On the graph theorem for Lagrangian minimizing tori

We study the graph property for Lagrangian minimizing submanifolds of the geodesic flow of a Riemannian metric in the torus \begin{document}$ (T^{n},g) $\end{document} , \begin{document}$ n>2 $\end{document} . It is well known that the transitivity of the geodesic flow in a minimizing Lagrangian submanifold implies the graph property. We replace the transitivity by three kind of assumptions: (1) \begin{document}$ r $\end{document} -density of the set of recurrent orbits for some \begin{document}$ r>0 $\end{document} depending on \begin{document}$ g $\end{document} , (2) \begin{document}$ r $\end{document} -density of the limit set, (3) every point is nonwandering. Then we show that a Lagrangian, minimizing torus satisfying one of such assumptions is a graph.


1.
Introduction. The theory of Lagrangian submanifolds that are invariant by a Hamiltonian flow has two well known statements called Birkhoff Theorems for Lagrangian invariant tori. These statements. which are actually open problems with many partial interesting answers, can be viewed as higher dimensional versions of the celebrated Birkhoff Theorems for invariant curves of measure preserving twist maps of the annulus. Let us recall briefly the context that these problems appear. Let L : T M −→ R be a convex, superlinear Lagrangian (or optic Lagrangian) defined in the tangent space T M of an n dimensional C ∞ manifold M . Let π : T M −→ M , π(p, v) = p, be the canonical projection.The first statement asserts that each continuous, Lagrangian torus W ⊂ T T n that is invariant by the Euler-Lagrange flow of L and that is minimizing, i.e., each orbit in W projects into a global minimizer of the Lagrangian action of L (see Section 1 for details), is in fact the graph of a section of the tangent bundle. Namely, the canonical projection restricted to W is a homeomorphism.
The second statement says that the same property is valid for a continuous, Lagrangian torus W , invariant by the Euler-Lagrange flow of L and that is homologous to the zero section. Notice that both problems seek to show that the set of points in the Lagrangian torus where the canonical projection is not injective is empty.
If we assume in addition that the Lagrangian torus in the statements is smooth, this is equivalent to show that the set of singular points of the canonical projection restricted to the Lagrangian torus is empty.
The above statements are conjectures, and the best understood of them is the second one. Let us make a brief account of the results known up to date, to our best knowledge For two dimensional tori, Bialy [9] proved that a Lagrangian torus homologous to the zero section which is invariant by the geodesic flow of a reversible Finsler metric and has no periodic orbit is a graph of the canonical projection. Carneiro-Ruggiero [16] showed the same result for C 1 tori dropping the assumption on the periodic orbits. Thus, a C 1 Lagrangian torus homologous to the zero section invariant by the geodesic flow of a reversible Finsler metric in T 2 is a graph, our second statement is a true theorem in this case. In higher dimensions, Bialy-Polterovich [11], [12], Polterovich [38] showed that a C 3 Lagrangian torus of sufficiently high energy that is invariant by the Euler-Lagrange flow of an optic Lagrangian is a graph of the canonical projection provided that the torus is homologous to the zero section and that the flow in the torus is chain recurrent. The smoothness assumption on the Lagrangian torus is very important, it allows the application of powerful tools of symplectic topology to study the singular set of the restriction of the canonical projection to the Lagrangian torus. The proof in this case relies strongly on a result proved by Viterbo [40] which essentially implies that the Maslov cycle of a C 3 , Lagrangian, invariant torus homologous to the zero section is trivial. Without any assumption on the dynamics of the flow in the Lagrangian torus, using graph selectors Marie Claude Arnaud [2] shows that if the torus is exact and Hamiltonianly isotopic to the zero section then it is a graph. See also a generalization of this result, by Bernard and dos Santos, for Lipschitz Lagrangian torus in [8]. For further generalizations using Floer homology by Amorim, Oh and dos Santos see [1].
For the case of the geodesic flow, In the book [36], the graph property is proven using a theorem due to Viterbo in [40], under the assumptions that the torus is homologous to the zero section and the set of nonwandering points of the restriction of the flow coincides with the whole of the torus.
The first of the above statements is much less explored in the literature. It was proven for C 1 Lagrangian invariant tori of Riemannian metrics in T 2 by Bialy and Polterovich [10] and Bangert [6] in the early 1980's; the statements holds even for continuous invariant tori without closed orbits. It was proved for C 1 Lagrangian, invariant, nonsingular tori in the Mañé critical level of an optic Lagrangian in T 2 by Carneiro and Ruggiero [15] (2004). By the way, in [16] it is given an example of a continuous, invariant minimizing torus with periodic orbits that is not a graph of the canonical projection. As far as we know, no further progress has been made in the subject (even assuming high smoothness of the Lagrangian invariant tori) in higher dimensions. Although the assumptions in both statements are related, the variational assumption in the first one is different in nature from the topological assumption in the second one.
