{2}} ∥Y (ψ_{s}(p))∥ ds]^{2} ≤ |t_{2} − t_{1}| ∫_{t_{1}}^{t_{2}} ∥Y (ψ_{s}(p))∥^{2} ds.

Thus

d(ψ_{t_{1}}(p), ψ_{t_{2}}(p))^{2} ≤ |t_{2} − t_{1}| | f(ψ_{t_{1}}(p)) − f(ψ_{t_{2}}(p)) |. (9)

Let I = [0, α[ a maximal interval of definition of t ↦ ψ_{t}(p). Suppose that a ≤ f(ψ_{t}(p)) ≤ b for 0 ≤ t < α < ∞. Let t_{n} ↑ α. By inequality (9), n ↦ ψ_{t_{n}}(p) is a Cauchy sequence on [a ≤ f ≤ b]. Since [a ≤ f ≤ b] is complete, it has a limit q = lim_{n} ψ_{t_{n}}(p) = ψ_{α}(p). Since f is C^{1}, we can extend the solution t ↦ ψ_{t}(p) at t = α. This contradicts the definition of α.

Write c = c(f, F). We shall prove that for all ε > 0 there is a critical value c_{ε} such that c − ε < c_{ε} < c + ε. Then, using ε = \frac{1}{n}, the Palais-Smale condition implies that c is a critical value.

Suppose that there are no critical points on A(ε) := [c − ε ≤ f ≤ c + ε]. The Palais-Smale condition implies that there is δ > 0 such that ∥∇f(p)∥ > δ for all p ∈ A(ε).

If f(p) ≤ c + ε, let

τ(p) := inf{ t > 0 | s ↦ ψ_{s}(p) is defined on [0, t] and f(ψ_{t}(p)) ≤ c − ε }.

Since s ↦ f(ψ_{s}(p)) is decreasing, by Lemma 22, either τ(p) = +∞ or τ(p) < α(p) and f(ψ_{τ(p)}(p)) ≤ c − ε. Since ∥∇f∥ > δ on A(ε), by equation (8),

c − ε ≤ f(ψ_{t}(p)) ≤ f(p) + t δ^{2} ≤ c + ε + t δ^{2} for 0 ≤ t ≤ τ(p).

Thus τ(p) ≤ 2ε/δ^{2} for all p ∈ A(2ε). Observe that the implicit function theorem applied to the function F (x, t) = f(ψ_{t}(x)) implies that p ↦ τ(p) is differentiable because \frac{∂}{∂t}F (x, t) = ∇f · X = − ∥∇f∥ < −δ.

By the definition of c(f, F) there exists F ∈ F such that

sup_{x∈F} f(x) ≤ c + ε.

Then

sup_{x∈F_{τ}} f(x) = sup_{x∈F} f(ψ_{τ(p)}(p)) ≤ c − ε.

Since τ is an admisible time, this contradicts the definition of c(f, F).

From Proposition 21 we derive, taking F to be the family of sets of the form {p} with p ∈ X, the following

Corollary 23. Let f : X → ℝ be a C^{2} function for which [f ≤ b] is complete for every b. Suppose that f is bounded from below and satisfies the Palais-Smale condition. Then f has a global minimum.

It is convenient to obtain a further refinement of the corollary above which will be useful in the next section.

Corollary 24. Let X be a connected manifold. Let f : X → ℝ be a C^{2} function for which [f ≤ b] is complete for every b, satisfying the Palais-Smale condition. Suppose that p_{1} is a relative minimizer of f and suppose that f admits a second relative minimizer p_{2} ≠ p_{1}. Then,

1. either there exists a critical point p of f which is not a relative minimum or

2. p_{1} and p_{2} can be connected in any neighborhood of the set of relative minimizers p of f with f(p) = f(p_{1}). Necessarily then f(p_{1}) = f(p_{2}).

Proof. A detailed proof can be found in [31, Theorem 10.3]. The idea is to apply again the minimax principle; this time F is the family of subsets of the form x([0, 1]) where x is a curve joining p_{1} to p_{2}.

