Global surfaces of section with positive genus for dynamically convex Reeb flows

We establish some new existence results for global surfaces of section of dynamically convex Reeb flows on the three-sphere. These sections often have genus, and are the result of a combination of pseudoholomorphic methods with some elementary ergodic methods.


Introduction and main results
Let (z 0 , z 1 ) be complex coordinates in C 2 , S 3 = {|z 0 | 2 + |z 1 | 2 = 1}, and α 0 be the standard Liouville form (−i/4) Σ jzj dz j − z j dz j . The standard symplectic form on C 2 is ω 0 = dα 0 . The fibers of the Hopf fibration are the periodic Reeb orbits of the contact form λ 0 on S 3 induced by α 0 . Let us call the Reeb flow of λ 0 the Hopf flow. The contact structure ξ 0 = ker λ 0 is called standard.
The contact form λ 0 is the first example of a dynamically convex contact form. In S 3 , a contact form λ is said to be dynamically convex if all periodic orbits have Conley-Zehnder index ≥ 3 when computed in a global dλ-symplectic frame of ker λ. This notion was introduced by Hofer, Wysocki and Zehnder (HWZ) in [21].
One can show quite explicitly that all finite collections of periodic orbits of the Hopf flow span some global surface of section, see [2]. It is natural to ask if this property remains true for all dynamically convex Reeb flows on S 3 , in particular for all strictly convex energy levels in (C 2 , ω 0 ) ( [21,Theorem 3.4]). This might be too ambitious to try to prove, and one may be led to naively think that it is easy to find a counterexample. There is, however, another This article is part of the topical collection "Symplectic geometry -A Festschrift in honour of Claude Viterbo's 60th birthday" edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk. natural way to generalize this property of the Hopf flow to the Reeb flows of all dynamically convex contact forms on S 3 . Since the Hopf fibers are unknotted with self-linking number −1, one might ask if all finite collections of periodic Reeb orbits of this kind span some global surface of section. This is our first result. Theorem 1.1. Let L be any link formed by periodic Reeb orbits of a dynamically convex contact form on S 3 whose components are unknotted with selflinking number −1. Then, L bounds a global surface of section for the Reeb flow.
There are no hidden genericity assumptions on the contact form. The genus of these sections will typically explode with the number of boundary orbits. Moreover, there is no need to specify the contact structure since only the standard one can be defined by a dynamically convex contact form on S 3 , see [19].
A proof relying exclusively on pseudoholomorphic curves would be complicated by the fact, originally observed in [14], that transversality fails for curves with genus which are everywhere transverse to the flow. The solution proposed to this problem in [14] is to consider a perturbation of the holomorphic curve equation which corrects the transversality problem, but seriously complicates the compactness theory (see [1,6,7]). However, dealing with genus is unavoidable since the links covered by Theorem 1.1 typically have positive Seifert genus. A proof without pseudoholomorphic curves seems out of reach since dynamical convexity is an assumption only on the periodic orbits, and holomorphic curve techniques have proven to be one of the very few-if not the only-effective methods for finding surfaces of section under assumptions of this kind.
Let us outline the argument. The main step is the result from [24] stating that every component of a link L as in Theorem 1.1 bounds a disk-like global surface of section. At this point ergodic methods come to aid via asymptotic cycles. We use the statement from [26] refining a celebrated result due to Fried [11]. The disks can be used to check the hypotheses of [26,Theorem 1.3]. Each disk has uniformly bounded return time, hence all invariant measures in S 3 \L positively hit the sum of the cohomology classes dual to each disk. Finally, positivity of rotation numbers follows from dynamical convexity.

Question 1. Is the Reeb flow of every dynamically convex contact form on S 3 right-handed?
A positive answer is probably very hard to obtain, even in finitedimensional families of interesting flows such as those appearing in Celestial Mechanics. One is then tempted to look for examples to give a negative answer, but they might not exist. In the context of the 3-body problem, we refer to [35] for a discussion of a version of this question, and to the book [10] by Frauenfelder and van Koert for a discussion on global surfaces of section, including a related conjecture of Birkhoff. The existence of genus zero global surfaces of section with prescribed binding orbits has been clarified in [31]. In [9], it is shown that geodesic flows on S 2 with curvatures pinched by some explicit constant lift to right-handed Reeb flows on S 3 . Remark 1.3. The dynamical convexity assumption is essential in Question 1, as one can easily check. It is, however, more subtle to rule out specific types of global surfaces of sections when dynamical convexity is dropped. For instance, in [33] one finds examples of contact forms on S 3 without disk-like global surfaces of section. The situation in higher dimensions is still wide open, but in [34], there are interesting new constructions for the spatial circular restricted 3-body problem.