The graph theorem for minimizing tori in dimension 2 follows from a combination of calculus of variations and the well understood dynamics of nonsingular flows in the 2-torus. Since the seminal works of Morse and Hedlund it is known that accumulation properties of globally minimizing geodesics determine their intersection properties: canonical projections of minimizing recurrent orbits are laminations in the two-torus.
On the other hand, in higher dimensions, the nonwandering set of a nonsingular flow in the torus might be extremely more complicated than in dimension two, giving no clue at all about the accumulation properties of minimizing orbits in a Lagrangian invariant torus. It is not difficult to show that a C 1 , Lagrangian, invariant torus that is a graph must be minimizing.
So the two categories, minimizing and homologous to the zero section, are closely connected. We might think that both graph problems could be solved by applying the same sort of techniques. However, the minimizing assumption does not give any hint about the triviality of the Maslov cycle.
The first main result of the paper relates the geometry to the dynamics, as a sufficient condition to have the graph property: Theorem A. Let (T n , g) be a C ∞ Riemannian metric in the torus, and let W ⊂ T 1 T n be a C 2 invariant, Lagrangian torus. There exists r > 0 depending on the supremum of sectional curvatures of g and the supremum of the seccional curvatures of W with respect to the Sasaki metric such if W is minimizing and the limit set of W is r-dense, then the canonical projection π : T 1 T n −→ T n restricted to W is a diffeomorphism.
Given a metric space (X, d) we say that a subset Y is r-dense in X if the open ball of radius r around each point of X contains an element of Y . The next main contribution of the paper is another extension of a well known result in the context of Lagrangian tori homologous to the zero section.
Theorem B. Let (T n , g) be a C ∞ Riemannian metric in the torus, and let W ⊂ T 1 T n be a C 1 minimizing Lagrangian torus such that every point is non-wandering for the restricted flow. Then W is a graph.
Theorems A and B extend to minimizing tori well known results that are known for invariant tori which are homologous to the zero section, after the work of Polterovich [38]. The difference from Theorem A to Theorem B relies essentially on the regularity of the Lagrangian torus. In Theorem A, we require C 2 regularity and r-density for the limit set, while in Theorem B we drop the regularity to C 1 but we assume a somehow stronger hypothesis on the dynamics (every point is nonwandering, notice that every point in the limit set is nonwandering).
The proofs of Theorems A and B can be extended to Finsler metrics in the torus, therefore to Euler-Lagrange flows in sufficiently high energy levels of Tonelli Lagrangians. We would like to point out that the known partial results of Birkhoff's theorems do not imply either Theorem A or Theorem B.
Indeed, a Lagrangian invariant tori homologous to the zero section is known to be a graph if the geodesic flow in W is chain recurrent or if every point is nonwandering as we mentioned before. Neither the r-density of the recurrent set of W or the limit set for some r > 0 imply either of the above properties.
Moreover, the theory of the Maslov cycle of invariant Lagrangian tori homologous to the zero section (see Section 2 for details ) cannot be applied to the case of minimizing tori. The minimizing assumption does not imply a priori that the torus is homologous to the zero section.
In the proof of Theorem B we apply a general result that is interesting in itself, a topological description of the singular set of a C 1 , Lagrangian invariant torus.
Theorem C. Let (M n , g) be a C ∞ compact n-dimensional Riemannian manifold, let φ t : T 1 M −→ T 1 M be the geodesic flow, and let W ⊂ T 1 M be a C 1 compact, invariant Lagrangian submanifold. Then the set of points S(W ) where the canonical projection is not regular has the following property: there exists τ > 0 such that for every θ ∈ S(W ) there exists a connected open neighborhood B(θ) ⊂ W satisfying the following properties: 1. There exists > 0, depending on θ, such that B(θ) is a flow box where Σ(θ) ⊂ S(W ) contains θ and is a continuous cross section of the geodesic flow.
For the definition of a topological cross section we refer to Section 4. We would like to thank the geometry group of the University of Avignon and specially Professor Marie-Claude Arnaud for interesting and fruitful discussions about the subject while this research work was in progress.

2.
Preliminaries. Let us introduce some notations. T M is the tangent bundle of a C ∞ manifold M , T p M is the tangent space of M at a point p, the unit tangent bundle of a Riemannian manifold (M, g) is T 1 M , and the canonical projection π : T M −→ M is the map π(p, v) = p, where (p, v) is a point in T M in canonical coordinates. We shall always assume that M is complete. The universal covering of M will be denoted byM .