We conclude this section with the following suggestive remark. Let F be the family of all subsets F of T^{∗}M of the form F = {(x, d_{x}u) : x ∈ M} where u ∈ C^{∞}(M, ℝ). Then

c(H, F) = inf_{F ∈F} sup_{(x,p)∈F} H(x, p)

= inf_{u∈C^{∞}(M,ℝ)} sup_{x∈M} H(x, d_{x}u)

= c(L),

where the last equality is proved in [9]. Hence Mañé’s critical value resembles a critical value of H as a smooth function even though in general it is not a critical value of H as a smooth function. This resemblance shows that the name “critical value” for c(L) is appropriate and explains the (intentional) similarities in our notation.

6. Applications to the reduced action functional

6.1. Periodic Orbits. In this subsection we show the existence of periodic orbits in every nontrivial free homotopy class and every energy level above c_{u}(𝕃), where 𝕃 : T N → ℝ is a convex Lagrangian quadratic at infinity and the manifold N is compact. Let σ be a nontrivial free homotopy class of closed loops in a compact manifold N. As before, define Ω_{σ} as the set of elements of ℝ^{+} × H^{1}(ℝ^{k}) of the form (b, x) where x([0, 1]) ⊂ N, x(0) = x(1) and x ∈ σ. Let 𝕃 : T N → ℝ be a convex and superlinear Lagrangian.

The action A_{𝕃} : ℝ^{+} × Ω_{σ} → ℝ is defined by

A_{𝕃}(b, x) = ∫_{0}^{1} b(𝕃(x(t), ẋ(t)/b)) dt.

The previous discussion about the Palais-Smale condition translates to the case of free loops in a non trivial free homotopy class with only minor changes. In particular if 𝕃 is quadratic at infinity and k > c_{u}(𝕃), then A_{𝕃+k} satisfies the Palais-Smale condition on ℝ^{+} × Ω_{σ}. However one has to use the compactness of N in the following lemma.

Lemma 25. Let k > c_{u}(𝕃) and (b_{n}, x_{n}) ∈ ℝ^{+} × Ω_{σ} such that A_{𝕃+k}(b_{n}, x_{n}) is bounded. Then {(b_{n}, x_{n})} has a converging subsequence in the C^{0} topology.

Let us prove first:

Lemma 26. If k ≥ c_{u}(𝕃), then

inf_{ℝ^{+}×Ω_{σ}} A_{𝕃+k} > −∞.

Proof. Fix x_{0} ∈ σ and let R be twice the diameter of N . Take C > 0 such that

S_{𝕃+k}(z) ≤ C

for all curves z : [0, R] → N such that |ż| ≤ 1. Let (b, x) be an arbitrary element of ℝ^{+} × Ω_{σ}. Let x_{1}(t) := x(t/b). Then there is a curve z, parametrized by arc length, with length not greater than R such that γ = x_{0} ∗ z ∗ x_{1} ∗ z^{−1} is homotopic to zero.

Since γ lifts as a closed curve we have

S_{𝕃+k}(γ) ≥ 0.

Then

S_{𝕃+k}(x_{1}) ≥ −S_{𝕃+k}(x_{0}) − 2C.

Since S_{𝕃+k}(x_{1}) = A_{𝕃+k}(x), we are done.

Proof of Lemma 25. Observe that if k > c := c_{u}(𝕃) then,

A_{𝕃+k}(b_{n}, x_{n}) = A_{𝕃+k}(b_{n}, x_{n}) + (k − c)b_{n},

hence if A_{𝕃+k}(b_{n}, x_{n}) is bounded it follows that {b_{n}} is also bounded. Since the manifold N is compact by the same arguments in the proof of Tonelli’s Theorem in Ω_{σ} and by the Arzela-Ascoli Theorem there is a convergent subsequence of x_{n} in the C^{0} topology (cf. [8, 6, 24]) and hence Lemma 25 follows.

Using Corollary 23 of the previous section we can obtain right away the following theorem:

Theorem 27. Let N be a closed manifold and let 𝕃 : T N → ℝ be a convex Lagrangian quadratic at infinity. Then for every k > c_{u}(𝕃) and every nontrivial free homotopy class σ there is a periodic orbit of 𝕃 in σ having energy k that minimizes the action of A_{𝕃+k} on ℝ^{+} × Ω_{σ}.