Our second result is closely connected to the following question. [21]) Is the minimal period among closed Reeb orbits of a dynamically convex contact form on S 3 equal to the contact area of some disk-like global surface of section ?

Question 2. (HWZ
Starting from a nondegenerate dynamically convex contact form on S 3 , Hutchings and Nelson [32] were able to implement the construction of the chain complex of Cylindrical Contact Homology (CCH), originally introduced by Eliashberg, Givental and Hofer in [8]. The arguments from [32] rely on elementary pseudoholomorphic curve methods. Invariance of the resulting homology is delicate and requires sophisticated technology, for instance, one can use the Polyfold Theory introduced by Hofer, Wysocki and Zehnder; see [15] for a survey. It will be shown in [30] that elementary methods are still enough to get invariance of CCH in its lowest degree. This is enough to get the first spectral invariant c CCH 1 well defined. Also in [30], it will be shown that c CCH 1 is the action of some periodic orbit with Conley-Zehnder index 3 realized as the asymptotic limit of a pseudoholomorphic plane. Hence we get the following consequence of a combination of Corollary 1.5 below with some of the results from [30]: "The spectral invariant c CCH 1 of a nondegenerate dynamically convex contact form on S 3 is the contact area of some global surface of section." Theorem 1.4. Let λ be a contact form on S 3 that is both nondegenerate and dynamically convex up to action C. Suppose that a periodic Reeb orbit P = (x, T ) satisfies T ≤ C and is the asymptotic limit of a fast finite-energy plane. Then, the knot x(R) spans a global surface of section for the Reeb flow.
In the theorem above and the corollary below a periodic Reeb orbit is a pair P = (x, T ), where x is a periodic trajectory of the Reeb flow and T > 0 is a period, not necessarily the primitive one. Corollary 1.5. Let λ be a contact form on S 3 that is both nondegenerate and dynamically convex up to action C. Suppose that a periodic Reeb orbit P = (x, T ) satisfies T ≤ C, CZ(P ) = 3, and is the asymptotic limit of a finiteenergy plane. Then the knot x(R) spans a global surface of section for the Reeb flow.
Proof. The equality CZ(P ) = 3 implies that any finite-energy plane asymptotic to P is fast. Now apply Theorem 1.4.
The proof of Theorem 1.4 is based on a certain class of pseudoholomorphic planes called fast, but also uses ergodic methods (asymptotic cycles) [11,12,26,39,40]. Fast planes were originally introduced in [25] and later used in [23,24,[27][28][29] to prove several existence results on global surfaces of section. Roughly speaking, an end of a plane is in some sense a gradient trajectory of the action functional, and the results from [17] basically say that the approach to the periodic orbit is governed by an eigenvector of an operator that plays the role of the Hessian of the action, the so-called asymptotic operator. The term "fast" refers to the fact that the eigenvalue of this asymptotic eigenvector has the same winding number of the most negative eigenvalue allowed, hence the approach is roughly the fastest it can be; see Definition 2.8. Remark 1.6. The global sections obtained from Theorem 1.4 and Corollary 1.5 may have genus. Note that P is not assumed to be simply covered, but still the global sections obtained are Seifert surfaces for the knot x(R).

Preliminaries
Let λ be a contact form on a 3-manifold M . The contact structure is denoted by ξ = ker λ.

Periodic orbits, asymptotic operators and Conley-Zehnder indices
The Reeb vector field X λ of λ is implicitly defined by Its flow φ t is called the Reeb flow. Let us fix a marked point on every periodic trajectory of φ t . A periodic Reeb orbit is a pair P = (x, T ), where x : R → M is a periodic trajectory of φ t such that x(0) is the marked point, and T > 0 is a period. It is not required that T is the primitive period. The set of periodic orbits will be denoted by P(λ). If T 0 > 0 is the primitive period of x then Vol. 24 (2022) Global surfaces of section with positive Page 5 of 21 45 k = T /T 0 ∈ N is called the covering multiplicity of P . The contact form λ is said to be nondegenerate up to action C ∈ (0, +∞] if 1 is not in the spectrum of dφ T | x(0) : ξ| x(0) → ξ| x(0) , for all P = (x, T ) ∈ P(λ) such that T ≤ C. When C = +∞, we simply say that λ is nondegenerate.
