On the local systolic optimality of Zoll contact forms

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.


Introduction
Metric systolic geometry A classical problem in Riemannian geometry consists in bounding the length of the shortest closed geodesic on a closed Riemannian manifold (W, g) by the volume of this manifold. In other terms, one asks if the systolic ratio of (W, g), i.e. the scaling invariant quantity ρ sys (W, g) := ℓ min (g) n vol(W, g) , where n = dim W and ℓ min (g) denotes the minimum of the length of all closed geodesics on (W, g), is bounded from above on the space of all Riemannian metrics on W . The first investigations on this problem go back to Loewner, who in a course given at Syracuse University in 1949 proved that the systolic ratio of the two-torus is maximized by the flat torus that is obtained as the quotient of R 2 by a lattice generated by two sides of an equilateral triangle (see [Ber03, Section 7.2.1.1] for two different proofs Loewner's result). Shortly afterwards, Pu [Pu52] showed that the systolic ratio of the projective plane is maximized by the Fubini-Study metric. A very general result, still in the framework of non-simply-connected manifolds, for which one can obtain closed geodesics by minimizing the length of non-contractible closed curves, was obtained by Gromov [Gro83]: The systolic ratio of any essential manifold is bounded from above by a constant depending only on the dimension. Here, a closed manifold W is called essential if its fundamental class is non-zero in the Eilenberg-MacLane space K(π 1 (W ), 1) of its fundamental group.
The first result about simply connected manifolds is due to Croke [Cro88], who showed that the systolic ratio of the two-sphere is bounded from above. Interestingly, the round metric does not maximize ρ sys (S 2 , ·), whose supremum is currently unknown, but it is a local maximizer, together with all Zoll metrics on S 2 , i.e. metrics all of whose geodesics are closed and have the same length (see [ABHS17] for the local maximality of Zoll metrics among suitably pinched metrics on S 2 and [ABHS18] for the case of an arbitrary Zoll metric on S 2 ). The question whether the systolic ratio of a simply connected manifold of dimension larger than two is bounded from above is open, even for spheres. Equally open is the boundedness of the systolic ratio of non simply connected non-essential manifolds, such as for instance S 2 × S 1 : The minimal length of a non-contractible closed curve can be arbitrarily large on any non-essential manifold of unit volume, see [Bab93], but this does not exclude the existence of short contractible closed geodesics.
Consider now a Finsler metric on the closed n-dimensional manifold W , i.e. a positively 1-homogeneous function F : T W → [0, +∞) that is smooth and positive outside of the zero section and such that the second fiberwise differential of F 2 is positive definite outside of the zero section. The systolic ratio of (W, F ) is the quantity where ℓ min (F ) denotes the length of the shortest closed geodesic on (W, F ) and vol(W, F ) is the Holmes-Thompson volume of (W, F ), which we normalize so that it coincides with the usual Riemannian volume when F = √ g is Riemannian.
Both Gromov's and Croke's results about the boundedness of the systolic ratio in the Riemannian setting extend to the Finsler setting. Indeed, bounds on the Riemannian systolic ratio imply bounds on the Finsler one by a combined use of Loewner ellipsoids and the Rogers-Shephard inequality in convex geometry, see [APBT16].

Contact systolic geometry
In [APB14],Álvarez-Paiva and Balacheff proposed to extend questions from metric systolic geometry to the broader setting of contact geometry and Reeb dynamics, in which one can take advantage of a larger symmetry group. We recall that a co-oriented contact structure ξ on the closed (2n−1)-dimensional manifold M is a maximally non-integrable, co-oriented hyperplane distribution ξ ⊂ T M. We call any one-form α on M such that ξ = ker α a contact form supporting the contact structure ξ. In this case, the top-degree form α∧dα n−1 is nowhere vanishing. Therefore, α ∧ dα n−1 is a volume form on M, and the volume of M with respect to it is denoted by vol(M, α) :=ˆM α ∧ dα n−1 .
It is then natural to define the systolic ratio of (M, α) as ρ sys (M, α) := T min (α) n vol(M, α) ∈ (0, +∞], where T min (α) denotes the minimum of the periods of all closed orbits of R α . Here, T min (α) is defined to be +∞ if R α does not have any closed orbit. Note, however, that the Weinstein conjecture, which has been confirmed for many contact manifolds, asserts that any Reeb vector field on a closed manifold has closed orbits, so ρ sys (M, α) is expected to be always a finite number. An important source of examples is given by starshaped hypersurfaces in the cotangent bundle T * W of any closed n-dimensional manifold W . Here, a hypersurface M ⊂ T * W is said to be starshaped if every ray in each cotangent fiber emanating from the origin meets M transversally at exactly one point, and we take as contact form on M the restriction of the Liouville form p dq defined on the ambient manifold T * W . If such a hypersurface is fiberwise strictly convex, then it can be seen as the unit cotangent sphere bundle S * F W of a Finsler metric on W . The Reeb flow of the associated contact form α F is then precisely the geodesic flow of F . In particular, T min (α F ) coincides with ℓ min (F ) and the two volumes are related by the identity vol(S * F W, α F ) = n! ω n vol(W, F ), where ω n denotes the volume of the Euclidean n-ball. Therefore, the Finsler systolic ratio of (W, F ) coincides up to a multiplicative constant with the contact systolic ratio: ρ sys (S * F W, α F ) = 1 n! ω n ρ sys (W, F ).
While in the metric case one studies the systolic ratio on W as a function of the metric F , in the contact case it is natural to consider the systolic ratio on (M, ξ) as a function of the contact form α supporting ξ. This is indeed an interesting problem, as the space of such contact forms is infinite dimensional, being parametrised by positive smooth functions f on M via f → α = f α * , where α * is a fixed contact form. We can then investigate global and local properties of the systolic ratio on this space. As far as global properties are concerned,Álvarez-Paiva and Balacheff asked whether the systolic ratio is bounded from above on the space of contact forms supporting ξ. This question was given a negative answer: Any closed contact manifold (M, ξ) admits contact forms of arbitrarily large systolic ratio. This was first proven for the tight three-sphere in [ABHS18], for arbitrary contact three-manifolds in [ABHS19], and in full generality in [Sag18]. In particular, without the convexity assumption a starshaped hypersurface in T * W can have an arbitrarily high systolic ratio, for every closed manifold W .
As far as local properties are concerned, a special role is played by Zoll contact forms, that is, contact forms such that all Reeb orbits are closed and have the same period. Zoll contact forms were introduced by Reeb in [Ree52] under the name of "fibered dynamical systems with an integral invariant" and are also called "regular" in the subsequent literature, but we prefer the term "Zoll", which we borrow from metric geometry: As recalled above, Zoll metrics are those Riemannian or Finsler metrics all of whose geodesics are closed and have the same length.
Alvarez-Paiva and Balacheff studied critical points of ρ sys on the space of contact forms supporting the contact structure ξ and showed that if α is a critical point then it is Zoll. Indeed, if the Reeb flow of a contact form α has an orbit that does not close up within the minimal period T min (α), then all orbits nearby do not close up before T min (α), and one can modify α near this orbit and change the volume at first order while keeping T min (α) constant. See [APB14][Theorem 3.4] for more details.
Zoll contact forms have an easy description that is due to Boothby and Wang [BW58] (see also [Gei08,Section 7.2]): If α is a Zoll contact form on M and T is the common period of all its Reeb orbits, then the quotient of M by the free S 1 -action defined by the Reeb flow is a symplectic manifold (B, ω), and the pull-back of ω by the projection map is (1/T )dα. Moreover, the cohomology class [ω] is integral and is the Euler class of the circle bundle M → B. It follows that the systolic ratio of a Zoll contact form α is the inverse of a positive integer: is the Euler number of the circle bundle M → B. For instance, the standard contact form on S 2n−1 is Zoll with common period π and systolic ratio 1, and the corresponding circle bundle is the Hopf fibration S 2n−1 → CP n−1 . Actually, the Hopf fibration gives a universal model for all Zoll contact forms: The restriction of it to the inverse image of any closed symplectic submanifold of CP n−1 defines a Zoll contact form, and any Zoll contact form with common period π can be produced in this way, by choosing n large enough (see [APB14][Theorem 3.2 and references therein).