The geodesic flow of (M, g) in T 1 M will be denoted by φ t : T 1 M −→ T 1 M . The canonical one form of the geodesic flow will be α, the pull back by the Legendre transform of the Liouville form on T * M and the canonical two-form ω = dα. The two-form ω is symplectic (non-degenerate and closed) and preserved by φ t . The tangent space T θ T M is the orthogonal (with respect to the Sasaki metric) sum of the horizontal subspace H θ , the kernel of the connection map, and the vertical subspace V θ = Ker(D θ π). The subspace N θ ∈ T θ T 1 M of vectors which are orthogonal to the geodesic vector field X(θ) is preserved by Dφ t for every t ∈ R, and we have Many forthcoming definitions and results have natural generalizations to Tonelli Lagrangians: a C k , k ≥ 3 Lagrangian L : T M → R is a Tonelli Lagrangian if it is strictly convex and superlinear (i.e., lim v →+∞ L(p,v) v = +∞ for every v ∈ T p M , for every p ∈ M ) when restricted to each tangent space T p M , p ∈ M . A Lagrangian L defined in T M has a natural lift in TM that we shall denote byL.
2.1. Lagrangian submanifolds. Suppose that the dimension of M is n. A subspace L θ of T θ T M is called Lagrangian if its dimension is n and the restriction of ω to L θ is vanishes. In the notation of symplectic geometry, L θ is Lagrangian for ω if it is an isotropic subspace (i.e., ω vanishes in L θ × L θ ) with maximal dimension (n). A continuous vector subbundle L of T M is called invariant if it is is invariant by the action of dφ t for every t ∈ R.
A smooth submanifold W ⊂ T M is called Lagrangian if it is isotropic of dimension n, that is canonical symplectic the two-form ω in T p W vanishes for every p ∈ W . A smooth Lagrangian submanifold W is called a graph if the canonical projection π restricted to W is a diffeomorphism. Let us denote by S, the singular set, the set of points θ ∈ W where the canonical projection is singular, namely, where T θ W ∩ V θ is a nontrivial subspace.

2.2.
Jacobi fields and conjugate points. Invariant Lagrangian subspaces can be described in terms of Jacobi fields and symmetric operators with very special properties, let us recall briefly this description for geodesic flows.
Let γ θ be a geodesic of (M, g). Taking a parallel frame {e i (t)}, i = 1, 2, .., n − 1 of vector fields along the geodesic γ θ (t) = π(ϕ t (θ)), which are orthogonal to γ θ (t), the subspace of perpendicular Jacobi fields along γ θ is obtained as the image of a one parameter family of linear operators J (t) given by (n − 1) × (n − 1) matrix solutions of the Jacobi equation where the derivatives are covariant derivatives of (M, g) along γ θ (t), and K(t) is the matrix of sectional curvatures is a subbundle defined along the orbit ϕ t (θ) and there exist n − 1 linearly independent Jacobi fields J i along the geodesic γ θ such that L(t) is generated by (J i (t), J i (t)), i = 1, ..., n−1, for every t ∈ R. Such Jacobi fields are perpendicular to γ θ (t) for every t, so there exists a matrix solution of the Jacobi equation associated to this basis and hence, whenever this solution is invertible it defines a solution of the Riccati equation U (t) associated to a Lagrangian bundle W (t) At these points, the subspace L(t) is the graph of U (t) viewed as a linear operator U (t) : H ϕt(θ) −→ V ϕt(θ) . The graph representation of W (t) fails whenever W (t) intersects V ϕt(θ) nontrivially. Equivalently, when there exists a nontrivial linear combination J(t) of the J i (t)'s in γ θ such that J(t 0 ) = 0 for some t 0 . The Riccati equation is a well known tool to study conjugate points along geodesics.
Definition 2.1. Let γ θ be a geodesic of (M, g). Two points γ θ (t), γ θ (s), t < s are conjugate along γ θ if there exists a nontrivial Jacobi field J(r) of γ which vanishes at r = t and r = s. The geodesic γ θ : (a, b) −→ M is said to have no conjugate points if every Jacobi field of γ θ has at most one zero in (a, b). Equivalently, The next two results are considered classical in the theory of geodesics without conjugate points. The first one is a straightforward consequence of the work of Green ([26] pages 1, 2).
In particular, 1. For any > 0 that is smaller than the length of I = (a, b), there exists

Global minimizers and conjugate points.
We recall the definition of minimizing Lagrangian submanifold.
The action of the Lagrangian L in an absolutely continuous curve c : In order to define the notion of minimizing submanifold, let us consider the abelian covering M a of a manifold M that is the quotient of the universal covering by the subgroup of π 1 (M ) generated by commutators xyx −1 y −1 . The abelian covering projection π a : M a −→ M is a local homeomorphism and the first integer homology group has a natural representation in the set of automorphisms of M a . Any Lagrangian L : T M −→ R lifts to a Lagrangian L a : T M a −→ R in a natural way. In particular, if (M, g) is a Riemannian manifold, the metric g lifts to a metric g a in M a that is locally isometric to g. As in the case of the universal covering, the first homology group acts a discrete subgroup of isometries of (M a , g a ). Of course, if the fundamental group is abelian (as in the torus case), M a coincides withM .
We say that an absolutely continuous curve γ : R −→ M is a global minimizer if for every a < b ∈ R, and any liftγ of γ in M a , given an absolutely continuous curve δ : 4. An invariant submanifold W is called minimizing if any lift to the abelian cover of the canonical projection of every trajectory in W is a global minimizer.