There is an unbounded operator on L 2 (x(T ·) * ξ) associated with a pair (P, J), where P = (x, T ) ∈ P and J : ξ → ξ is a dλ-compatible complex structure. Here ∇ is a symmetric connection on T M and ∇ t denotes the associated covariant derivative along the loop t ∈ R/Z → x(T t). This is called the asymptotic operator. It does not depend on the choice of ∇. It is self-adjoint when L 2 (x(T ·) * ξ) is equipped with the inner product Its spectrum is discrete, consists of eigenvalues whose geometric and algebraic multiplicities coincide, and accumulates at ±∞. It turns out that λ is nondegenerate if, and only if, 0 is never an eigenvalue of an asymptotic operator. The eigenvectors are nowhere vanishing sections of x(T ·) * ξ since they solve linear ODEs. Hence they have well-defined winding numbers with respect to a dλ-symplectic trivialization σ of x(T ·) * ξ. The winding number is independent of the choice of eigenvector of a given eigenvalue. This allows us to talk about the winding number wind σ (ν) ∈ Z of an eigenvalue ν with respect to σ. For every k ∈ Z, there are precisely two eigenvalues satisfying wind σ = k, multiplicities counted and, moreover, . These properties are independent of σ. These properties of the asymptotic operator have been established in [18]. Given any δ ∈ R we set Note that this is defined also in degenerate situations. A contact form λ is dynamically convex up to action C ∈ (0, +∞] if c 1 (ξ, dλ) vanishes on π 2 → H 2 , and every contractible 1 P = (x, T ) ∈ P(λ) satisfying T ≤ C also satisfies CZ 0 σ disk (P ) ≥ 3. Here, σ disk is a trivialization that extends to a capping disk. If C = +∞, then we say that λ is dynamically convex.  [21]. The assumption that c 1 (ξ, dλ) vanishes on spheres implies that the homotopy class of σ disk does not depend on the choice of a capping disk.

Pseudoholomorphic curves in symplectizations
From now on, we assume M is closed. Let J be a compatible complex structure on the symplectic vector bundle (ξ, dλ). Hofer [13] considers an almost complex structure J defined on R × M by where X λ and ξ are seen as R-invariant objects in R × M . Then J is Rinvariant. Consider a closed Riemann surface (S, j), a finite set Γ ⊂ S and a pseudoholomorphic map satisfying a finite-energy condition where the supremum is taken over the set of φ : It turns out that every puncture is positive, negative or removable, and that u can be smoothly extended across a removable puncture; see [13]. Let z ∈ Γ and let K be a conformal disk centered at z, i.e., there is a biholomorphic map ϕ : (K, j, z) → (D, i, 0). Then K \ {z} admits positive holomorphic polar coordinates (s, t) ∈ [0, +∞) × R/Z defined by (s, t) ϕ −1 (e −2π(s+it) ), and negative holomorphic polar coordinates (s, t) ∈ (−∞, 0] × R/Z defined by (s, t) ϕ −1 (e 2π(s+it) ). [13]) Let z ∈ Γ be a nonremovable puncture, and (s, t) be positive holomorphic polar coordinates at z. For every sequence s n → +∞ there exist a subsequence s nj and P = (

Theorem 2.2. (Hofer
From now on, we denote by the projection along X λ .

Theorem 2.3.
(HWZ [17]) Suppose that λ is nondegenerate up to action C, and that z is a nonremovable puncture of a finite-energy curve u = (a, u) in Vol. 24 (2022) Global surfaces of section with positive Page 7 of 21 45 Remark 2.4. The orbit P is called the asymptotic limit of u at z.