The main results
Knowing that critical points of the systolic ratio are Zoll contact forms it is natural to wonder if the converse is also true, and if so, what is the local behavior of the systolic ratio in a neighborhood of a Zoll contact form. The main result of [APB14] goes in this direction: It says that if α t is a one-parameter deformation of the Zoll contact form α 0 , then either the function t → ρ sys (α t ) has a local maximum at t = 0, or α t is tangent up to infinite order to the space of Zoll contact forms at t = 0. See [APB14][Theorem 2.9] for the precise statement. Therefore, we are led to ask: Are Zoll contact forms local maximizers of the systolic ratio, with respect to some reasonable topology on the space of contact forms? The aim of this paper is to give an affirmative answer to this question in the C 3 -topology of contact forms.
Theorem 1 (Local systolic maximality of Zoll contact forms). Let α 0 be a Zoll contact form on the closed manifold M. For all C > 0 there exists δ C > 0 such that, if we define the C 3 -neighborhood N C of α by with equality if and only if α is Zoll.
The local systolic maximality of Zoll contact forms in the C 3 -topology is already known in dimension three: It was first proven for M = S 3 in [ABHS18] and then for arbitrary three-manifolds in [BK19] (see also [BK19b] for a generalization to odd symplectic forms on three-manifolds and [BK19c] for an application to magnetic flows on surfaces). The proofs in [ABHS18] and [BK19] build on the fact that a closed orbit with minimal period of a contact form that is close to a Zoll one on a three-manifold is the boundary of a global surface of section for the Reeb flow. Global surfaces of section bounded by closed orbits are peculiar of dimension three, and we do not see a way of applying this approach to the higher dimensional case.
The proof of Theorem 1 will be based instead on a normal form for contact forms close to Zoll ones. More precisely, it will use the following theorem, that is the second main result of this paper.
Theorem 2 (Normal Form). Let α 0 be a Zoll contact form on the closed manifold M. There is δ 0 > 0 such that if α is a contact form on M with α − α 0 C 2 < δ 0 , then there exists a diffeomorphism u : M → M such that where: (i) S is a smooth positive function on M that is invariant under the Reeb flow of α 0 ; (ii) f is a smooth function on M; (iii) η is a smooth one-form on M satisfying ı Rα 0 η = 0; Moreover, for every integer k ≥ 0 there is a monotonically increasing continuous function The proof of Theorem 2 is based on a normal form for vector fields close to vector fields inducing a free S 1 -action that is due to Bottkol [Bot80], which we include, in the form that is needed here, as Theorem 2.1. In Appendix B, we exhibit a proof of Bottkol's theorem following an idea we learned in [Ker99][Proposition 3.4].
The relevant condition in Theorem 2 is the fourth one. Indeed, any one-form β can be decomposed as β = Sα 0 + η + df, with S, η and f as in (i), (ii) and (iii): Define S(x) to be the integral of β on the closed orbit of R α 0 through x, so that the one-form β − Sα 0 has zero integral on every orbit of R α 0 and hence differs by a differential from a one-form vanishing on R α 0 (see Lemma 1.3).
Being invariant under the flow of R α 0 , the function S descends to a smooth function on the quotient B of M by the free S 1 -action defined by this flow. Condition (iv) implies that the function S is a variational principle for detecting closed orbits of R α of short period, that is, those closed orbits that bifurcate from the (2n − 2)-dimensional manifold of closed orbits of R α 0 . Indeed, the following result is easy to prove (see Section 3).
Proposition 1 (Variational principle). Let α 0 be a Zoll contact form on the closed manifold M and let π : M → B be the corresponding S 1 -bundle. Let β be a contact form on M of the form where S, η and f satisfy the conditions (i)-(iv) of Theorem 2. Denote by S : B → R the function that is defined by S = S • π. Then for every critical point b of S the circle π −1 (b) is a closed orbit of R β of period S(b)T min (α 0 ). Moreover, β is Zoll if and only if the function S -or equivalently the function S -is constant.
Theorem 2 and Proposition 1 immediately imply that any contact form α that is C 2close to the Zoll contact form α 0 has at least as many closed orbits as the minimal number of critical points of a smooth function on B. Indeed, the image by the diffeomorphism u of Theorem 2 of the circles π −1 (b) corresponding to critical points b ∈ B of S are closed orbits of R α .
For instance, if α is a contact form on S 2n−1 that is C 2 -close to the standard Zoll contact form whose Reeb trajectories defines the Hopf fibration S 2n−1 → CP n−1 , then R α has at least n closed orbits of period close to π. Proving this and more general multiplicity results for closed orbits bifurcating from manifolds of closed orbits was Bottkol's original motivation for his normal form. See also [Wei73b], [Wei77], [Mos76], [Gin87], [Gin90], [Ban94] and [BR94] for other approaches to this question.
Besides producing a finite dimensional variational principle, the power of the normal form appearing in Theorem 2 lies in the fact that it yields the following useful formula for the volume.
Proposition 2 (Volume formula). Assume that α 0 is a Zoll contact form on the (2n − 1)dimensional closed manifold M and let β be a one-form on M of the form where S and f are smooth functions on M and η is a one-form satisfying for some endomorphism F : T * M → T * M lifting the identity. Then Moreover, for every c > 0 and ǫ > 0 there exists δ > 0 such that if then p j C 0 < ǫ for every j = 1, . . . , n − 1.
It is now easy to see how Theorem 2, Proposition 1 and Proposition 2 lead to the proof of the sharp systolic inequality of Theorem 1. Indeed, for every C > 0 we can find a positive number δ C such that if α belongs to the neighborhood N C defined in Theorem 1, then α can be put in the normal form β = u * α of Theorem 2 by a diffeomorphism u, and furthermore the function s → p(x, s) of Proposition 2 is strictly increasing on the interval [min S, max S] for every x ∈ M. This fact, together with the fact that the principal coefficient of the polynomial map p is 1 and all the other coefficients have vanishing integral, implies the estimate By Proposition 1, the Reeb vector field of α has a closed orbit of period (min S)T min (α 0 ), and hence T min (α) ≤ (min S)T min (α 0 ).
The above two inequalities imply the desired sharp systolic bound The fact that α is Zoll if and only if the function S is constant, see again Proposition 1, implies that the equality holds in the above estimate if and only if α is Zoll. See Section 5 for a detailed proof. At the end of that section we also discuss a lower bound for the maximal period of "short" closed orbits.

Three applications of Theorem 1
We conclude this introduction with three corollaries of the local systolic maximality of Zoll contact forms.
Finsler geodesic flows. The first corollary is immediate and consists in applying Theorem 1 to the contact form α F on S * F W that is induced by a Finsler metric F on W .
Corollary 1. Let F 0 be a Zoll Finsler metric on the closed manifold W . Then F 0 has a C 3 -neighborhood U in the space of all Finsler metrics on W such that with equality if and only if F is Zoll.
In dimension two, this theorem follows from known results: The only surfaces admitting Zoll Finsler metrics are S 2 , for which this result was proven in the already mentioned articles [ABHS18] and [ABHS19], and RP 2 , for which the result immediately follows by lifting the metric to S 2 . Actually, reversible Zoll metrics on RP 2 are global maximizers of the systolic ratio among reversible Finsler metrics, as proven by Ivanov in [Iva11]. In higher dimensions, the local sharp systolic inequality of Corollary 1 appears to be a new result, even for Riemannian perturbations of simple rank-one spaces, such as the round S n or the round RP n . In particular, this corollary gives a positive answer to the local version of Question 5.3 in Berger's survey paper [Ber70].
Symplectic capacity of convex domains. Our next corollary concerns the behavior of symplectic capacities on convex bodies in R 2n . Recall that a (normalized) symplectic capacity on the vector space R 2n , endowed with its standard symplectic structure ω 0 , is a function c : P(R 2n ) → [0, +∞] that satisfies the following conditions: (c3) Homogeneity: c(λA) = λ 2 c(A) for all λ > 0.
Many non-equivalent symplectic capacities have been constructed in this and in more general settings, but for convex bodies many of them have been shown to coincide: This is the case of the first of the Ekeland-Hofer capacities (see [EH89]), of the Hofer-Zehnder capacity (see [HZ90]), of the Viterbo capacity (see [Her04]) and of the capacity coming from symplectic homology (see [AK19] and [Iri19]). Following a common usage, we shall refer to the common value of these capacities on convex bodies as Ekeland-Hofer-Zehnder capacity and denote it by c EHZ . The crucial fact about it is that when the convex body C ⊂ R 2n is a neighborhood of the origin and has a smooth boundary, then the minimal period of closed orbits on ∂C with respect to the Reeb flow induced by the contact form α C := λ 0 | ∂C , where λ 0 is the homogeneous primitive of ω 0 , that is the one-form In [Vit00], Viterbo formulated a challenging conjecture relating symplectic capacities and volume: If c : P(R 2n ) → [0, +∞] is any symplectic capacity and C ⊂ R 2n is a convex body, then c(C) n ≤ vol(C, ω n 0 ), with equality if and only if C is symplectomorphic to a ball. Note that vol(C, ω n 0 ) is n! times the Euclidean volume of C. This conjecture has been shown to be asymptotically true, that is, valid up to a multiplicative constant that is independent on the dimension, in [AAMO08]. Moreover, its validity in the sharp form for the Ekeland-Hofer-Zehnder capacity would imply the Mahler conjecture in convex geometry, see [AAKO14].
Thanks to Theorem 1, we can prove the sharp version of this conjecture for the Ekeland-Hofer-Zehnder capacity assuming the convex body C to be C 3 -close to a ball.
Corollary 2. There is a C 3 -neighborhood B of the ball in the space of smooth convex bodies in R 2n such that with equality if and only if C is symplectomorphic to a closed ball.
For n = 2, this is proven in [ABHS18]. The inequality in the above corollary is an immediate consequence of Theorem 1, thanks to identity (1) and to Stokes theorem, which gives us the identity vol(∂C, α C ) = vol(C, ω n 0 ). In order to characterize the equality, we need to show that if the Reeb flow of (∂C, α C ) is Zoll, then C is symplectomorphic to a ball. In [ABHS18,Proposition 4.3] this is proven for n = 2 in full generality for all starshaped domains C. In higher dimension, many of the ingredients of that proof break down and we do not known if the result holds true, but we are able to recover it for domains that are C 3 -close to the ball by a combined use of Moser's trick and generating functions. See Proposition 6.1 below.
Symplectic non-squeezing in the intermediate dimensions. Our last corollary concerns a local generalization to intermediate dimensions of Gromov's non-squeezing theorem [Gro85]. Recall that this theorem can be stated in the following way: If P V is the symplectic linear projection onto a symplectic two-dimensional subspace V ⊂ R 2n (i.e. linear projection along the symplectic orthogonal) and ϕ : R 2n → R 2n is a symplectomorphism, then In other words, the two-dimensional shadow of a symplectic ball has a large area, see [EG91]. In [AM13] it was shown that higher dimensional shadows of symplectic balls can have arbitrarily small volume: If P V is the symplectic linear projection onto a symplectic 2k-dimensional subspace V ⊂ R 2n with 1 < k < n and ǫ is any positive number, then there exists a symplectomorphism ϕ : On the other hand, if Φ : R 2n → R 2n is a linear symplectomorphism, then the volume of the shadow of the image of the ball B 2n by Φ is given by the identity where the function w associates to any 2k-dimensional real subspace W ⊂ R 2n ∼ = C n the number w(W ) := |ω k 0 [w 1 , . . . , w 2k ]| k! |w 1 ∧ · · · ∧ w k | , with w 1 , . . . , w 2k basis of W.
By the Wirtinger inequality, w(W ) ≤ 1 and equals 1 if and only if W is a complex subspace, so the above identity implies the sharp inequality for the linear symplectomorphism Φ and tells us that the equality holds if and only if Φ −1 (V ) is a complex subspace. See [AM13] and Theorem 7.1 below.
In [AM13], some evidence to the conjecture that the above sharp inequality should hold also for nonlinear symplectomorphisms that are close enough to linear ones was given. Thanks to Theorem 1, we can confirm this conjecture for C 3 -closeness.
Corollary 3. There is a C 3 loc -neighborhood W of the set of linear symplectomorphisms in the space of all smooth symplectomorphisms of R 2n such that the following holds: If 1 ≤ k ≤ n and P V is the symplectic linear projection onto a symplectic 2k-dimensional subspace V ⊂ R 2n then vol(P V (ϕ(B 2n )), ω k 0 | V ) ≥ π k for every ϕ ∈ W .
For k = 2, a slightly weaker version of this result was proven in [ABHS18] (there, the order of quantifiers is different, and the neighborhood W depends on the choice of the linear symplectic subspace V ). In the analytic category, a related result for arbitrary k is proven in [Rig15].
It is interesting to observe that, in contrast to the above result, other inequalities of a similar flavor are known to fail in the intermediate dimensions, even locally. For instance, Gromov studied the higher homological systoles of metrics on CP n having the same volume as the Fubini-Study metric g 0 and showed that the 2-systole of CP 2 is locally maximized by g 0 (and all its quasi-Kähler deformations), whereas for 2 ≤ k ≤ n − 1 there are metrics on CP n that are arbitrary close to g 0 and have a strictly larger 2k-systole. See [Gro96,Section 4].
Corollary 3 is proven in Section 7 below. Here we wish to remark that the validity of the Viterbo conjecture for the Ekeland-Hofer-Zehnder capacity would imply the conclusion of Corollary 3 for all symplectomorphisms ϕ such that ϕ(B 2n ) is convex. Indeed, this follows from the fact that the Ekeland-Hofer-Zehnder capacity of the image of a convex body C ⊂ R 2n with respect to the linear symplectic projection P V is not smaller than the Ekeland-Hofer-Zehnder capacity of C: where the capacity on the left-hand side is acting on subsets of the symplectic vector space (V, ω 0 | V ). The above inequality follows from the characterizations of the Ekeland-Hofer-Zehnder capacity via Clarke duality, see e.g. [AM15, Theorem 4.1 (v)].