Following a notation given by Morse in [35], we say that an absolutely continuous curve γ : R −→ M is a type A minimizer if for every a < b ∈ R, and any liftγ of γ inM , given an absolutely continuous curve δ : aL (γ(t),γ (t))dt. For Riemannian Lagrangians , type A minimizers of the Lagrangian action are globally minimizing geodesics in the universal covering (M ,g). It is easy to show that global minimizers are type A minimizers, the converse is not true in general. This implies that global minimizers have no conjugate points.
In particular, every trajectory in W projects onto a geodesic that has no conjugate points.
3. Recurrence and regularity of the canonical projection restricted to a Lagrangian minimizing torus. The purpose of this section is to show that nontrivial dynamics in an invariant Lagrangian torus combined with minimizing properties imply regularity of the canonical projection. We start by recalling what happens at recurrent points of tori which are homologous to the zero section. Based on the results in this case we shall look at the limit set of Lagrangian minimizing tori and extend such results by different methods.
Recall that a point θ ∈ T 1 M is forward (resp. backward) recurrent for the geodesic flow if given any open neighborhood B of θ there exists t B > 1 (resp.
A point θ is recurrent if it is positively and negatively recurrent. We shall analyze the problem in each of the assumptions of the main Theorem.
3.1. The Maslov cycle. One of the classical approaches to study the points of a Lagrangian invariant submanifold where the canonical projection is not regular is the theory of the Maslov cycle.
Recall that Λ(n) = U (n) O(n) , the set of all Lagrangian subspaces in a symplectic space is called the Lagrangian Grassmann manifold.
For a symplectic manifold (N, ω) we can construct the Lagrangian Grassmann bundle Π : Λ(N ) → N , the bundle over N whose fibers consist of all the Lagrangian subspaces. That is, for each point x ∈ N we consider all Lagrangian subspaces of T x N . Let Σ ⊂ N be a 2n − 1 connected submanifold. Let Λ(Σ) be the induced bundle π −1 (Σ). For a fixed section E, the Maslov cycle is the subbundle of Λ(Σ) whose fiber over a point x consists of all Lagrangian subspaces which intersects the fiber E θ non trivially.
In our case, the symplectic form on T M induced by the Legendre transform defined the Riemannian metric. Let us consider the Lagrangian Grassmannn Λ(T M ) and the subbundle Λ(T 1 M ) over the unit tangent bundle.
where Λ k (T 1 M ) is the bundle of Lagrangian subspaces of T M whose intersection with the vertical subspace has dimension k.
The Maslov cycle Λ V is a subset of the Grassmannian G of Lagrangian subspaces of the fiber bundle Λ(T M ). Each of the sets Λ k (T M ) is a smooth submanifold of codimension k(k+1) 2 , so Λ is a stratified manifold. There is a natural lift of the geodesic flow φ t to the Grassmannian G. Namely, ) defines a flow in G. This flow commutes with the projection P : G −→ T 1 M given by P (θ, E) = θ: P • φ G t = φ t • P . A C k invariant Lagrangian submanifold Σ ⊂ T 1 M lifts to a C k−1 , φ G t -invariant submanifoldΣ of the same dimension. The next result pointed out by Polterovich [38] is an important application of the work of Viterbo [40]: Theorem 3.2. Let W be a C 2 Lagrangian torus invariant by the geodesic flow of (T n , g). If W is homologous to the zero section then recurrent orbits of the lifted flow φ G t restricted to the liftW of W in the Grassmannian do not meet the Maslov cycle.
A brief sketch of the proof of Theorem 3.2 is the following: Viterbo shows that the Maslov cycle of a Lagrangian invariant torus homologous to the zero section is trivial. From this fact it is possible to deduce that the tangent space along compact, "almost closed" pieces of recurrent orbits of a C 2 Lagrangian invariant torus homologous to the zero section cannot cross the vertical subbundle. Therefore, such orbits are regular for the canonical projection.
3.2. Lagrangian, invariant, minimizing tori. Now, let us consider a C 2 Lagrangian minimizing torus. Viterbo's result might not apply in this case since we know nothing about the triviality of the Maslov cycle of the torus. Instead, we have the following Arnold's generalization to Lagrangian invariant subspaces of the Sturm-Liouville theorem for second order, ordinary differential equations ( [5], first Corollary of Section 1), that describes the intersections of a Lagrangian invariant subspace with the Maslov cycle.
is not zero is bounded above by n. In particular, if W is a C 1 minimizing, Lagrangian torus invariant by the geodesic flow of (T n , g), we have that for every θ ∈ W , the number of times where T φt(θ) W meets the vertical subspace is at most n.
Proof. Recall briefly that the existence of l > 0 is a consequence of Rauch comparison Theorem.