Theorem 2.6. (HWZ [17], Mora-Donato [36], Siefring [38]) Suppose that λ is nondegenerate up to action C > 0, and that z is a nonremovable puncture of sign = ±1 of a finite-energy curve u = (a, u) with Hofer energy E( u) ≤ C. Let (s, t) be positive or negative holomorphic polar coordinates at z when = +1 or = −1, respectively. Consider any Martinet tube Ψ : N → R/Z×D for the asymptotic limit P of u at z, and s 0 If z(s, t) does not vanish identically, then the following holds. There exist r > 0 and an eigenvalue ν of the asymptotic operator of (P, J) satisfying ν < 0, such that: • There exist c, d ∈ R and a liftθ : where k is the covering multiplicity of P . • There exists an eigenvector of ν, represented as a nowhere-vanishing vector field v(t) in the frame {∂ x1 , ∂ x2 } along P , such that The alternative z(s, t) ≡ 0 can be expressed independent of coordinates as saying that the end of the domain of u corresponding to the puncture is mapped into the trivial cylinder over the asymptotic limit. In this case, we say that u has trivial asymptotic behavior at the puncture. Otherwise, the asymptotic behavior is said to be nontrivial at the puncture. be shown that if π λ • du does not vanish identically then its zeros are isolated and count positively. Theorem 2.6 further implies that there are finitely many zeros in this case. HWZ [18] define to be the algebraic count of zeros in case π λ • du does not vanish identically. Fix a dλ-symplectic trivialization σ of u * ξ. Let z be a puncture of u with asymptotic limit P = (x, T ). The asymptotic behavior described in Theorem 2.6 allows one to deform σ so that it extends to a trivialization of x(T ·) * ξ. Let wind ∞ ( u, z, σ) ∈ Z be defined to be the winding of the asymptotic eigenvalue of u at z with respect to the extension of σ to x(T ·) * ξ. Finally, we consider where Σ + denotes a sum over the positive punctures, and Σ − is a sum over the negative punctures. Standard degree theory shows that For the next two definitions consider a finite-energy plane u = (a, u) : (C, i) → (R × M, J) and assume that λ is nondegenerate up to action E( u). By Stokes' theorem, ∞ must be a positive puncture, and the similarity principle implies that Definition 2.9. The covering multiplicity cov( u) of the plane u is the covering multiplicity of its asymptotic limit. Proof. That u is an immersion transverse to X λ follows from (4) and (5). If u is not somewhere injective, then it covers another plane via a polynomial map of degree ≥ 2, but this forces u to have critical points, in contradiction to u being an immersion; here, we used that the Cauchy-Riemann equations force a critical point to be a zero of the derivative of u.

Asymptotic cycles
Here, we explain the basics on asymptotic cycles, and state the main result from [26]. Let φ t be a smooth flow on a smooth, closed, oriented and connected 3-manifold M , and let L be a link consisting of (nonconstant) periodic orbits. The set of φ t -invariant Borel probability measures on M \ L is denoted by P φ (M \ L). Fix an auxiliary Riemannian metric g on M . If p ∈ M \ L is recurrent and the sequence T n → +∞ satisfies φ Tn (p) → p, then we denote by k(T n , p) loops obtained by concatenating to φ [0,Tn] (p) a g-shortest path If γ is the periodic orbit given by a connected component of L, then ξ γ = T M| γ /T γ is a rank-2 vector bundle over γ. It carries an orientation induced by the ambient orientation and the flow orientation on γ. A positive frame of ξ γ allows one to identify ξ γ γ × C R/T γ Z × C, where T γ > 0 is the primitive period. If t is the coordinate on R/T γ Z (given by the flow) and θ ∈ R/2πZ is the polar angle on C * then {dt, dθ} is a basis of H 1 ((ξ γ \0)/R + ; R). With the aid of any exponential map the class y induces a class in this homology group that can be written as pdt+qdθ. The coefficients p, q ∈ R depend only on y and on the chosen frame. If u is a nonzero vector in ξ γ , then using the frame we can write dφ t · u r(t)e iθ(t) with smooth functions r(t) > 0, θ(t). The rotation number is independent of the choice of frame, and of the vector u.
The following statement is a refinement of a result due to Fried [11], see also Sullivan [40].