A few facts about differential forms
In this section, we fix some notation and we discuss some results about differential forms that will be used in the proof of the normal form of Theorem 2.
We denote by Λ k M the vector bundle of alternating k-forms on the manifold M and by Ω k (M) the space of smooth sections of this bundle, i.e. differential k-forms on M. The vector bundle Λ 1 M is the cotangent bundle T * M.
The C k -norms of differential forms on M are induced by the choice of some arbitrary but fixed Riemannian metric on M. When estimating such norms, we will use the symbol " " to mean "less or equal up to a multiplicative constant depending on k".
Alternatively, bounds will be given in terms of moduli of continuity. By modulus of continuity we mean here a monotonically increasing continuous function such that ω(0) = 0. This has the advantage that we can conclude the smallness of the output from the smallness of the input and the boundedness of the output from the boundedness of the input in one go.
The first lemma allows us to bound the pullback of differential forms. Its proof is standard and is contained in Appendix A.
and for every α ∈ Ω j (M), 0 ≤ j ≤ d, the following bounds hold: for every integer k ≥ 0, where for k = 0 the term du C k−1 in (1.2) is set to be zero.
The second lemma allows us to bound the distance of two Reeb vector fields in terms of the corresponding contact forms. The proof is given in Appendix A.
Lemma 1.2. Let M be a closed manifold of dimension 2n − 1 with contact form α 0 . Then there exists δ > 0 and a sequence of moduli of continuity ω k such that The last lemma of this section is a splitting result for one-forms on M whose integrals over the Reeb orbits of a Zoll contact form vanish.
Then β splits as where η ∈ Ω 1 (M) satisfies ı Rα 0 η = 0 and f ∈ Ω 0 (M). Moreover, this splitting can be chosen in such a way that for every integer k ≥ 0 the following bounds hold: Proof. By assumption, the function h := ı Rα 0 β has integral zero along each orbit of R α 0 . This implies the existence of a function f ∈ Ω 0 (M) such that ı Rα 0 df = h. This fact can be proven in the following way. Let {ρ j } j=1,...,N be a smooth partition of unity on B, where each ρ j is supported in an open set B j that is a trivializing domain for the S 1 -bundle π.
Then the function h j := (ρ j • π)h is supported in π −1 (B j ). If we identify π −1 (B j ) with B j × S 1 in such a way that R α 0 is identified with the vector field ∂ θ , θ being the variable in S 1 , then the assumption on h implieŝ Then the formula defines a smooth function f j on M that is supported in π −1 (B j ) and satisfies Since the sum of the functions h j is h, we see that Then the one-form η := β − df satisfies the desired relation Let k ≥ 0 be an integer. By differentiating (1.5) we get the bounds where T 0 denotes the period of the orbits of R α 0 . Together with the identities we deduce the bound By the definition of η and the above bound we have Since the C k -norm of ı Rα 0 β can be bounded by the C k -norm of β, the above inequality implies the bound which concludes the proof.

Normal form for contact forms close to a Zoll one
In [Bot80], Bottkol constructed a normal form for vector fields X on a manifold M which are close to a vector field X 0 having a submanifold of periodic orbits with the same minimal period and satisfying a suitable non-degeneracy assumption. In the proof of Theorem 2, we shall use the following version of Bottkol's theorem concerning the case in which the manifold of periodic orbits of X 0 is the whole M.
Theorem 2.1. Let M be a closed manifold and X 0 a vector field on M all of whose orbits are periodic and with the same minimal period T 0 . Then there exists δ > 0 such that for every vector field X on M with X − X 0 C 1 < δ there is a diffeomorphism u : M → M, a vector field V on M, a smooth function h : M → R, and a linear automorphism Q : T M → T M lifting the identity such that: Moreover, for every k ≥ 0, there is a modulus of continuity ω k such that is calculated at points of the unit sphere bundle of M.
Here, L X 0 denotes the Lie derivative along X 0 and g is an arbitrary metric on M that is invariant under the S 1 -action defined by X 0 . In Appendix B, we give a complete proof of the above version of Bottkol's theorem and we discuss it further.
This section is devoted to the proof of the normal form for contact form closes to a Zoll one stated in Theorem 2 from the Introduction.
Proof of Theorem 2. Let α 0 be a Zoll contact form on M with associated S 1 -bundle denoted by π : M → B. Let δ > 0 be the number obtained in Theorem 2.1 taking X 0 = R α 0 . By Lemma 1.2, there exists δ 0 > 0 such that and we can apply Theorem 2.1 to X = R α . We get a smooth diffeomorphism u : M → M, a smooth vector field V on M satisfying a bundle linear morphism Q : T M → T M lifting the identity and a smooth function By choosing the S 1 -invariant metric g so that R α 0 is orthogonal to the contact distribution ker α 0 , we obtain that V takes values into ker α 0 . Thanks to (2.1) and Lemma 1.2, u, V , Q and h satisfy the bounds for every integer k ≥ 0, where the ω k 's are suitable moduli of continuity. In the following argument, we will need to successively replace the ω k 's by larger and larger moduli of continuity, but in order to keep the notation simple we will denote these new functions by the same symbol ω k . By (2.4), u is C 1 -close to the identity when α − α 0 C 2 is small. In particular, up to reducing the size of the positive number δ 0 in (2.2), we may assume that where r is the positive number given by Lemma 1.1. Let us consider now the one-form β := u * α, so that R β = u * R α and (2.3) can be rewritten as For every k ≥ 0, we can bound the C k -norm of the difference β − α 0 using Lemma 1.1 by , where for k = 0 the undefined term du k C k−1 is set to be zero. Using (2.4), we then get a bound of the form Similarly, Lemma 1.1 implies that the C k -norm of the two-form We define the function S ∈ Ω 0 (M) by where T 0 is the common period of the orbits of R α 0 . By construction, the function S is invariant under the action of the Reeb flow of α 0 , i.e. L Rα 0 S = 0. From (2.7) we obtain that S is close to the constant function 1: Denote by φ t α 0 the flow of R α 0 and by its orbit through x ∈ M. Then the function S has the form where S is the action functional defined by the one-form β, i.e.
The Gateaux differential of S at the curve γ is for every tangent vector field ξ along γ. The chain rule implies that the differential of S has the form The above integrand vanishes if β = α 0 , so this identity and (2.8) imply the bound which, together with (2.9) for k = 0, implies By the definition of S, the one-form β − Sα 0 satisfieŝ so by Lemma 1.3 it splits as where η ∈ Ω 1 (M) satisfies ı Rα 0 η = 0 and f ∈ Ω 0 (M). Moreover, the same lemma gives us the bounds for every k ≥ 0. From (2.7) and (2.9) we obtain bounds of the following form for the C k -norm of β − Sα 0 , for every k ≥ 0: (2.14) Now we wish to estimate the C k -norm of the one-form dı Rα 0 (β − Sα 0 ). We have Applying the push-forward operator by u to (2.3) we obtain By plugging the above formula into (2.15) we obtain the identity (2.16) By (2.4), the vector field Y has the bound and hence we have so (2.4) and (2.17) imply a bound of the form The above estimate, together with the identity (2.16) and the bounds (2.4) for h and (2.11) for dS imply a bound of the form Thanks to the above estimate, (2.13) and (2.14) imply the following bounds for the oneform η and the function f in the splitting of β − Sα 0 : The differential of η is the two-form and its C k -norm can be estimated by the triangle inequality as follows: The above expression, together with (2.8), (2.9) and (2.11), shows that the C k -norm of dη satisfies So far, we have proven that the diffeomorphism u puts α into the desired normal form We now turn to the proof of (iv) and of the last bound. Contracting equation (2.19) by the vector field R α 0 and using (2.6), we find (2.21) Now we wish to show that V (x), which we recall belongs to ker α 0 (x), depends linearly on dS(x), for every x ∈ M. From (2.10) and (2.6) we obtain and hence, using the fact that V is invariant under the action of the flow φ α 0 , because We conclude that where B is the following bilinear form on T M: Notice that the above expression gives us the alternating bilinear form dα 0 if dβ = dα 0 and Q = id. Therefore, (2.4) and (2.8) imply that B is close to B 0 = dα 0 : Consider now the restriction of B to ker α 0 × ker α 0 and let B : ker α 0 → (ker α 0 ) * be the corresponding bundle morphism, which is defined by where ·, · denotes the duality pairing. Now observe that the morphism B 0 associated to B 0 = dα 0 is invertible as dα 0 is non-degenerate on ker α 0 . Then, (2.23) tells us that, up to reducing the size of the positive number δ 0 from (2.2), the morphism B is invertible with Identifying (ker α 0 ) * with the subspace of T * M consisting of one-forms vanishing on R α 0 , we see that (2.22) can be rewritten as (2.25) From (2.21) we conclude that We can therefore uniquely define the endomorphism F : and we obtain the desired identity If F 0 is the endomorphism corresponding to B 0 , the tautological identity From the definition of F , we see that the bounds on Q, B −1 and dβ established in (2.4), (2.24), (2.8) imply that concluding the proof of the last bound and hence of Theorem 2.