We conclude this subsection by showing:

Proposition 28. Let c_{u}(𝕃) < k < c_{a}(𝕃). Then there is a periodic orbit γ of 𝕃 which is homologically trivial and also has S_{𝕃+k}(γ) < 0.

Proof. Since k < c_{a}(𝕃) there is a closed curve α such that

S_{𝕃+k}(α) < 0.

Such a curve cannot be homotopically trivial, otherwise, we lift it to the universal covering as a closed curve having negative action, contradicting the condition c_{u}(𝕃) < k. Let σ be the (non trivial) homotopy class of α.

By the previous theorem

A_{𝕃+k} : ℝ^{+} × Ω_{σ} → ℝ

has a global minimum (b, x) which is a periodic orbit with energy k. If we set y(t) := x(t/b), then

A_{𝕃+k}(b, x) = S_{𝕃+k}(y) ≤ S_{𝕃+k}(α) < 0,

as desired.

6.2. Periodic orbits of Anosov energy levels. In this last subsection we show the following theorem. It was proved for geodesic flows by W. Klingenberg [20].

Theorem 29. Let N be a closed manifold and let 𝕃 : T N → ℝ be a convex superlinear Lagrangian. Suppose that the Euler-Lagrange flow of 𝕃 restricted to the regular energy level 𝔼^{−1}(k) is Anosov. Then in any non-trivial free homoopy class there is a unique closed orbit of 𝕃 with energy k.

Proof. We proved in [9] that if the Euler-Lagrange flow of 𝕃 restricted to the regular energy level 𝔼^{−1}(k) is Anosov then k > c_{u}(𝕃). Let σ be a non-trivial free homotopy class. Without loss of generality we can assume that 𝕃 is quadratic at infinity and hence A_{𝕃+k} satisfies the Palais-Smale condition on ℝ^{+} × Ω_{σ}. By Theorem 27 we know that there exists a closed orbit of 𝕃 with energy k that minimizes A_{𝕃+k} on ℝ^{+} × Ω_{σ}. The next lemma shows in fact that every closed orbit with energy k in the homotopy class σ has this minimizing property. We will postpone its proof until we complete the proof of the theorem.

Lemma 30. Every closed orbit of 𝕃 with energy k in the homotopy class σ is a minimum of A_{𝕃+k} on ℝ^{+} × Ω_{σ}.

Now suppose that we have two geometrically different closed orbits γ_{1} and γ_{2} of 𝕃 with energy k in the free homotopy class σ. By Lemma 30 all the critical points of A_{𝕃+k} on ℝ^{+} × Ω_{σ} are minimizers and hence in Corollary 24 the second alternative holds. This contradicts that γ_{1} (or γ_{2}) is hyperbolic.

Proof of Lemma 30. Let γ be a closed orbit of 𝕃 with energy k in the free homotopy class σ. Let W^{s}(γ) be the weak stable leaf of γ for the corresponding Hamiltonian flow and let W̃^{s}(γ) be its lift to the universal covering. We proved in [9] that W̃^{s}(γ) is the graph of an exact 1-form. This means that there exists u : Ñ → ℝ such that W̃^{s}(γ) = {(x, d_{x}u) : x ∈ Ñ} and since W̃^{s}(γ) is contained in the energy level k we have that H(x, d_{x}u) = k for all x ∈ Ñ. By the relation between H and L we have

L(x, v) − d_{x}u(v) + k ≥ 0, (10)

and equality holds if and only if L_{v}(x, v) = d_{x}u, i.e. when (x, v) belongs to the inverse image of W̃^{s}(γ) under the Legendre transform which is the same as the lift of the weak stable leaf for the Euler-Lagrange flow on T N.

Let γ̃ be a lift of the closed curve γ : [0, T] → N to Ñ. Let η : [0, T_{1}] → N be a closed curve in the free homotopy class σ and let η̃ be a lift to Ñ. Let a : [0, 1] → Ñ be a curve such that a(0) = γ̃(0) and a(1) = η̃(0). Let φ : Ñ → Ñ be the covering transformation that takes γ̃(0) to γ̃(T). Let b_{n} : [0, 1] → Ñ be the curve φ^{n}a^{−1}. Using (10) we get:

∫_{a∗η̃∗···∗φ^{n}η̃∗b_{n}} L − du + k ≥ 0 = ∫_{γ̃∗···∗φ^{n}γ̃} L − du + k.