According to Theorem 3.3, if T φt(θ) W = D θ φ t T θ W meets non trivially the vertical bundle more than n + 1 times t i in an interval (a, b), then the subspaces D θ φ t (V θ ) will meet the vertical bundle at least twice in the same interval (a, b). This contradicts the absence of conjugate points in the geodesic γ θ (t).
Corollary 3.5. Let W be a C 1 minimizing, Lagrangian torus invariant by the geodesic flow of (T n , g). There exists α > 0 with the following property: given θ ∈ W there exists b θ ∈ R such that the angle subtended by the vertical subspace V φt(θ) and T φt(θ) W is at least α for every t > b θ + 1.
Proof. Let θ ∈ W . By Corollary 3.4 there exists a compact interval [a θ , b θ ] which contains all the intersections of T φt(θ) W with the vertical bundle. Let us suppose that b θ is the maximum of the values of t where T φt(θ) W meets the vertical bundle. Let U (t) be the solution of the Riccati equation associated to W defined along φ t (θ). By Eberlein's Theorem 2.3, if −k 0 < 0 is a lower bound for the sectional curvatures of (T n , g), we have thatŨ (t) = U (b θ + t) satisfies | g(Ũ (t)X, X) |≤ k 0 coth(k 0 t) for every t > 0, and every unit vector X ∈ T γ(b θ +t) M perpendicular to γ (t). Since the subspace T φt(θ) W ∩ N γ(t) is the graph of U (t), which is a bounded operator for t > 1, elementary linear algebra implies that there exists α > 0 such that for every t > 1 we have that the angle between the vertical bundle and T φt(θ) W is at least α,for every t > b θ + 1.
Lemma 3.6. Let W be a C 1 minimizing, Lagrangian torus invariant by the geodesic flow of (T n , g). Then every recurrent orbit is regular for the canonical projection. Moreover, if α > 0 is the number defined in Corollary 3.5, the angle from T φt(θ) W and V φt(θ) is at least α for every t ∈ R.
Proof. Let θ ∈ W be a forward recurrent point. Since every point of its orbit is recurrent, then every point is accumulated by positive iterates of φ t (θ) for t > 0 arbitrarily large. According to Corollary 3.5, for t > b θ + 1 the angle subtended by T φt(θ) W and V φt(θ) is at least α > 0. Therefore, there is no t 0 such that T φt 0 (θ) W ∩ V φt 0 (θ) = {0}, for otherwise, there would be a sequence t n → ∞, with φ tn (θ) → φ t0 (θ), where T φt n (θ) W tends to a Lagrangian subspace that is transversal to V φt 0 (θ) . Since T W is a continuous bundle, the subspaces T φt n (θ) W tend to T φt 0 (θ) W and therefore, the latter subspace must be transversal to the vertical bundle. The argument extends easily to backward recurrent points. 4. Graph property of Lagrangian minimizing tori in a neighborhood of the limit set. Combining the results of the previous section we get, Proposition 4.1. Let (T n , g) be a C ∞ Riemannian metric in the torus and let W be a C 1 Lagrangian invariant torus. Suppose that W is either homologous to the zero section or minimizing. Then 1. The Riccati solution U θ (t) associated to a recurrent orbit φ t (θ) is well defined for every t ∈ R. 2. There exists r > 0 such that the r-tubular neighborhood B r ⊂ W of the closure of the set of recurrent orbits of W is regular.
Proof. Theorems 2.2, 2.3 together with Theorem 3.2 and Lemma 3.6 imply that recurrent points of W are regular and the underlying geodesics have no conjugate points. Moreover, the tangent space of W never meets the vertical bundle along recurrent orbits and there is a positive lower bound α for the angle subtended by T W and the vertical bundle along recurrent orbits. Since W is C 1 and compact, this implies that there exists an open neighborhood of radius r > 0 of the closure of the set of recurrent points of W where T W is transversal to the vertical fibration. Therefore, the canonical projection restricted to the r-tubular neighborhood B r of recurrent points is regular.
Next, let us extend the above result to the limit set of a Lagrangian minimizing torus.
Recall that the limit set L(W ) is the union of the forward or ω-limit set and the backward or α-limit set of W by the action of the geodesic flow. The ω-limit set of W , ω(W ), is the union of the ω-limit sets of points, namely, given θ ∈ W , the ω-limit set of θ, ω(θ), is the set of points η such that there exists a sequence of numbers t n → +∞ with η = lim n→+∞ φ tn (θ). The α-limit set α(W ) is defined analogously replacing t n → +∞ by t n → −∞.
Lemma 4.2. Let (T n , g) be a C ∞ Riemannian metric in the torus and let W be a C 1 Lagrangian invariant minimizing torus. Then 1. The Riccati solution U θ (t) associated to an orbit φ t (θ) in the limit set of W is well defined for every t ∈ R. 2. There exists r > 0 such that the r-tubular neighborhood B r ⊂ W of the closure of the limit set L(W ) of W is regular.
