The codisc radius capacity

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing inequality concerning symplectic embeddings of the boundaries of two balls of equal radius into the open unit ball. If the interior components of the image spheres are disjoint, then the radii are less than the square root of one half. Furthermore, we introduce the spherical variant of the relative Gromov radius and prove finiteness for monotone Lagrangian tori in symplectic vector spaces.


Introduction
A symplectic manifold (V, ω) is a smooth 2n-dimensional manifold V together with a non-degenerate closed 2-form ω. The most important examples of symplectic manifolds are R 2n with dx ∧ dy := dx 1 ∧ dy 1 + . . . + dx n ∧ dy n , the total space of cotangent bundles π : T * Q → Q of smooth manifolds Q with dλ can , where (λ can ) u = u • T π is the Liouville 1-form on T * Q, and all Kähler manifolds with its Kähler form. A symplectomorphism is a diffeomorphism ϕ that preserves the symplectic form ω in the sense that ϕ * ω = ω. Symplectomorphisms can be obtained from compactly supported Hamiltonian functions H on V as the time-1-map of the Hamiltonian vector field X H which is defined via i XH ω = −dH. The canonical lift of diffeomorphisms on a smooth manifold Q to T * Q gives another class of examples. The first observation, which is due to Liouville, is that a symplectomorphism preserves the symplectic volume 1 n! ω n of a symplectic manifold (V, ω). A natural question is to what extend a symplectormorphism is more special than a volume preserving diffeomorphism. In [26] Gromov gave the following answer. Consider the cylinder Z R = {x 2 1 + y 2 1 < R} × R 2n−2 of radius R. Observe that for any radius r there exists a linear volume preserving diffeomorphism of R 2n that maps the open ball B r of radius r into Z R . But if B r embeds into Z R symplectically Gromov's non-squeezing theorem implies r ≤ R, see [26, 0.3.A]. More generally, if only a neighbourhood of the sphere S 2n−1 r = ∂B r in R 2n embeds symplectically into Z R it was show in [24,45,46,51]  holds. We call the restriction of a symplectic embedding of a neighbourhood of S 2n−1 r ⊂ R 2n a symplectic embedding of the sphere S 2n−1 r .
In [26, 0.3.B] Gromov proved that if two open balls B r1 and B r2 embed symplectically into B R with disjoint images then the packing inquality r 2 1 + r 2 2 ≤ R 2 holds. Ziltener asked whether a packing obstruction holds for symplectic embeddings of spheres S 2n−1 r1 and S 2n−1 r1 such that bounded components of the complements of the images are disjoint. If n = 2 this follows from the packing inequality for balls because the existence of embeddings of S 3 r1 ⊔ S 3 r2 into R 4 imply the existence of embeddings of B 4 r1 ⊔ B 4 r2 , see [26, 0.3.C]. In Section 2.2 we will prove the following: embeds symplectically into B R ⊂ R 2n such that the bounded components of the complements of the images are disjoint. Then r 2 1 + r 2 2 < R 2 . Symplectic embeddings of balls B r ⊂ R 2n into symplectic manifolds (V, ω) exist by Darboux's theorem. Any point has a neighbourhood symplectomorphic to B r provided r > 0 is sufficiently small, cf. [33]. A similar statement is due to Weinstein [48]. Consider a closed Lagrangian submanifold L of (V, ω), which is a submanifold such that ω| T L = 0. A neighbourhood of the zero section L in the cotangent bundle T * L is symplectomorphic to a neighbourhood of L in (V, ω). A natural question is how large such a neighbourhood can be.
Consider for example a closed Lagrangian submanifold L of R 2n . We provide L with the metric g induced from R 2n . Denote by D * r L the r-codisc bundle of L, which is the subset of T * L consisting of all covectors of length at most r. By Weinstein's neighbourhood theorem [48] D * r L embeds symplectically into R 2n for r > 0 sufficiently small identifying the zero section of T * L with L. We estimate the largest radius r such that the r-codisc bundle embeds symplectically in terms of the volume vol(L) and the length of the shortest non-trivial closed geodesic inf(g), which is bounded from below by the injectivity radius of the metric g. The estimate is obtained using Viterbo's isoperimetric inequality [47], see Section 3.2.