Observe that the curves a ∗ η̃∗ ⋯ ∗ φ^{n}η̃∗ b_{n} and γ̃ ∗ ⋯ ∗ φ^{n}γ̃ have the same end points, hence

∫_{a}(L + k) + ∫_{b_{n}}(L + k) + n ∫η̃(L + k) ≥ n ∫γ̃(L + k)

Since ∫_{b_{n}}(L + k) is independent of n, diving by n and letting n → ∞ we obtain:

S_{𝕃+k}(η) ≥ S_{𝕃+k}(γ).

7. Appendix: an example of a Lagrangian with h_{c} = +∞.

Let L : T ℝ^{2} → ℝ be L(x, v) = \frac{1}{2} |v|^{2} + ψ(x), where | · | is the euclidean metric on ℝ^{2} and ψ(x) is a smooth function with ψ(x) = \frac{1}{|x|} for |x| ≥ 2, ψ > 0 and ψ(x) = 2 for 0 ≤ |x| ≤ 1.

Then

c(L) = − inf ψ = 0,

because if γ_{n} : [0, T_{n}] → ℝ^{2} is a smooth closed curve with length ℓ(γ_{n}) = 1, |γ_{n}(t)| ≥ n for t ∈ [0, T_{n}] and energy E(γ_{n}) = \frac{1}{2} |γ̇_{n}|^{2} − ψ(γ_{n}) ≡ 0, then

c(L) ≥ − inf_{n>0} A_{L}(γ_{n}) ≥ − ∫_{0}^{T_{n}} \frac{1}{2} |γ̇_{n}|^{2} + ψ(γ_{n})
= − ∫_{0}^{T_{n}} |γ̇_{n}|^{2} ≥ −\sqrt{\frac{2}{n}} ⟶ 0.

On the other hand,

c(L) = − inf {A_{L}(γ) | γ closed } ≤ 0,

because L > 0.

Observe that since L > 0 and on compact subsets of ℝ^{2}, L > a > 0, then we have that

d_{c}(x, y) = Φ_{c}(x, y) > 0 for all x, y ∈ ℝ^{2} with x ≠ y.

Hence Σ̂(L) = ø.

Suppose that h_{c}(0, 0) < +∞. For every x ∈ ℝ^{2} there exists a vector v ∈ T_{x}ℝ^{2} such that the solution of the Euler-Lagrange equation with initial conditions (x, v) is semistatic in forward time [9]. Let v be such a vector in T_{0}ℝ^{2} and write x_{v}(t) = (r(t), θ(t)) in polar coordinates about the origin 0 ∈ ℝ^{2}. Then lim inf_{t→+∞} r(t) = +∞ because otherwise the orbit of v would lie on a compact subset of E^{−1}(0) and then ø ≠ ω −limit(v) ⊆ Σ̂(L) = ø (see [8] for a proof of the fact that the ω − limit set of a forward semistatic orbit is contained in the static set). Note that for any t with r(t) ≥ 2 we have:

|ẋ_{v}(t)| = \sqrt{\frac{2}{r(t)}}

and

L(ϕ_{t} v) = |ẋ_{v}(t)|^{2} = \sqrt{\frac{2}{r(t)}} |ẋ_{v}(t)|.

Let T_{n} → +∞ be such that r(T_{n}) → +∞. Hence there is n_{0} such that for all n ≥ n_{0}, r(T_{n}) ≥ 2. Since L + c = L > 0, then

h_{c}(0, 0) ≥ ∫_{0}^{+∞}(L(ϕ_{t}(v)) + c) dt

≥ lim sup_{T_{n}} ∫_{T_{n_{0}}}^{T_{n}} \sqrt{\frac{2}{r(t)}} |ẋ_{v}(t)| dt

≥ lim sup_{T_{n}} ∫_{T_{n_{0}}}^{T_{n}} \sqrt{\frac{2}{r(t)}} |ṙ| dt

≥ lim sup_{T_{n}} ∫_{T_{n_{0}}}^{T_{n}} \sqrt{\frac{2}{r}} ṙ dt

= lim sup_{n} ∫_{T_{n_{0}}}^{r(T_{n})} \sqrt{\frac{2}{r}} dr = +∞.

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