Proof. So let θ ∈ L(W ) ∩ S(W ), and let η ∈ W such that the orbit of η accumulates θ, namely, there is a sequence t n → ∞ such that lim n→∞ φ tn (η) = θ. By Corollary 3.4 the subspaces D η φ t (V η ) just meet the vertical bundle at a finite number of t k , 0 ≤ k ≤ n. So there exists t P > 0 such that for every t ≥ t P . Since t n → ∞ there exists t n0 > t P such that t n − t n0 > 1 for every n > n 0 . Therefore, by Theorem 2.3 there exists C(1, k 0 ) > 0 such that the norm of the Riccati operator U (φ tn 0 +t (η)) representing T φt n 0 (η) W is bounded above by C(1, k 0 ) for every t > t n0 . Hence, by the the continuity of the tangent bundle of W we conclude that the same happens with the Riccati operator of W at θ. Since every point of the orbit of θ is in the limit set, the same happens at every point of the orbit. So the solution of the Riccati equation U θ (t) is well defined for every t ∈ R and hence (by Corollary 3.5) the tangent space of W at θ forms an angle greater than α with the vertical bundle, where α > 0 is given in Corollary 3.5. By the compactness and the smoothness of W , the transversality of the tangent space of W with respect to the vertical fibres extends to the closure of L(W ). Since the closure of the limit set is compact, there exists r > 0 such that in an open neighborhood of radius r > 0 of the closure every point is regular for the canonical projection.

4.1.
On the geometry of Lagrangian minimizing tori at the limit set. In this subsection we look at the following problem: in what extent the geometry of a Lagrangian invariant torus as an isometrically embedded submanifold of T 1 T n depends on the geometry of its canonical projection? The question is natural and motivated by the fact that the Riccati operator associated to the Lagrangian torus is related to sectional curvatures of (T n , g) at places where the torus is a graph. Let us explain in detail.  Let (M, g) be a compact C ∞ Riemannian manifold of dimension n and let k 0 > 0 be such that −k 0 is a lower bound for the sectional curvatures of (M, g). Then, given C > 0 there exists a constant K C = K C (k 0 ) > 0 such that every C 3 Lagrangian minimizing submanifold W ⊂ T 1 M whose C 3 norm is bounded above by C has the following property: Suppose that θ ∈ W is a point in the limit set. Then the supremum of the absolute values of sectional curvatures of W at θ with respect to the Sasaki metric is bounded by K C .
Proof. By assumption, there exists an open neighborhood U ⊂ W of θ where the canonical projection is a diffeomorphism. So we have a local solution of the Hamilton-Jacobi equation H(q, d q f ) = 1 where d q f = α q is the canonical projection of the canonical one form of the geodesic flow restricted to W . The canonical projection π(U ) of U is an open subset of M where the form α is exact, its Reeb flow is a unit flow by geodesics, the projections of the orbits of U . Let X be the unit vector field in π(U ) tangent to the Reeb flow, then the levels f = c are co-dimension one C 3 submanifolds that are perpendicular to X and invariant by the Reeb flow in the following sense: if ψ t = π(φ t ) is the Reeb flow (the projection of the restriction of φ t to U ) then for p ∈ π(U ) we have ψ t (f −1 (p)) = f −1 (ψ t (p)) as long as ψ t (p) belongs to π(U ).
Claim. The matrix of the second fundamental form of a level f −1 (c) at a point x = π(x, v), (x, v) ∈ U , is the matrix solution of the Riccati equation associated to the geodesic γ (x,v) . This is perhaps well known by specialists, but we recall briefly the proof of this fact. The unit vector field X is the unit normal field of f −1 (c), and the geometry of the family of levels is quite similar to the geometry of horospheres in manifolds without conjugate points (where the role of the function f is played by a Busemann function). Pesin in [37], Theorem 6.1, shows that the matrix solution U (t) of the Riccati equation of a geodesic γ (p,v) (t) in the universal covering is the matrix of the normal curvatures of the horosphere of γ (p,v) (t), namely, the matrix of the second fundamental form of the immersion f −1 (c) where f a Busemann function. The same proof extends without changes to our case since the picture of the levels f −1 (c) and the Reeb flow is totally analogous.
The sets π −1 (f −1 (c)) define a C 2 foliation of U ⊂ W by codimension one submanifolds. Since the canonical projection is a Riemannian submersion from T 1 M endowed with the Sasaki metric to (M, g), we can apply the curvature formulae of Riemannian submersions (see Kowalsky [29]) to estimate the sectional curvatures of π −1 (f −1 (c)) at θ with respect to the Sasaki metric. These sectional curvatures depend on the sectional curvatures of f −1 (c) and its first derivatives. By the bounds on the Riccati operator given in Theorem 2.3, the fact that f −1 (c) is C 3 and the compactness of π(L(W )) we conclude the statement.
Corollary 4.4. Let (T n , g) be a compact C ∞ Riemannian manifold and let K C = K C (k 0 , C) > 0 be given in Lemma 4.3. Then given a C 3 Lagrangian minimizing torus, the number r > 0 given in Lemma 4.2 depends only on K C .