Proof of Theorem 1.1
The main input in the proof is the following statement proved with pseudoholomorphic curves. Here, there are no hidden genericity assumptions, the only assumption is that of dynamical convexity. A disk-like global surface of section D spanned by some unknotted, self-linking number −1 periodic orbit γ = ∂D obtained from the above result has the following property: the first return time Since D is a global surface of section, it follows from (7) that there exists Let γ 1 , . . . , γ N be a collection of unknotted, self-linking number −1 periodic Reeb orbits. These orbits are taken as knots, i.e., primitive orbits, oriented by the flow. Consider a disk-like global surface of section D i spanned by γ i , provided by Theorem 3.1, oriented in such a way that the identity ∂D i = γ i takes orientations into account. Algebraically counting intersections with D i induces a cohomology class y i ∈ H 1 (S 3 \ γ i ; R). Denoting inclusion maps by ι j : S 3 \ ∪ i γ i → S 3 \ γ j we get a cohomology class Denote also which are positive integers since all D i are global surfaces of section. Let T i denote the primitive period of γ i . With i fixed consider a small smooth compact neighborhood N i and a smooth, orientation preserving, diffeomorphism Ψ i : Here D ⊂ C denotes the unit disk oriented by the complex orientation. Up to twisting, we may assume that Ψ i is aligned with D i , i.e., if > 0 is small then the linking number of the loop t → Ψ −1 i (t, ) with γ i is equal to zero. Denote by re iθ the polar coordinates on D i . It follows that with respect to the basis {dt It follows from this and from the definition of the rotation number (6) that where (9) was used in the third line. We claim that this limit is strictly positive. This will follow from CZ(γ i ) ≥ 3 together with sl(γ i ) = −1. Here we write CZ for the Conley-Zehnder index in a global dλ-symplectic frame of (ξ 0 , dλ). In fact, the global dλ-symplectic frame of ξ 0 rotates sl(γ i ) = −1 turns with respect to a dλ-symplectic of ξ| γi aligned with D i . It turns out that there exists α i ∈ R such that CZ(γ Hence, we are done checking which is (iii-a) in Theorem 2.11. Now, we check (iii-b). Let μ ∈ P φ (S 3 \ ∪ i γ i ) be arbitrary. As explained in subsection 2.3, there exists a Borel set E ⊂ M \ ∪ i γ i contained in the set of recurrent points such that μ(E) = 1, and for all p ∈ E the limits lim n→∞ y, k(T n , p) /T n exist independent of the sequence T n → +∞ satisfying φ Tn (p) → p and define a function f μ,y ∈ L 1 (μ) whose integral is μ · y. Since each D i \ γ i is transverse to the flow, we conclude that μ(E \∪ i D i ) = 1. Fix p ∈ E \ ∪ i D i and a sequence T n → +∞ satisfying φ Tn (p) → p. Then using the (positive) transversality of the flow with all the surfaces D i \ γ i where τ i is the return time function of D i . Recall that sup τ i < +∞ (7). Plugging (13) into (12) we obtain Taking the limit as n → ∞ and we are done checking (iii-b). A direct application of Theorem 2.11 concludes the proof of Theorem 1.1.

Proof of Theorem 1.4
Let P = (x, T = m 0 T 0 ) be a periodic Reeb orbit, with multiplicity m 0 , of a defining contact form λ on (S 3 , ξ 0 ). Here T 0 denotes the primitive period of x. Throughout this section we denote by τ a global dλ-symplectic trivialization of ξ 0 . Assume that λ is dynamically convex up to action T , and also that λ is nondegenerate up to action T .  (16) holds for every p ∈ S 3 \ x(R) and every T ≥ 0.
We first show that Proposition 4.1 can be used to check the hypothesis of Theorem 2.11 for the periodic orbit x(R) and the cohomology class counting linking numbers with it. Theorem 1.4 follows as a consequence.
Proof that Theorem 1.4 follows from Proposition 4.1. Let y ∈ H 1 (S 3 \ x(R); R) be the cohomology class that counts linking number of loops in Here, we ignore Z-coefficients and work with R-coefficients. If we compactify C to a disk D C by adding a circle at ∞ then u induces a capping diskū : D C → S 3 for P such that the class in is the fundamental class in H 2 (D C , ∂D C ; Z) induced by the complex orientation. Observe that u(C) \ x(R) has measure zero with respect to any μ ∈ P φ (S 3 \ x(R)) since it is transverse to the flow. Hence, in view of the discussion in Sect. 2.3, we get a Borel set E ⊂ S 3 \ x(R) such that μ(E) = 1 and every point p ∈ E has the following properties: (a) p is recurrent. (b) The limits lim n→+∞ link(k(T n , p), x(R)) T n = lim n→+∞ y, k(T n , p) T n exist independent of T n → +∞ satisfying φ Tn (p) → p (and of auxiliary Riemannian metrics), and define a function in L 1 (μ) whose integral is equal to the intersection number μ · y. (c) p ∈ u(C). Hence, using the transversality between u and the Reeb vector field, for every p ∈ E we can estimate m 0 link(k(T n , p), x(R)) = m 0 y, k(T n , p) for all n large enough, where T n → +∞ satisfies φ Tn (p) → p. With the aid of Proposition 4.1 we can estimate from (17) (18) which implies, by definition of intersection numbers, that Condition ρ y (x(R)) > 0 follows immediately from CZ(P 0 ) ≥ 3 where P 0 = (x, T 0 ) is the simply covered periodic orbit underlying P . Theorem 1.4 now follows from a direct application of Theorem 2.11 since y is dual to any Seifert surface for x(R); here, it was used that the ambient space is S 3 .