The variational principle
In this Section, we prove Proposition 1 from the Introduction, namely the variational principle for contact forms in normal form, and we discuss some consequences of it and Theorem 2.
Proof of Proposition 1. Assume that α 0 is a Zoll contact form on M and β is a contact form on M of the form By differentiating (3.1) and contracting along R α 0 we obtain the identity Let b ∈ B be a critical point of S. Then the circle π −1 (b) consists of critical points of S, and the above identity shows that ı Rα 0 dβ vanishes on this circle. Therefore, R β is parallel to R α 0 on π −1 (b), and hence π −1 (b) is a closed orbit of R β . Its period iŝ Assume now that β is Zoll. Therefore, all its closed orbits have the same period, and in particular this is true for the closed orbits corresponding to the maxima and minima of S on B. The above formula for the periods then forces max S = min S, i.e. S -or equivalently S -is constant. Conversely, assume that S and S are constantly equal to a positive number S 0 . Then all the points in B are critical for S and hence each circle π −1 (b) is a closed orbit of R β of period S 0 T min (α 0 ). This shows that β is Zoll.
As observed in the Introduction, Theorem 2 and Proposition 1 immediately imply a multiplicity result for closed orbits of Reeb flows close to Zoll ones that goes back to Weinstein [Wei73b]. Denoting by σ prime (α) the prime spectrum of α, i.e. the set of periods of the non-iterated closed orbits of R α , we can complement this result with a spectral rigidity result and state it as follows.
Corollary 3.1. Let α 0 be a Zoll contact form on the closed manifold M with closed orbits of common period T 0 , and let π : M → B be the corresponding S 1 -bundle. For every ǫ > 0 there exists δ > 0 such that every contact form α with α − α 0 C 2 < δ has at least as many closed Reeb orbits with period in the interval (T 0 − ǫ, T 0 + ǫ) as the minimal number of critical points of a smooth function on B. Moreover, if for such a contact form α the set contains only one element, then α is Zoll.
Proof. If α − α 0 C 2 < δ with δ small enough, Theorem 2 gives us a diffeomorphism u : M → M such that u * α = β, with β of the form (3.1). Up to choosing δ small enough, we also obtain Denote by S : B → R the induced function on B. By Proposition 1, for every critical point and hence u(π −1 (b)) is a closed orbit of R α of the same period. This proves the first statement. If the prime spectrum of α has just one element in the interval (T 0 − ǫ, T 0 + ǫ) then S must be constant, and hence α is Zoll.
The second statement in the corollary above is a local version, in arbitrary dimension, of a spectral rigidity phenomenon that has been recently proven by Cristofaro-Gardiner and Mazzucchelli in dimension three, see [CGM19, Corollary 1.2]: Any contact form α on a closed three-manifold whose prime spectrum consists of a single element is Zoll. The proof of the latter result uses embedded contact homology.
The vector field R α might of course have many other closed orbits of very large period, but it is natural to ask whether all the closed orbits of R α of period close to T 0 are determined by the variational principle S. This is indeed true, provided that α is C 3 -close to α 0 : For every ǫ > 0 there exists ρ > 0 such that if α − α 0 C 3 < ρ then every noniterated closed orbit of R α has either period larger than 1/ǫ or contained in the interval (T 0 − ǫ, T 0 + ǫ), and in the latter case it is of the form u(π −1 (b)) for some critical point b of S. This follows from the more general Proposition B.2 that is proved in Appendix B, thanks to the identity (2.25).
Remark 3.2. Let α 0 be a Zoll contact form on the closed manifold M. Then for every other Zoll contact form α that has orbits of the same period as α 0 and is C k+1 -close enough to it with k ≥ 1 there exists a diffeomorphism u : M → M that is C k -close to the identity and conjugates the two Reeb flows: This fact follows from the structural stability of free S 1 -actions, whose proof is not difficult (see e.g. [BK19b, Lemma 4.7])), but can also be deduced from the results of this section. Indeed, thanks to Theorem 2 the fact that α is C k+1 -close to α 0 implies the existence of a diffeomorphism u : M → M which is C k -close to the identity and brings α in the normal form u * α = S dα 0 + η + df.
Since u * α is Zoll, the function S is constant by Proposition 1. Then by (2.25) the vector field V is identically zero, and (2.3) gives us the identity The function h is constant along each orbit of R α 0 , and since the orbits of R α and R α 0 have the same period, h must be identically equal to one.

The volume formula
In this section, we wish to prove Proposition 2 from the Introduction. In the proof we need the notion of dual endomorphism on the space of alternating forms. If M is a d-dimensional manifold, then the vector bundle Λ d M is one-dimensional and the wedge product induces a non-degenerate pairing for every k = 0, 1, . . . , d. Therefore, every endomorphism F : We now proceed with the proof of the volume formula.
Proof of Proposition 2. Let α 0 be a Zoll contact form on the (2n − 1)-dimensional closed manifold M. Our first aim is to compute the integral M β ∧ dβ n−1 for the one-form β := S dα 0 + η + df, where S, f ∈ Ω 0 (M) and η ∈ Ω 1 (M) satisfies for some endomorphism F : T * M → T * M lifting the identity. An elementary computation, involving only the identity ı Rα 0 η = 0 and Stokes theorem, shows that (4.1) For the reader's convenience, this computation is carried on explicitly at the end of this subsection, see Lemma 4.1 below.
Observe that the operator ξ → α 0 ∧ ı Rα 0 ξ acts as the identity on (2n − 1)-forms. Therefore, the forms appearing in the last sum of (4.1) can be manipulated as follows: Here we have used the fact that η vanishes on R α 0 . Now we can use the assumption on dη and replace ı Rα 0 dη in the above expression by F [dS]. Using also the definition of the dual operator F ∨ at the beginning of this section, we can go on with the chain of identities and obtain Multiplication of the above form by S j−1 gives us By plugging the above identities into the last sum of (4.1) we obtain the following expression: where τ j is the (2n − 2)-form By Stokes theorem we can turn this formula intô This formula can be rewritten aŝ where p(x, s) := s n + j=1 p j (x)s j and the functions p j ∈ Ω 0 (M) are defined by Since the right-hand side is an exact (2n − 1)-form, the function p j integrates to zero when multiplied by α 0 ∧ dα n−1 0 , as stated in Proposition 2. There remains to check the last statement about the C 0 -norm of the functions p j . Namely, we must prove that for any ǫ > 0 there exists δ > 0 such that if then p j C 0 < ǫ for every j = 1, . . . , n − 1.
Assume that (4.2) holds for some positive number δ, whose size will be specified in due time. Then the (2n − 2)-form τ j and its differential for a suitable constant b 0 . Using the Leibniz formula, (4.2) implies also the bound for a suitable constant b 1 . The estimates on the morphism F in (4.2) give analogous bounds for the dual morphism F ∨ , i.e.
for a suitable constant b 2 . Then the Leibniz formula together with (4.3) and (4.4) yield The second bound in (4.3) and the above one show that, by choosing δ small enough, the C 0 -norm of both dτ j and d(F ∨ [τ j ]) can be made arbitrarily small. By definition of the densities p j , this implies that we can find a positive number δ, depending on c, such that (4.2) implies p j C 0 < ǫ ∀j = 1, . . . , n − 1.
This concludes the proof.
We conclude this subsection by reproducing the computations leading to identity (4.1).
Proof. We set γ := Sα 0 + η, so that α = γ + df . Then dγ = dα and and we will now compute the right-hand side of this equality. The differential of γ is the two-form dγ = dS ∧ α 0 + Sdα 0 + dη, and its (n − 1)-th wedge power is the (2n − 2)-form Wedging this form with γ we obtain the (2n − 1)-form (4.6) The forms with j different from n − 1 in the first sum above can be rewritten as Therefore, the first sum in (4.6) can be rewritten as first sum in (4.6) = S n α 0 ∧ dα n−1 By plugging the above expression into (4.6) and by summing the first sum of the formula above with the third sum in (4.6), from which we isolate the term with j = 0, we obtain the identity Now we examine the first sum in the above expression. The coefficient of its j-th term is n j , by the addition formula for binomial coefficients, and the term with j = n−1 vanishes, because both η and dα 0 vanish on R α 0 . By incorporating the term η ∧ dη n−1 into this sum, we get the identity We now integrate over M and use Stokes theorem when integrating the second sum. We obtain: By using again the addition formula for binomial coefficients and by shifting the indices in both sums we find the identitŷ that is precisely (4.1) thanks to (4.5).