4.2.
Proof of Theorem A. We now conclude the proof of Theorem A: Proposition 4.1, Corollary 4.4 and the assumption in Theorem A yield that W is regular for the canonical projection. Hence, it is a covering map. The following well known result due to Arnold [27] implies that W is a Lagrangian graph.
Theorem 4.5. Let S ⊂ T 1 T n be a closed C 2 Lagrangian submanifold such that the restriction of the canonical projection to S is a covering map. Then the restriction of the canonical projection to S is a diffeomorphism.

5.
On the structure of the singular set. Now let us proceed to prove Theorem C. As we mentioned before, we already know that the singular set of the canonical projection restricted to a Lagrangian invariant submanifold of T 1 M can be represented by the intersections of the orbits of the lifted geodesic flow in the Grassmannian of Lagrangian subspaces with the Maslov cycle. The Maslov cycle is not a manifold in general as in the case of surfaces, so it is not clear that the Maslov cycle provides a cross section of the lifted geodesic flow.
We shall suppose through the section that W is a C 1 Lagrangian invariant submanifold for the geodesic flow of (M, g) where M is a compact C ∞ manifold with dimension n.
A topological cross section C ⊂ W for the geodesic flow restricted to W is a C 0 , (n − 1)-dimensional submanifold of W with the following property: there exists δ > 0 such that is an open neighborhood of W homeomorphic to C × (−δ, δ). We say that a subset S ⊂ N of a smooth manifold N is a C 0 k-dimensional submanifold if S is a Hausdorff topological space endowed with the relative topology and if there exists an atlas of S whose local parametrizations are homeomorphisms defined in open subsets of R k .
Let us denote by singular set S(W ) the set of points of W where the canonical projection is not regular. The first simple remark about the singular set is Proof. The Gauss map θ ∈ W → T θ W is continuous, therefore S(W ), the pre-image of the Maslov cycle, is closed. Theorem 2.3 gives us an estimate for the geometry of solutions of the Riccati equation associated to invariant Lagrangian subspaces that meet the vertical subbundle. It tells us that the geometry of certain reference functions (namely, f a (t) = √ k 0 coth(k 0 (a − t))) "tames" the behavior of the solution of the Riccati equation in neighborhoods of singularities.
Lemma 5.2. Let (M, g) be a C ∞ compact Riemannian manifold. Let θ ∈ T 1 M , let L θ ⊂ H θ ⊕ V θ be a Lagrangian subspace and let U (t) be the Riccati operator associated to dφ t (L θ ) (wherever it is defined). Suppose that for two numbers b < c we have consecutive intersections of dφ t (L θ ) with the vertical bundle, namely, Proof. The proof is straightforward from Theorem 2.3 but it is important for the sequel. Let ) is strictly decreasing for every t = b, it is positive if t > b and negative if t < b, then we have Since the function f (t) is odd we get C(b, c, ε) = √ k 0 coth(k 0 ( )). The dependence on b, c is implicit because depends on b, c. Lemma 5.3. Let (M, g) be a compact Riemannian manifold and let l > 0 be such that every geodesic γ : (0, l) −→ M has no conjugate points. Let W ⊂ T 1 M be a compact C 1 Lagrangian invariant submanifold. Given θ ∈ S(W ) there exist 0 < < l 2 , a local cross section Σ θ for the geodesic flow restricted to W containing φ − (θ), an open neighborhood B(θ) ⊂ W of θ homeomorphic to Σ θ × (−2 , 2 ), and a continuous positive function t 1 : Σ θ −→ R such that such that φ r0 (σ) ∈ B(θ(σ)).
Proof. Let θ ∈ S(W ). By Corollary 3.4 there exists > 0 such that the point θ is the unique point of intersection of the piece of orbit φ t (θ), t ∈ (−3 , 2 ) with S(W ). We can assume without loss of generality that < l 2 . 1 2 ). Let Σ ⊂ B ρ (θ) be a local cross section for the geodesic flow restricted to W containing φ − (θ). We can take for instance ρ > 0 suitably small, a (n − 1) dimensional plane P ⊂ T φ− (θ) W perpendicular to the direction of the geodesic flow with respect to the Sasaki metric and let

Since the set of regular points of the canonical projection is open in W , there exists an open ball
where exp η is the exponential map at η ∈ T 1 M . Let t 1 : Σ −→ R be given by the first positive time t 1 (η) ∈ (− , l 2 ) where φ t 1 (η) (η) ∈ S(W ). By the choice of , t 1 (φ − (θ)) = .