To complete the proof of Theorem 1.4 we need to establish Proposition 4.1. The rest of this section is concerned with the proof of Proposition 4.1.
Let us denote by P 0 = (x, T 0 ) the simply covered periodic orbit underlying P , and recall that m 0 denotes the covering multiplicity of P = (x, T = m 0 T 0 ). For every k ≥ 1, we denote P k 0 = (x, kT 0 ). In particular P = P m0 0 . Consider the set M fast (P, J) of equivalence classes of fast finite-energy planes asymptotic to P , where two planes u, v are equivalent if there exist A ∈ C * , B ∈ C and c ∈ R such that v c (z) = u(Az + B) holds for every z ∈ C.
Here v c denotes the translation of v by c in the R-component. Equivalence classes are denoted by [ u].
It is possible to build a Fredholm theory for M fast (P,J) modeled on sections of the normal bundle, using Sobolev or Hölder spaces [3]. Fix a number δ < 0 in the spectral gap of the asymptotic operator associated to (P, J) between eigenvalues with winding number 1 and 2 with respect to τ . This is possible since CZ 0 τ (P ) ≥ 3. Note that α <δ τ (P ) = 1 and α ≥δ τ (P ) = 2. Let An important fact is that automatic transversality holds, i.e., D u at a fast plane u is always a surjective Fredholm operator. Let us prove this fact. There is no loss of generality to deform the normal bundle so that it coincides with u * ξ 0 over C \ B R (0), R 1. A dλ-symplectic trivializing frame of the normal bundle induces, up to homotopy, a dλ-symplectic trivialization σ N of x(T ·) * ξ 0 which winds +1 with respect to the global frame τ . Moreover, a nontrivial section ζ ∈ ker D u admits an asymptotic behavior governed by an eigensection of the asymptotic operator associated with an eigenvalue ν < δ, see [23, Theorem 6.1], or [38,Theorem A.1]. Hence, ζ does not vanish near ∞ and the total algebraic count of zeros of ζ is equal to the winding number of ν with respect to σ N , which is equal to wind τ (ν) − 1 ≤ α <δ τ (P ) − 1 = 1 − 1 = 0. But the equation D u ζ = 0 allows us to use Carleman's similarity principle to say that zeros are isolated and count positively. The important conclusion is that ζ never vanishes. Since the Fredholm index is 2, we would find 3 linearly independent sections of the kernel if D u were not surjective. But the normal bundle is two-dimensional, and hence a nontrivial linear combination of them would have to vanish at some point, which gives the desired contradiction.

Remark 4.2.
Arguments like the one used above to prove automatic transversality statements were explored in [16,41], see also [25].
It follows from the above discussed automatic transversality that M fast (P, J) can be given the structure of a smooth, Hausdorff and secondcountable one-dimensional manifold.

Remark 4.3.
Under our standing assumption that λ is nondegenerate up to action T one can show that the topology on M fast (P, J) inherited from the functional analytic set-up used for the Fredholm theory coincides with the topology of C ∞ loc -convergence. There are situations where this can also be proved dropping nondegeneracy [24,29].
Consider a sequence u n : (C, i) → (R × S 3 , J) of fast finite-energy planes asymptotic to P . Since λ is assumed to be nondegenerate up to action T we can apply the SFT compactness theorem [3] to get, up to selection of a subsequence, that u n SFT-converges to a stable holomorphic building u. Since u is a limit of planes it can be conveniently described as a directed, rooted tree T . Each vertex v corresponds to a finite-energy map with a unique positive puncture ∞. The finite set Γ v consists of the negative punctures of u v . The top level of this building corresponds to the root r, and consists of a single finite-energy map u r which is asymptotic to P at its positive puncture ∞. Edges are always assumed oriented as going away from the root. An edge e from the vertex v to the vertex v corresponds to a negative puncture of u v . The asymptotic limit u v at the negative puncture corresponding to e is equal to the asymptotic limit of u v at its positive puncture. The leaves correspond precisely to the vertices v such that u v is a plane (Γ v = ∅).