Theorem 1.2. There exists a positive constant ρ n such that for any symplectic embedding of D * r L into R 2n the codisc radius r satisfies The Gromov radius is defined by see [26]. Due to Gromov's non-squeezing theorem the quantity c = c B defines a symplectic capacity, i.e. according to Ekeland-Hofer [19], c satisfies the following conditions: (Monotonicity) c(V, ω) ≤ c(V ′ , ω ′ ) provided that there exists a symplectic embedding (V, ω) ֒→ (V ′ , ω ′ ). is a symplectic capacity. This follows from the spherical non-squeezing theorem, see [24,45,46,51].
We introduce a quantity that measures the size of symplectic neighbourhoods of Lagrangian submanifolds. For subsets U in V the codisc radius capacity is defined by where the supremum is taken over all closed Lagrangian submanifolds L ⊂ U of (V, ω), over all Riemannian metrics g on L, and over all r > 0 such that D * r (g)L embeds symplectically into (V, ω) mapping the zero section onto L. By Weinstein's neighbourhood theorem [48] the codisc radius capacity c D (U ) is positive for all open subsets U and all closed Lagrangian submanifolds U = L of (V, ω). In Section 3.4 we will prove the following: Theorem 1.3. The codisc radius capacity c D is a special capacity such that is the open unit ball, and A more sensible method to measure the size of a symplectic neighbourhood of a closed Lagrangian submanifold L of R 2n was introduced by Barraud, Biran and Cornea [5,6,9,10]. A symplectic embedding of the open ball B r ⊂ R 2n of radius r into R 2n is called to be relative to L if the real part D r = B r ∩ R n of the ball is mapped to L and if the complement B r \ D r is mapped to R 2n \ L. The relative Gromov radius is defined by c B (L) = sup{πr 2 | ∃ relative symplectic embedding (B r , D r ) ֒→ (R 2n , L)}, see [5,6,9,10]. A Lagrangian submanifold L of R 2n is called monotone if the Liouville class of L and the Maslov class of L are positively proportional. Finiteness of c B (L) follows for monotone Lagrangian tori L with [10, Theorem 1.2.2], cf. [13,18]. As pointed out by McDuff [37, p. 125] it is not known whether the relative Gromov radius is finite in general.
Similar to the spherical capacity [51] we consider symplectic embeddings of neighbourhoods U ⊂ R 2n of S 2n−1 r = ∂B r into R 2n such that the induced neighbourhood U ∩ R n of the equatorial sphere S n−1 r = ∂D r is mapped to L. We call those embeddings to be relative to L if no point from U \ R n is mapped to L. The relative spherical Gromov radius is defined by Notice that c B (L) ≤ s(L). We show finiteness of s(L) for monotone Lagrangian tori. The proof is given in Section 4.1.
Theorem 1.4. Let 2n ≥ 4. The relative spherical Gromov radius is finite for any monotone Lagrangian torus in R 2n .

Packing with empty balls
The aim of this section is to prove Theorem 1.1.
2.1. Stretching the neck. We equip R 2n with the symplectic form dx ∧ dy. A closed hypersurface M is of restricted contact type in R 2n provided there exists a primitive 1-form of dx ∧ dy that restricts to a contact form α on M . In particular (M, α) is a contact manifold. We denote by inf(α) the minimal period of a closed Reeb orbit on (M, α). Because (M, α) appears as a hypersurface of restricted contact type in the present context inf(α) is the minimal positive action of a closed characteristic on M .
Proof. The proof is an application of the compactness result in [31]. We assume the contact form α to be generic in the sense that 1 is not an eigenvalue of the linearized Poincaré return map for all closed Reeb orbits on (M, α). If α is not generic we replace (M, α) by the graph of a positive function on M inside a symplectic tubular neighbourhood (−ε, ε) × M, d(e s α) . In view of [30, Proposition 6.1] there is a dense set of positive functions on M such that the contact form obtained by restriction of e s α to its graphs is generic. An application of the Arzelà-Ascoli theorem, see [33], and the Liouville flow induced by α allow to undo the perturbation.