By Lemma 2.3 the Riccati operator U ηn (t) associated to the tangent space of W along the orbit φ t (η n ) satisfies for every t ∈ ( − 7τ0 8 , + 7τ0 8 ). Since W is C 1 , the Riccati operators associated to W depend continuously on ψ ∈ W . Moreover, lim n→∞ η n = φ − (θ), so lim n→∞ φ (η n ) = θ. By the invariance of the solutions of the Riccati equation by the action of the geodesic flow we would have that for every t ∈ (− 7τ0 8 , 7τ0 8 ) which yields that the tangent space of W at θ is "far" from the vertical subbundle, contradicting the assumption that θ ∈ S(W ). Now, let Σ θ = Σ δ . Claim 1 implies that t 1 : Σ θ −→ R is continuous at φ − (θ).
If necessary, we can take above smaller in a way that the map F : So the open neighborhood of the statement is B(θ) = ∪ t∈(−2 ,2 ) φ t (Σ θ ) and items (1), (2), (3) follow from the Claims. Item (4) follows from the existence of the local cross section Σ θ . Indeed, an orbit that intersects B(θ) must intersect the cross section Σ θ and by items (1), (2), (3) it must intersect the singular set S(W ).
Proof. Let us endow Σ(θ) with the restriction of the topology of W . It is clear that Σ(θ) is a Hausdorff space and that ψ θ is a continuous bijection due to the continuity of the function t 1 ( Lemma 5.3 ) and the injectivity of the geodesic flow. The image Σ(θ) of ψ θ is the graph of t 1 , namely Σ(θ) = {(η, t 1 (η)), η ∈ Σ θ } in the local coordinates Σ θ × (−2 , 2 ). According to the definition, Σ(θ) is a C 0 submanifold of codimension 1 in W , the composition of the inverse of ψ θ with a local chart of Σ θ is a local continuous chart for Σ(θ).
By the Brouwer's invariance of domain theorem, the set is an open set and therefore, Σ(θ) is a topological cross section for the restriction of the geodesic flow to W by the definition.
Corollary 5.4 gives a local description of part of the singular set S(W ) for every C 1 Lagrangian, invariant torus. Around every point θ ∈ S(W ) there is an open neighborhood B(θ) of W such that B(θ)∩S(W ) contains a n−1 continuous submanifold that is a topological cross section for the geodesic flow. The structure of the singular set might be more complicated, it might look in general like a "ramified" continuous, codimension one submanifold (see the figure below). 6. Nonwandering points and the proof of Theorem B. The purpose of the section is to show Theorem B. A point θ ∈ T 1 M is nonwandering for the geodesic flow if given an open neighborhood N of θ there exists t N ∈ R with | t N |≥ 1 such that φ t N (N ) ∩ N is nonempty. Every recurrent point is of course nonwandering but the converse is not true in general.
The main idea is to apply Theorem C to look at the singular set S(W ) as a sort of weak or generalized cross section.
Suppose that S(W ) is nonempty, let B(θ) for θ ∈ S(W ) be the set given by Proof. Let us argue by contradiction. Suppose that such a point θ ∈ S(W ) does not exist. By continuity of the geodesic flow the set of points θ ∈B(θ i1 ) whose orbits φ t (θ) intersect anyB(θ i2 ) for t ≥¯ is an open subset R i1,i2 ⊂B(θ i1 ). By the choice of¯ we have that i 1 is different from i 2 , and by the contradiction assumption we have that the union is open and dense inB(θ i ) for every i and represents the set of points in the union of the neighborhoodsB(θ i ) that meet the singular set at least twice. Let P i,j : R i,j −→ Σ(θ j ) be the Poincaré map giving the first intersection of an orbit of a point in the domain R i,j of P i,j with the cross section Σ(θ j ). The union of the images of the maps P i,j is R 1 = ∪ i,j P i,j (R i,j ) and by assumption, this set is also open in the union of the sections Σ(θ i )'s and therefore their orbits intersect the union of the balls B(θ i )'s in an open set. Now, by the contradiction assumption, the set Γ 1 ∩ R 1 = ∪ i,j (P i,j (R i,j ) ∩ Γ 1 ) = ∪ i,jRi,j is open and dense in R 1 and therefore, the set is open and dense in the union of the ballsB(θ i ) and represents the set of points that meet the singular set at least 3 times.
We can continue by induction defining the open sets R i1,i2,i3 = P i1,i2 (R i1,i2 ) ∩B(θ i3 ) R 2 = ∪ i,j,k P j,k (R i,j,k ) Γ 1 ∩ R 2 = ∪ i,j,k (P i,j (R i,j,k ) ∩ Γ 1 ) = ∪ i,j,kRi,j,k that is an open set by the contradiction assumption, Γ 3 = ∪ i,j,k P −1 i,j (R i,j,k ) that is open and dense in the union of the ballsB(θ i ) and consists of the points whose orbits meet at least 4 times the singular set, and so on.
By this procedure we get that given any m ∈ N the set of orbits of W that meet the singular set at least m times is open and dense in the union of the balls B(θ i ). This clearly contradicts the fact that the torus W is minimizing since we know that this number must be less than n.  To finish the proof of Theorem B, the assumption and the Claim imply that S(W ) is empty and therefore by Theorem 4.5 the torus W is a graph.