Proof. SFT compactness allows us to find A n ∈ C * , B n ∈ C and c n ∈ R such that the planes w n (z) = c n · u n (A n z + B n ) converge to u v in C ∞ loc (C \ Γ v ). Here, c n · u n denotes the translation by c n in the R-component.
The frame τ can be used to represent the maps (s, t) → π λ (∂ s w n ) by smooth maps ζ n : [s 0 , +∞) × R/Z → C satisfying a Cauchy-Riemann type equation. Carleman's similarity principle implies that either ζ n vanishes identically on [s 0 , +∞) × R/Z, or its zeros are isolated and count positively. It can not vanish identically since the w n are planes. By Theorem 2.6 ζ n (s, t) does not vanish when s is large enough and for every n we have lim s→+∞ wind(ζ n (s, ·)) = lim s→+∞ wind(π λ (∂ s w n )(s, ·)) = wind ∞ ( w n ) = wind ∞ ( u n ).
If s > s 1 is large enough then wind(ζ n (s, ·)) − wind(ζ n (s 1 , ·)) is the algebraic count of zeros of ζ n on [s 1 , s] × R/Z. Since this count is nonnegative we get for all n ≥ n 0 . Hence, as desired. Proof. Suppose that u * v dλ > 0. At the negative punctures z ∈ Γ v of u v , we have wind ∞ ( u v , z, τ) ≥ 2 since the asymptotic limits at these punctures are periodic Reeb orbits with action less than T and hence, by assumption, satisfy CZ 0 τ ≥ 3. By the previous lemma together with (4) and (5) we arrive at Thus, Γ v = ∅ and v is a leaf. Proof. Case (i) is handled by the previous lemma. Let us now argue for (ii). By Lemma 4.4 if v is a leaf then it is a plane satisfying wind ∞ ( u v ) ≤ 1.
, wind ∞ ( u v ) = 1 and u v is a fast plane, and by (i) it is clearly asymptotic to a cover of P 0 .
For every 1 ≤ k ≤ m 0 , we consider M fast (P k 0 , J) the moduli space of fast finite-energy planes asymptotic to P k 0 , defined as before. For each k there is a suitable choice of negative weight placed precisely at the spectral gap between eigenvalues of the asymptotic operator associated to (P k 0 , J) with winding 1 and 2 in a global frame. With these weights one builds a Fredholm theory as before, and there is automatic transversality. The space M fast (P k 0 , J) becomes a 1-dimensional smooth, second countable Hausdorff manifold. Moreover, the induced topology coincides with the topology induced by C ∞ loc -convergence. Proof. We work under the assumption of Theorem 1.4 that M fast (P = P m0 0 , J) is nonempty. If M fast (P, J) is compact then there is nothing to be proved. If M fast (P, J) is not compact then some sequence in M fast (P, J), represented by fast planes u n , will SFT-converge to a building u with more than one level. This means that the corresponding tree T does not consist of a single vertex (the root), and by Corollary 4.6 every leaf v must be a fast finite-energy plane asymptotic to P kv 0 , for some k v ∈ {1, . . . , m 0 −1}. The reason for the strict inequality k v < m 0 is that the root must have at least two negative punctures: otherwise the root corresponds to a trivial cylinder, which is ruled out by stability of the limiting building. Pick any leaf v, denote m 1 = k v . Hence the moduli space M fast (P m1 0 , J) of fast planes asymptotic to P m1 0 is not empty. If M fast (P m1 0 , J) is compact, then we are done with the proof. If not we proceed just as above to find 1 ≤ m 2 ≤ m 1 − 1 such that M fast (P m2 0 , J) is non empty. After a finite number of steps k ≥ 0 this process stops and we find 1 ≤ m k ≤ m 0 such that M fast (P m k 0 , J) is nonempty and compact.
From now on m is given by the previous lemma, that is, M fast (P m 0 , J) is a nonempty, compact, smooth and Hausdorff 1-dimensional manifold, i.e., a finite collection of circles.