For notational convenience we assume R = 1 so that (M, α) is a hypersurface of restricted contact type in the open unit ball B. Invoking an argument used in [25, Corollary 3.7] we find a primitive 1-form λ of dx ∧ dy which is equal to 1 2 (x dy − y dx) on a neighbourhood of R 2n \ B such that λ| T M = α. Collapsing the boundary sphere S 2n−1 to the hyperplane CP n−1 at infinity yields a symplectic embedding B ⊂ CP n , where CP n is provided with the Fubini-Study symplectic form ω. Recall that that CP n is a monotone symplectic manifold, and that through any two distinct points in CP n it passes a unique complex line. With [26, 0.2.B] we have that for any compatible almost complex structure on CP n and any pair of distinct points p 1 ∈ W 1 and p 2 ∈ W 2 there exists a possibly non-unique holomorphic sphere through p 1 and p 2 , which is homologous to CP 1 .
For each N ∈ N we choose an almost complex structure J N which is equal to the complex structure of CP n restricted to CP n−1 . Moreover, in a neighbourhood of M the almost complex structure J N is subject to the process of stretching the neck: A neighbourhood of M ⊂ B is symplectomorphic to [−ε, ε] × M, d(e s α) for ε > 0. We assume that the points p 1 and p 2 inside W are contained in the complement of this neighbourhood. Denote by V the concave filling cut out of CP n by (M, α). We form a symplectic manifold By the above discussion we find for each N ∈ N a J N -holomorphic map We quote the compactness result on [31, p. 192-193], which applies to the present situation because M is of restricted contact type in B. Therefore, formulated in the language of [11], a subsequence of w N converges to a holomorphic building. The lowest level of the building, which corresponds to components in W ∪ [0, ∞)×M , consists of finite energy planes only, cf. [31,p. 193,Fig. 14]. At least two of them, say u 1 and u 2 , pass through the points p 1 and p 2 , resp. Moreover, the total Hoferenergy see [31]. Here the supremum is taken over all smooth strictly increasing functions τ on [−ε, ∞) that agree with e s on [−ε, −ε/2] and tend to 1 as s → ∞. The symplectic Because the primitive λ extends to τ α on the cylindrical end the finite energy planes u 1 and u 2 are asymptotic to closed Reeb orbits of period less or equal to its Hofer-energy, see [28] and cf. [25,Lemma 6.3]. In other words, we have found closed Reeb orbits in each component of (M 1 , α 1 ) ⊔ (M 2 , α 2 ) having period T 1 and T 2 , resp., such that T 1 + T 2 < π. . Therefore, Theorem 1.1 follows.

Superadditivity. Smooth boundaries of bounded convex domains K in R 2n
are of restricted contact type. Moreover, the action-capacity representation theorem for the Hofer-Zehnder capacity c HZ implies that c HZ (K) is the minimal positive action of a closed characteristic on ∂K, see [32]. Because c HZ has inner regularity Theorem 2.1 yields that for disjoint convex subsets K 1 and K 2 of B R . Artstein-Avidan and Ostrover [4] proved that for open disjoint subsets U 1 and U 2 of B R . In particular, c HZ tends to zero on B 1 \ B 1−ε as ε → 0 while the spherical capacity [51] as well as the orbit capacity [25,24] are equal to π for all ε ∈ (0, 1). Hence, neither the spherical capacity nor the orbit capacity equal the Hofer-Zehnder capacity.

Codisc radii.
In the proof of Theorem 2.1 the existence of finite energy surfaces contained in the symplectic filling W follows without making use of the restricted contact type property of M ⊂ B. In order to obtain the estimates on the periods it suffices that the filling W is exact. Hence, Theorem 2.1 continues to hold e.g. for images of codisc bundles.