The systolic inequality
The first aim of this section is to put together Theorem 2, Proposition 1 and Proposition 2 to prove the local systolic maximality of Zoll contact forms of Theorem 1. We follow the argument that we already sketched in the Introduction.
Proof of Theorem 1. Let C > 0 be an arbitrary constant. Let α be a contact form on M such that α − α 0 C 2 < δ 0 , where δ 0 is given by Theorem 2. Then, we can find a diffeomorphism u : M → M such that where S ∈ Ω 0 (M) is invariant under the flow of R α 0 , f ∈ Ω 0 (M), and η ∈ Ω 1 (M) satisfies for a suitable endomorphism F : T * M → T * M. Moreover, the bounds hold for every k ≥ 0. We set β := u * α, and we observe that it suffices to prove the systolic inequality for β because both the volume and the minimal period of Reeb orbits are invariant under diffeomorphisms: vol(M, β) = vol(M, α), T min (β) = T min (α).
We apply Proposition 2 to β and find functions p j : M → R for j = 1, . . . , n − 1 with zero average with respect to the volume form α 0 ∧ dα n−1 Assume now that α − α 0 C 3 < C. The last three bounds in (5.1) for k = 1 yield Take now c = ω 1 (C) and ǫ = n 2(2 n −n−1) in Proposition 2 and obtain a corresponding δ > 0 such that for every j = 1, . . . , n − 1 We now choose δ C such that ω 0 (δ C ) < min{1/2, δ}, so that for α − α 0 C 2 < δ C we get If the equality holds in (5.7), then it must hold also in (5.5) and hence S ≡ S 0 is constant. By Proposition 1 β is Zoll. Conversely, assume that β is Zoll. Then all of its orbits have the same minimal period. By Proposition 1 S is constant. In this case, the inequalities in (5.5) and in (5.6) are equalities. Therefore, (5.7) is an equality. This concludes the proof of the theorem.
We conclude this section by discussing a lower bound for the maximal period of "short" periodic orbits that can be proven by an easy modification of the argument described above.
Recall that σ prime (α) denotes the prime spectrum of the contact form α, i.e. the set of the periods of all its non-iterated closed Reeb orbits. Fix some number τ > T 0 . By Corollary 3.1 we can find a C 2 -neighborhood U τ of α 0 in the space of contact forms on M such that for every α ∈ U τ the set σ prime (α) has non-empty intersection with the interval (0, τ ]. Therefore, the function is well defined on U τ . Then an easy modification of the above proof allows us to show the following lower bound for T max (α, τ ).
Theorem 5.1. Let α 0 be a Zoll contact form on the closed manifold M with orbits of period T 0 and let τ > T 0 . Then for all C > 0 there exists δ τ,C > 0 such that the C 3 -neighborhood of α 0 is contained in U τ and for every α ∈ N τ,C we have with equality if and only if α is Zoll.
Indeed, in order to get this bound it is enough choose δ τ,C so small that which implies that a circle at which S achieves its maximum is a closed orbit of R β of period less than τ , and to replace (5.5) by the inequality which is an equality if and only if S is constant.

Zoll starshaped hypersurfaces
Endow R 2n with coordinates (x 1 , y 1 , . . . , x n , y n ), with the Liouville one-form and with the symplectic form The restriction of λ 0 to the unit sphere S 2n−1 ⊂ R 2n is the Zoll contact form α 0 , all of whose orbits have period π. By starshaped domain we mean here an open set of the form where f : S 2n → R is a smooth positive function. With this notation, the unit ball B 2n of R 2n is the set B 1 . The C k -distance of the starshaped domains B f and B g is by definition the C k -distance of the smooth functions f and g on S 2n−1 . The one-form λ 0 restricts to a contact form on the boundary of the starshaped domain B f , and the radial projection pulls this contact form back to the contact form f 2 α 0 on S 2n−1 : The aim of this section is to prove the following result.
Proposition 6.1. There is a C 3 -neighborhood B of the unit ball B 2n in the space of starshaped domains such that if A belongs to B and λ 0 | ∂A is Zoll with all orbits of period π, then there exists a symplectomorphism of (R 2n , ω 0 ) mapping B 2n onto A.
Corollary 2 from the Introduction is an immediate consequence of Theorem 1 and this proposition. The proof of the latter makes use of the following lemma.
Lemma 6.2. There is a C 3 -neighborhood B of the unit ball B 2n in the space of starshaped domains such that if A belongs to B and λ 0 | ∂A is Zoll with all orbits of period π, then there exists a diffeomorphism ψ : S 2n−1 → ∂A that is C 1 -close to the inclusion S 2n−1 ֒→ R 2n and satisfies ψ * (λ 0 | ∂A ) = λ 0 | S 2n−1 .
Proof. Assume that the starshaped domain A = A f ⊂ R 2n is C 3 -close to B 2n . Then the contact form α := ρ * (λ 0 | ∂A ) = f 2 α 0 on S 2n−1 is C 3 -close to the standard contact form α 0 , all of whose orbits are closed with period π. Here, ρ : S 2n−1 → ∂A denotes the radial projection z → f (z)z, which, by assumption, is C 3 -close to the inclusion S 2n−1 ֒→ R 2n . Since λ 0 | ∂A is Zoll with all orbits of period π, so is α. As observed in Remark 3.2, the fact that both α and α 0 are Zoll with orbits of the same period and are C 3 -close implies that there exists a diffeomorphism u : S 2n−1 → S 2n−1 that is C 2 -close to the identity and conjugates the respective Reeb vector fields: Therefore, the contact forms α 1 := u * α and α 0 share the same Reeb vector field R α 0 . Moreover, α 1 and dα 1 = u * (dα) are C 1 -close to α and dα, and hence to α 0 and dα 0 (see Lemma 1.1). We claim that there exists a diffeomorphism v : S 2n−1 → S 2n−1 that is C 1 -close to the identity and satisfies v * α 1 = α 0 . (6.1) This claim implies that the diffeomorphism ψ := ρ • u • v is C 1 -close to the identity and satisfies as we wished to show. There remains to construct the diffeomorphism v. This can be done by Moser's trick. The one-forms α t := tα 1 + (1 − t)α 0 are all contact forms for every t ∈ [0, 1] with the same Reeb vector field. The contact structure ker α t depends smoothly on t ∈ [0, 1] and since dα t is non-degenerate on it we can find a smooth family of vector fields Y t , t ∈ [0, 1], on S 2n−1 such that Y t ∈ ker α t , ı Yt dα t | ker αt = (α 0 − α 1 )| ker αt . (6. 2) The C 1 -closeness of α 1 to α 0 and of dα 1 to dα 0 implies that Y t is C 1 -small, uniformly in t ∈ [0, 1]. Moreover, Let φ t be the path of diffeomorphisms of S 2n−1 that is defined by integrating the vector field Y t : Notice that φ t is uniformly C 1 -close to the identity. We claim that so that the diffeomorphism v := φ 1 satisfies (6.1) and is C 1 -close to the identity. The above identity is clearly true for t = 0, so it is enough to check that Using Cartan's identity, we compute where we used (6.3) and the fact that Y t belongs to ker α t . This proves (6.4) and concludes the proof of the lemma.
Proof of Proposition 6.1. Thanks to the above lemma we find a C 3 -neighborhood B of B 2n such that for every A ∈ B with λ 0 | ∂A Zoll of period π there exists a diffeomorphism ψ : S 2n−1 → ∂A such that ψ * (λ 0 | ∂A ) = λ 0 | S 2n−1 .
Moreover, by reducing the C 3 -neighborhood B the map ψ can be chosen to be arbitrarily C 1 -close to the inclusion S 2n−1 ֒→ R 2n . In particular, we may assume that We now extend ψ to a positively 1-homogeneous map on the whole R 2n by mapping rz with r > 0 and z ∈ S 2n−1 into rψ(z). This extension is still denoted by It is continuous on R 2n , smooth on R 2n \ {0}, maps B 2n onto A and satisfies In particular, it is a symplectomorphism of R 2n \ {0} onto itself. In order to conclude, we just need to smoothen it near the origin, by keeping it a symplectomorphism. Such a smoothing can be performed using generating functions, as we shall now explain.
We identify R 2n with C n by choosing the complex coordinate z = x + iy, where (x, y) ∈ R n ×R n are the coordinates introduced at the beginning of this section, in which the symplectic form ω 0 takes the form dx∧dy. The symplectic vector space (C n ×C n , ω 0 ⊕−ω 0 ) can be identified with the cotangent bundle T * C n = C n × (C n ) * by the linear symplectomorphism Here, T * C n is endowed with its standard symplectic structure dp ∧ dq, where q ∈ C n , p ∈ (C n ) * , and the dual space (C n ) * is identified with C n by the standard Euclidean product on C n . This linear symplectomorphism maps the diagonal ∆ of C n × C n onto the zero-section of T * C n . It is an explicit linear realization of the Weinstein tubular neighborhood theorem for the Lagrangian submanifold ∆ of (C n × C n , ω 0 ⊕ −ω 0 ). The graph of the symplectomorphism ψ| C n \{0} is a Lagrangian submanifold of (C n × C n , ω 0 ⊕ −ω 0 ), and hence gets mapped into a Lagrangian submanifold of T * C n by the map (6.6). The map dψ is 0-homogeneous and is arbitrarily C 0 -close to the identity, provided that B is sufficiently small. This implies that if B is small enough the graph of the symplectomorphism ψ| C n \{0} is mapped into the image of a positively 1-homogeneous Lagrangian section of T * (C n \ {0}), that is, to the graph of a positively 1-homogeneous closed one-form on C n \ {0}. Since C n \ {0} is simply connected (we are assuming that n > 1, because this proposition is trivially true for n = 1), this one-form is the differential of a positively 2-homogeneous smooth function S : C n \ {0} → R. From (6.6) we deduce that the symplectomorphism ψ satisfies The Hessian of S is positively 0-homogeneous and is C 0 -small. The function S extends continuously to the origin by setting S(0) = 0, but this extension is in general not smooth. In order to smoothen it, choose a smooth function σ : [0, +∞) → [0, 1] such that σ(r) = 0 for all r sufficiently small and σ(r) = 1 for every r ≥ 1/2. We then define a smooth real functionS on C n byS (z) := σ(|z|)S(z) ∀z ∈ C n .
The Hessian ∇ 2S ofS is C 0 -small when the one of S is C 0 -small, so up to reducing the size of B we can assume that ∇ 2S C 0 < 2. By the Banach fixed point theorem applied to the metric space C n , the identity uniquely defines a map ϕ : C n → C n , which is smooth because of the smooth dependence of the fixed point in the parametric Banach fixed point theorem. This map is a symplectomorphism with inverse ϕ −1 : C n → C n obtained by solving the equation Since ∇S(z) = ∇S(z) for every z ∈ C n with |z| ≥ 1/2, inequality (6.5) implies that ϕ(z) = ψ(z) for every z ∈ S 2n−1 . We conclude that ϕ is a symplectomorphism of R 2n mapping B 2n onto A.