Consider the space M fast 1 (P m 0 , J) of equivalence classes of pairs ( u, z) where u is a fast plane asymptotic to P m 0 and z ∈ C. Two pairs ( u 0 , z 0 ),  Proof. For every u ∈ M fast (P m 0 , J) nontrivial sections in the kernel of the linearized Cauchy-Riemann operator at u, with the appropriate weighted exponential decay, which represent elements in the tangent space, never vanish and u is an immersion.
Proof. Suppose that [ u n , z n ] represents a sequence in ev −1 (K). Up to reparametrization, translation in the R-component, and selection of a subsequence, we may assume that u n converges in C ∞ loc to some plane u representing an element of M fast (P m 0 , J). Let N be a neighborhood of x(R) such that K ∩ N = ∅. One can then invoke results on cylinders of small contact area from [22] to conclude that there exists R and n 0 such that if n ≥ n 0 and |z| ≥ R then u n (z) ∈ R × N . This implies that sup n |z n | ≤ R. Hence one can assume, up to selection of a subsequence, that z n → z for some z. It follows that [ u n , z n ]/R → [ u, z]/R.
defined by with the convention that the infimum of the empty set is +∞. Proof. From the transversality of u to the Reeb flow, and the asymptotic formula from Theorem 2.6, we conclude that given any [ v = (b, v)] ∈ M fast (P m 0 , J) and p ∈ S 3 , the set v −1 (p) ⊂ C is finite, and also that τ takes values on (0, +∞]. Suppose that p ∈ x(R) and ω(p) ∩ x(R) = ∅. By invariance of x(R) under the Reeb flow, the trajectory φ t (p) will spend arbitrarily long times in the future arbitrarily and uniformly close to x(R). Hence, the way in which it rotates around x(R) is governed by the linearized Reeb flow along x. Every plane v = (b, v) representing an element in M fast (P m 0 , J) is asymptotic to P m 0 according to an eigenvector of a negative eigenvalue of the asymptotic operator with winding +1 in a global frame; this information is encoded in wind ∞ ( v) = 1. Hence, in transverse polar coordinates aligned with the global frame the plane rotates 2π. After one period T = mT 0 the linearized flow rotates every transverse vector by an angle larger than 2π + Δ for some uniform Δ > 0. This information is encoded in CZ 0 τ (P m 0 ) ≥ 3. Hence after flow time of about 2π Δ +1 T any point nearby P 0 already returned once back to the plane. It follows that the return time is bounded from above for points near P 0 .
If ω(p)∩x(R) = ∅, then it follows from compactness of M fast (P m 0 , J) and transversality of the planes to the Reeb flow that for every [ v] ∈ M fast (P m 0 , J) the trajectory φ t (p) will hit v(C) in finite time.
So far we have proved that τ takes values on (0, +∞), and that τ is bounded near x(R). To conclude, we note that if {w 1 , . . . , w N } = u −1 (φ τ (p) (p)), then there are N local smooth germs of hitting times τ 1 , . . . , τ N near p. Then, τ can be locally bounded in terms of these germs. Remark 4.12. We observe that the finite energy planes produced by Corollary 4.7 can themselves be thought of as sorts of generalized surfaces of section where we allow for the possibility that the surface is an immersion rather than embedding. Indeed our proof shows that the projection to S 3 of every such plane is an immersion, transverse to the Reeb flow, and that the flow line through any given point in S 3 \ P 0 will hit the surface in forward and backward time. In the case that the plane is not an embedding, it follows from results in [18,37] that it must intersect its asymptotic limit, and thus in this case the plane will intersect the flow line through any given point in S 3 including points in P 0 .
We observe further that, since our proof shows that the evaluation map ev : M fast 1 (P m 0 , J)/R → S 3 is an immersion between manifolds of the same dimension, it is also a local diffeomorphism, so we can use ev −1 to lift the flow to the moduli space M fast 1 (P m 0 , J)/R, each component of which is diffeomorphic to C × S 1 . Moreover, since each plane in M fast 1 (P m 0 , J)/R is transverse to the flow, the resulting flow on M fast 1 (P m 0 , J)/R will be transverse to the disk-like fibers of the forgetful map M fast 1 (P m 0 , J)/R → M fast (P m 0 , J)/R. So although the surface of section provided by our theorem will in general have genus, the fast finite energy planes that we construct in the proof can themselves be used to visualize the dynamics as a return map on a disk.