Shadows of symplectic balls
In this section, we wish to prove Corollary 3 from the Introduction. Before starting with the proof, we need to discuss the linear symplectic non-squeezing theorem. The vector space R 2n is endowed with the standard symplectic form ω 0 , with the standard Euclidean product and with the standard complex structure, which is ω 0 -compatible and allows us to identify R 2n with C n . If 0 ≤ k ≤ n, we have the inclusions of the Grassmannian of complex k-subspaces into the Grassmannian of symplectic 2ksubspaces, and of the latter into the Grassmannian of all real 2k-subspaces. The smallest and the largest Grassmannians are compact, while the symplectic Grassmannian is an open neighborhood of Gr k (C n ) in Gr 2k (R 2n ).
The Wirtinger inequality states that for all 2k-uples of vectors in R 2n and, in the case of linearly independent vectors, the equality holds if and only if the vectors v 1 , . . . , v 2k span a complex subspace. Therefore, the formula where v 1 , . . . , v 2k denotes a basis of V , defines a non-negative function on Gr 2k (R 2n ) which is strictly positive precisely on Gr 2k (R 2n , ω 0 ) and achieves its maximum 1 precisely at Gr k (C n ): For every V ∈ Gr 2k (R 2n ) there holds w(V ) ≥ 0, with equality if and only if V / ∈ Gr 2k (R 2n , ω 0 ), w(V ) ≤ 1, with equality if and only if V ∈ Gr k (C n ). (7. 2) The function w is invariant under unitary transformations. Given V ∈ Gr 2k (R 2n , ω 0 ), we denote by P V the linear projection onto V along the symplectic orthogonal of V . The linear symplectic non-squeezing theorem can be stated in the following way, where B 2n denotes the unit ball in R 2n .
Theorem 7.1. For every element V ∈ Gr 2k (R 2n , ω 0 ) and every linear symplectomorphism In particular: In the equality case, the set P V Φ(B 2n ) is linearly symplectomorphic to a 2k-ball of radius 1: , ω 0 ). The proof of the above theorem can be obtained by an easy modification of the proof of [AM13, Theorem 1], but for the reader's convenience we include a full proof at the end of this section. See also [DdGP19] for another approach to the linear non-squeezing theorem: There, statement (i) above is proven by showing that P V Φ(B 2n ) always contains a symplectic 2k-ball of radius 1, by using the Williamson symplectic diagonalization and Schur complements.
We can now proceed with the proof of Corollary 3.
Proof of Corollary 3. We use the notation Symp(R 2n ) for the space of (nonlinear) symplectomorphisms ϕ : (R 2n , ω 0 ) → (R 2n , ω 0 ) and consider the function This function is continuous with respect to the C 0 loc -topology on Symp(R 2n ) and the standard topology of Gr 2k (R 2n , ω 0 ).
We fix a linear symplectomorphism Φ : R 2n → R 2n . In order to prove Corollary 3, it suffices to find a C 3 loc -neighborhood W 0 of Φ so that the set of V ∈ Gr 2k (R 2n , ω 0 ) such that Φ −1 (V ) is a complex linear subspace. By the compactness of the complex Grassmannian Gr k (C n ), G 0 is a compact subset of Gr 2k (R 2n , ω 0 ). If V belongs to G 0 , then we have the identity by statement (i) of Theorem 7.1, so composing Φ with a unitary map U V : R 2k → Φ −1 (V ) we obtain a linear symplectomorphism Ψ V : R 2k → V such that Let now B be the C 3 -neighborhood of B 2k in the space of smooth convex bodies in R 2k given by Corollary 2 from the Introduction. The set B has the property that By the compactness of G 0 there exists an open C 3 loc -neighborhood W 0 of Φ in Symp(R 2n ) and finitely many V 1 , . . . , V N ∈ G 0 such that the set is an open neighborhood of G 0 , and by (7.7) we obtain the following statement: For every (ϕ, V ) in W 0 × V there exists a linear symplectomorphism Ψ : V → R 2k such that ΨP V (ϕ(B 2n )) ∈ B. (7.8) If (ϕ, V ) belongs to W 0 × V then we find, thanks to (7.6), (7.8) and the fact that both the volume and the EHZ-capacity are invariant by the symplectomorphism Ψ, Now we can use the fact that the EHZ-capacity of the linear symplectic projection of a convex body C is not smaller than the EHZ-capacity of C, see e.g. [AM15, Theorem 4.1 (v)], and we obtain Putting the last two inequalities together, we have shown the inequality in (7.5) on W 0 ×V : Let us now consider the set which is compact thanks statement (ii) in Proposition 7.1. Let us shrink W 0 so that the implication holds. If (ϕ, V ) ∈ W 0 × V c , then this implication yields Thus, we have shown the inequality in (7.5) on W 0 × V c : Since V \ V is compact and f > π k on {Φ} × ( V \ V ), up to shrinking the neighborhood W 0 of Φ we may assume that (7.11) Inequalities (7.9), (7.10) and (7.11) yield the desired lower bound (7.5). This concludes the proof of Corollary 3 from the Introduction.
We conclude this section by proving Theorem 7.1.
Proof of Theorem 7.1. We first consider the special instance in which V is a complex subspace. In this case, the symplectic projector P V is orthogonal, and hence symmetric. We denote by A the surjective linear map Then where ⊥ denotes the Euclidean orthogonal complement, and the Euclidean volume vol 2k of this set can be expressed by the formula where A T : V → R 2n denotes the transpose of A with respect to the Euclidean product and ω 2k = π k /k! is the volume of the unit 2k-ball. Note that, denoting by J the standard complex structure of R 2n and using the fact that V is complex and P V is symmetric, we have (7.14) where the last equality follows from the fact that the automorphism Φ is symplectic. Let v 1 , . . . , v 2k be a basis of (ker A) ⊥ with |v 1 ∧ · · · ∧ v 2k | = 1. By (7.13) we have the chain of identities vol 2k (A(B 2n )) ω 2k (7.15) From (7.14) and from the fact that J is unitary we obtain Moreover, since P V is symmetric, there holds and hence, using the fact that Φ T is symplectic and V = A(R 2n ) is complex, we obtain where in the last identity we have used (7.12). The identities (7.15), (7.16) and (7.17) give us the following formula for the Euclidean volume of A(B 2n ): .
As V is complex, we deduce the desired identity for the ω k 0 -volume of P V Φ(B 2n ) = A(B 2n ): .
The case of a general symplectic subspace V ∈ Gr 2k (R 2n , ω 0 ) can be deduced from the above case as follows. Choose an ω 0 -compatible scalar product on R 2n such that the projector P V is orthogonal, and denote byB 2n andJ the corresponding unit ball and ω 0 -compatible complex structure, which satisfiesJ(V ) = V . Let Ψ : (R 2n , ω 0 ,J) → (R 2n , ω 0 , J) be a symplectic and complex linear isomorphism. Then Ψ is unitary from (R 2n ,J) to (R 2n , J), and hence Ψ(B 2n ) = B 2n . By applying (7.3) to the complex subspace V of (R 2n ,J) and to the linear symplectomorphism ΦΨ we obtain wherew denotes the function (7.1) on the symplectic and complex vector space (R 2n , ω 0 ,J) and in the last equality we have used again the fact that Ψ is unitary. This proves the identity (7.3) in general.
The first part of statement (i) and statement (ii) are now immediate consequences of this identity and (7.2). There remains to show that if Φ −1 (V ) is a complex linear subspace, then identity (7.4) holds. This identity can be deduced from (7.12) by the following chain of equalities: Here, the fact that the subspace Φ −1 (V ) is complex has been used in the last equality.
A Appendix: Estimates for differential forms can find a positive number r > 0 such that any map u : ..,N . Let α ∈ Ω j (M) and let u : M → M be a smooth map with dist C 0 (u, id) < r. Then is the sum of N smooth j-forms, the ith of which of which is supported in U ′′ i . Moreover, u maps U ′ i into U i and By means of the coordinate system ϕ i , ρ i α can be seen as a smooth j-form β i on R d supported in B ′′ and the restriction of u to U ′ i as a smooth map v i : B ′ → B with bounded derivatives of every order such that v * i β i is compactly supported in B ′ This localization argument allows us to reduce the proof of Lemma 1.1 to the following statement: For every β ∈ Ω j (R d ) with compact support and every smooth map v : B ′ → R d with bounded derivatives of every order we have for every k ≥ 0, where for k = 0 the undefined term dv C k−1 (B ′ ) in (A.1) is set to be zero. Indeed, there holds and inequality (1.2) in the statement of Lemma 1.1 follows from the fact that the quantity N i=1 ρ i α C k is a norm on Ω j (M) that is equivalent to α C k . Similarly, we get proving inequality (1.3) in the statement of Lemma 1.1. There remains to prove (A.1) and (A.2). We first deal with (A.1) in the case j = 0, i.e. β : R d → R is a compactly supported smooth real function, and argue inductively on k. In this case, (A.1) holds trivially for k = 0, and we assume that it holds for a certain integer k ≥ 0. We denote the standard basis of R d by {e j } j=1,...,d and multi-indices and partial derivatives by where the p j 's are non-negative integers. If |p| = k + 1 is of the form p = q + e j with |q| = k we find where the generalized binomial coefficient q r is the product of the binomial coefficients q i r i and v i denotes the components of v. From this identity and from the inductive assumption applied to the functions ∂ x i β we obtain Using again the inductive assumption, we deduce that the C k+1 -norm of v * β has the upper bound This concludes the proof of (A.1) for j = 0. The bound (A.1) for higher order forms follows from the case of functions by writing each smooth j-form as sum of the elementary j-forms β = f dx i 1 ∧ · · · ∧ dx i j and by using the identity Now we prove the bound (A.2), starting again from the case of a function β ∈ Ω 0 (R d ). We have We can now estimate the C k -norm of (A.3) using the above bound and the fact that the C k norm of a product is bounded by the product of the C k norms: which is a stronger version of (A.2) for j = 0. If j ≥ 1 and β is the elementary j-form By writing v i (x) = x i + w i (x) we can expand the term dv i 1 ∧ · · · ∧ dv i j and get the bound The identity (A.5) and the estimate (A.6), together with the bound (A.4) applied to the function f , imply By adding up over all elementary forms we obtain (A.2).
A.2 Proof of Lemma 1.2 Let J : Λ 2n−2 M → T M be the vector bundle isomorphism that is the inverse of the map and consider the bundle map If α is a contact form on M, then α∧dα n−1 = f α 0 ∧dα n−1 0 for some non-vanishing function f ∈ Ω 0 (M), and we have From the identity ı Rα α ∧ dα n−1 = dα n−1 we then obtain that the non-vanishing vector field K(dα) is parallel to R α and hence we find the following formula for the Reeb vector field of α: . (A.7) Since K(dα 0 ) = R α 0 , the (n − 1)-homogeneous map K satisfies for every k ≥ 0. We deduce the estimates from which we obtain the bounds for a suitable sequence of moduli of continuity ω ′ k . Let δ > 0 be such that ω ′ 0 (δ) ≤ 1/2. If the contact form α on M satisfies α−α 0 C 1 < δ then (A.9) implies that α(K(dα)) is uniformly bounded away from zero and we have bounds for a suitable sequence of moduli of continuity ω ′′ k . The desired estimates for the C k -norm of R α − R α 0 now follow from (A.7), (A.8) and (A.10).
B Appendix: Bottkol's theorem B.1 The statement of the theorem Let M be a smooth closed manifold and X 0 a smooth vector field on M all of whose orbits are periodic with the same minimal period T 0 . The flow φ t X 0 of X 0 induces a free S 1 -action on M.
We fix a Riemannian metric g on M such that the diffeomorphisms φ t X 0 are isometries for all t ∈ R. In order to construct a metric with this property, it is enough to start from any metric on M and average it on the orbits of φ X 0 .
For every integer k ≥ 0, we denote by X k (M) the vector space of C k vector fields on M endowed with the C k -norm induced by g. The symbol X(M) denotes the space of smooth vector fields on M.
Let now U ∈ X 0 (M) be a continuous vector field. First, we can average U on the orbits of X 0 , producing the following φ X 0 -invariant vector field: Second, we can define a continuous map where exp : T M → M denotes the exponential mapping associated to the metric g. The map u is C k if U is C k and is a C k diffeomorphism if U is C k and C 1 -small. Third, for every x ∈ M we denote by the linear map that is induced by the Jacobi fields along the geodesic t → exp(tU(x)), t ∈ [0, 1], vanishing at t = 0: where d v exp denotes the vertical differential of the exponential map: The map P (U) x is an isomorphism provided that U C 0 < r inj , where r inj denotes the injectivity radius of (M, g). The aim of this appendix is to discuss the proof and some consequences of the following result.
Moreover, for every integer k ≥ 1 we have the bound for some modulus of continuity ω k .
Under the stronger assumption that X is C 2 -close to X 0 , the existence of C 1 vector fields U, V and of a C 1 function h satisfying (i)-(v) was proven by Bottkol in [Bot80][Theorem 1 and Lemma A], building on ideas of Weinstein and Moser from [Wei73a, Wei73b,Mos76]. Actually, Bottkol's setting is more general: The flow of the vector field X 0 is T 0 -periodic only on a submanifold of M that satisfies a suitable non-degeneracy assumption. The fact that the C 2 -closeness assumption can be replaced by C 1 -closeness by adapting an argument from [Mos76] is explicitly observed in [Bot80].
Up to reducing the positive number δ in the above theorem, we can assume that U is sufficiently C 1 -small so that u = exp •U is a diffeomorphism. In this case, condition (i) can be rewritten as is a linear automorphism of T M lifting the identity. Note that the bounds on U from (B.1) imply for all k ≥ 1 that for suitable moduli of continuity ω k . In this way, we have obtained the formulation of Bottkol's theorem that we stated as Theorem 2.1 and used in the proof of the normal form.

B.2 Some consequences
Theorem B.1 can be used to prove the existence of closed orbits for vector fields X that are C 1 -close to the vector field X 0 , all of whose orbits are closed and have the same period T 0 . Indeed, denote by π : M → B the projection onto the quotient induced by the free S 1 -action given by the flow of X 0 . Conditions (iii) and (iv) in Theorem B.1 imply that there is a smooth vector field V on B such that dπ[V ] = V • π and V (x) = 0 if and only if V (π(x)) = 0. Let b ∈ B be a zero of V . Then V vanishes on the circle π −1 (b), and (B.2) implies that u * X is parallel to X 0 along this circle. Therefore, π −1 (b) is a closed orbit of u * X of period T = h(x)T 0 , where x is any point on π −1 (b) (by (v), h is constant on π −1 (b)). We conclude that the original vector field X has the periodic orbit u(π −1 (b)), which is close to π −1 (b) and has period T close to T 0 . Therefore, any zero of the vector field V on B correspond to a closed orbit of X that bifurcates from the manifold of closed orbits of X 0 and has period close to T 0 . In particular, if the Euler characteristic of B does not vanish, X must have closed orbits of this kind.
On the other hand, it is well known that, under the above assumptions on X 0 , the following fact holds true: For every ǫ > 0 there exists ρ > 0 such that if X − X 0 C 1 < ρ then all non-iterated closed orbits of X have period that is either contained in the interval (T 0 −ǫ, T 0 +ǫ) or larger than 1/ǫ, see [Ban86, Corollary 1]. It is then natural to ask whether the zeroes of the vector field V actually detect all the closed orbits of X with period in the interval (T 0 − ǫ, T 0 + ǫ). The next result says that this is indeed true, provided that X is C 2 -close to X 0 .
Proposition B.2. For every ǫ > 0 there exists ρ > 0 such that if X − X 0 C 2 < ρ then the following facts hold: (i) All the non-iterated closed orbits of X have period that is either contained in the interval (T 0 − ǫ, T 0 + ǫ), or is larger than 1/ǫ.
(ii) The closed orbits of X with period in the interval (T 0 − ǫ, T 0 + ǫ) are precisely the curves of the form u(π −1 (b)), where b is a zero of the vector field V .
Proof. Assume, without loss of generality, that T 0 = 1. As recalled above, statement (i) can be derived from [Ban86, Corollary 1]. However, we will deduce both (i) and (ii) simultaneously from Bottkol's Theorem B.1. Assume that for some positive number ρ whose size will be specified along the proof. Using the notation introduced above, we have by (B.1) and (B.3) In particular, if ρ is small enough we have Set Y := u * X, so that (B.2) gives us and from (B.5) we obtain the bound for a suitable modulus of continuity ω.
Since the diffeomorphism u conjugates the flows of Y and X, it suffices to prove the following fact: For every closed orbit γ : Indeed, if this is the case then V vanishes along γ, and the identity (B.7) implies that T agrees with the (constant) value of h on π −1 (b). Thus, |T − 1| < ǫ follows from (B.6).
The upper bound 1/ǫ on T guarantees that if Y is C 0 -close enough to X 0 then γ(R/T Z) remains close to some fiber π −1 (b) of the S 1 -bundle π : M → B. Therefore, by choosing ρ small enough we may assume that γ(R/T Z) is contained in a trivializing neighborhood π −1 (B 0 ) for such a bundle. We can then identify B 0 ⊂ B with an open set of R d−1 , where d = dim M, and π −1 (B 0 ) with the product in such a way that π is the projection onto the first factor and X 0 = ∂ θ , where θ denotes the variable in T := R/Z. By this identification, the closed orbit γ has components γ(t) = (β(t), θ(t)) ∈ B 0 × T.
By projecting the equationγ = Y (γ) (B.9) onto R d−1 we obtain the following equation for β : where A is the closed path of linear mappings From (B.5), (B.8) and (B.9) we deduce that the path A is C 1 -close to the constant path −π, and hence for a suitable modulus of continuity ω ′ . We denote by E the vector bundle over B 0 × T whose fibers are the (d − 1)-dimensional g-orthogonal complements of R∂ θ . If ρ is small enough, (B.11) implies that A(t) maps the fiber E γ(t) of E at γ(t) isomorphically onto R d−1 and, if we denote by the inverse of this restriction, we have for a suitable modulus of continuity ω ′′ . Since T ≤ 1/ǫ, the Poincaré inequality applied to the mapβ : R/T Z → R d−1 , which has vanishing integral, gives us (B.14) If we choose ρ so small that ω ′′ (ρ) is less than 2πǫ, (B.13) and (B.14) forceβ to be identically zero. Therefore, β(t) = b for every t ∈ R/T Z, for some b ∈ B 0 ⊂ B. Equation (B.10) implies that V vanishes on π −1 (b), and we conclude that where b is a zero of V , as we wished to prove.

B.3 The proof
The proof of Theorem B.1 we exhibit here is different from Bottkol's one: We obtain the triplet (U, V, h) with low regularity properties by a rather straightforward application of the inverse mapping theorem, building on an idea we learned from Kerman, see [Ker99][Proposition 3.4], and then we prove its smoothness, together with the bounds (B.1), by a standard argument that appears, for instance, in [Mos76]. Without loss of generality, we assume that the period of the flow φ X 0 is 1 and we denote by T := R/Z the 1-torus. We introduce the following space of continuous vector fields that are continuously differentiable along X 0 and have vanishing average on the orbits of X 0 : U := {U ∈ X 0 (M) | L X 0 U exists and is continuous on M, U = 0}.

The norm
U U := U C 0 + L X 0 U C 0 turns U into a Banach space. We denote by U inj the open subset of U consisting of those vector fields U ∈ U such that U C 0 < r inj .
We consider also the following space of continuous vector fields that are orthogonal to X 0 and φ X 0 -invariant: This is a closed linear subspace of X 0 (M), and hence a Banach space with the C 0 -norm. Finally, we consider the following space of continuous φ X 0 -invariant real functions on M: The space H is also Banach with the C 0 -norm. When evaluated at x ∈ M, the identity (i) in the statement of Theorem B.1 is an equality of vectors in T u(x) M. When U ∈ U inj we can rearrange this equality as an identity for vector fields on M by applying the inverse of the isomorphism P (U) x : T x M → T u(x) M to both sides. We obtain the identity This shows that the triplet (U, V, h) we are looking for is a zero of the following map and ∇ X 0 U = L X 0 U − ∇ U X 0 is a continuous vector field. Here, d h and d v denote the horizontal and vertical derivatives of maps defined on T M. The usual facts about composition operators imply that Φ X is continuously differentiable. Moreover, the map (X, U, V, h) → Φ X (U, V, h), resp. (X, U, V, h) → dΦ X (U, V, h) is continuous from where the space X(M) is given the C 1 -topology, into X 0 (M), resp. into the space of bounded operators from U × V × H to X 0 (M) endowed with the operator norm. The map Φ X 0 maps (0, 0, 1) to 0. Moreover, its differential at (0, 0, 1) has the form In the next lemma we show that this operator is an isomorphism.
Lemma B.3. The linear operator is an isomorphism.
where the second equality follows from partial integration. This equation defines a continuous vector field U which has zero average and satisfies (B.18). Thus, U and the pair (V, h) ∈ V ×H that is defined by (B.17), form the unique solution of (B.15). This equation implies that L X 0 U is continuous, so the vector field U belongs to U .
The regularity properties of Φ X discussed above and the invertibility of dΦ X 0 (0, 0, 1) allow us to apply the parametric inverse mapping theorem and conclude that there is a positive number δ and an open neighborhood N of (0, 0, 1) in U inj × V × H such that for every X ∈ X(M) with X − X 0 C 1 < δ the restriction of Φ X to N is a C 1 diffeomorphism onto an open neighborhood of 0 in X 0 (M). In particular, if X − X 0 C 1 < δ then there exists a unique (U, V, h) ∈ N such that Φ X (U, V, h) = 0.
Moreover, the inverse of Φ X | N depends continuously on X ∈ X(M) with respect to the C 1 topology and hence for a suitable modulus of continuity ω 0 . Up to reducing the size of δ and N , we may also assume that dΦ X (U, V, h) −1 ≤ c ∀(U, V, h) ∈ N , ∀X ∈ X(M) with X − X 0 C 1 < δ, (B.21) for a suitable positive number c.
There remains to prove that U, V and h are smooth, and that the bounds (B.1) hold. Indeed, smooth zeros of Φ X satisfy the conditions (i)-(v) of Theorem B.1 and hence the following lemma concludes the proof of this theorem.
Lemma B.4. The maps U, V and h are smooth and for every integer k ≥ 1 we have the bound for some modulus of continuity ω k .
Proof. Since the matter is local, it is enough to consider the special case in which M is a torus T d and X 0 is the constant vector field ∂ x 1 . In this case, L X 0 U is just ∂ x 1 U. In order to simplify the notation we set W := (U, ∂ x 1 U, V, h) : T d → R 3d+1 × R so that the map Φ X becomes a multiplication operator of the form Φ X (U, V, h) = ϕ X ( · , W ), for a suitable smooth map ϕ X : T d × R 3d+1 → R d .
Note that the differential of order k of ϕ X depends on the derivatives up to order k of the smooth vector field X. Differentiation yields the identity dΦ X ( We denote by τ y the translation operator by the vector y ∈ R d : The fact that (U, V, h) is a zero of Φ X implies that ϕ X (·, W ) = 0.
A first order expansion for y → 0 then gives us 0 = ϕ X ( · + y, τ y W ) − ϕ X ( · , W ) = ϕ X ( · + y, τ y W ) − ϕ X ( · , τ y W ) + ϕ X ( · , τ y W ) − ϕ X ( · , W ) = d 1 ϕ X ( · , τ y W ) which shows that the maps U, V and h are of class C 1 with where ∂ x i ϕ X := d 1 ϕ X [∂ x i ]. This shows also that ∂ x i U belongs to U , meaning that ∂ x 1 ∂ x i U exists and is continuous. If we set w 0 := (0, 1) ∈ R 3d × R, the fact that Φ X (0, 0, 1) = 0 reads ϕ X 0 (x, w 0 ) = 0 ∀x ∈ T d , and hence |∂ x i ϕ X (x, W (x))| ≤ |∂ x i ϕ X (x, W (x)) − ∂ x i ϕ X 0 (x, W (x))| + |∂ x i ϕ X 0 (x, W (x)) − ∂ x i ϕ X 0 (x, w 0 )| ≤ ∂ x i ϕ X − ∂ x i ϕ X 0 C 0 + ω W − w 0 C 0 for some modulus of continuity ω. Since X → d 1 ϕ X and X → W are continuous in the C 1 -norm of X, this inequality implies a bound of the form for a suitable modulus of continuity ω. This bound, together with (B.24) and (B.21) gives us a modulus of continuity ω such that for all i = 1, . . . , d The above inequality and (B.20) imply the case k = 1 in (B.22). By bootstrapping the above argument we obtain that U, V and h are smooth and satisfy (B.22) for all k ≥ 1.