Proof. We assume the splitting situation of the proof of Theorem 2.1 and consider again the sequence w N of J N -holomorphic spheres. The energy decomposes into the sum of two integrals. The first one is taken over where the first integral is taken over and the second over By [11, Lemma 9.2 and Theorem 10.3] there exists a subsequence of w N which converges to a holomorphic building. Its lowest level contains energy surfaces u 1 and u 2 such that p 1 ∈ im(u 1 ) and p 2 ∈ im(u 2 ). [11,Proposition 5.6] implies that these energy surfaces are asymptotic to finitely many periodic Reeb orbits on (M, α) with total period T . With [11,Lemma 9.1] their ω-energy Therefore, by smoothing out the integrand and employing an approximation argument as above, we see that the total period T is less than π. Remark 2.4. We consider a Riemannian manifold (L, g). Using the metric g we identify the tangent bundle of L with T * L. The canonical Liouville 1-form of T * L induces a contact form α = λ can | T S * r (g)L on the cosphere bundle S * r (g)L of radius r. According to [22] non-trivial closed geodesics on (L, g) and closed Reeb orbits on S * r (g)L are in one-to-one correspondence. The speed curvec of a closed geodesic c, which is parametrized proportional to arc length, with speed |ċ| = r is containd in S * r (g)L. It defines a closed Reeb orbit γ by reprarametrizingc by 1/r 2 . The action of γ and the length of c are related via The length of the shortest non-trivial closed geodesic on (L, g) is denoted by inf(g), which is bounded by the injectivity radius from below. We have Assume in the following that L decomposes into closed submanifolds L 1 ⊔ L 2 . The metric g defines Riemannian manifolds (L 1 , g 1 ) and (L 2 , g 2 ). If the closure of the codisc bundles D * r1 (g 1 )L 1 ⊔ D * r2 (g 2 )L 2 embed symplectically into B R such that the images are disjoint Corollary 2.2 implies If in addition (L, g) has no contractible closed geodesics we obtain in view of Remark 2.3 that 2r inf(g) ≤ inf ℓ (α) using the bundle projection. This implies In Section 3.4 we continue the discussion on the size of symplectically embedded codisc bundles.
2.5. More than two components. We consider a closed hypersurface We assume that the bounded components W 1 , . . . , W k of the complements of M 1 , . . . , M k are pairwise disjoint and that dx ∧ dy admits a primitive 1-form on the closure of W = W 1 ⊔ . . . ⊔ W k that restricts to contact forms α 1 , . . . , α k on M 1 , . . . , M k , resp. In Corollary 2.2 we considered the case k = 2. The proof of Theorem 2.1 and Corollary 2.2 generalizes to hypersurfaces M with k ≥ 3 connected components provided that there exists a holomorphic curve C through k generic points for any (generic) compatible almost complex structure. Therefore, Observe that the compactness result in [11] which we used in the above proofs applies to holomorphic curves of higher genus.
Definition 2.5. The smallest positive action of a closed characteristic on M divided by π is denoted by a k .
It follows form Corollary 2.2 that a k < 1/2 for all k.

The size of a Weinstein neighbourhood
The aim of this section is to prove Theorem 1.2 and Theorem 1.3.
3.1. The action-area inequality. Let (X, ω) be a symplectic manifold which is symplectically aspherical, i.e. the symplectic area S 2 f * ω vanishes for all smooth maps f : S 2 → X. We assume that (X, ω) is either closed, of bounded geometry in the sense of Gromov [26], or compact with convex contact type boundary. In the latter case we replace X by its completion so that (X, ω) has positive cylindrical ends as introduced in [11].
Let L ⊂ X be a closed Lagrangian submanifold. The Gromov width of L is defined by where σ(L, J) is the minimal symplectic area D u * ω of a non-constant Jholomorphic disc u : D → X with boundary on L, see [26]. The supremum is taken over all almost complex structures J that are tamed by ω and have adapted boundary or asymptotic conditions, resp. Notice that σ(L, J) = ∞ if no such disc exists and that σ(L, J) > 0 by Gromov's compactness theorem, cf. [21]. Denote by D * L the unit codisc bundle of L w.r.t. a Riemannian metric. On the unit cotangent bundle S * L the canonical Liouville 1-form λ can defines a contact form α = λ can | T S * L . The aim is to compare the Gromov width σ(L) with the minimal period of a closed Reeb orbit inf(α). Proof. The proof is based on a stretching the neck argument along the lines of Theorem 2.1. Denote by M the image of S * L and assume that the contact form α on M is generic. Identify D * L with its image W in X and denote the Liouville primitive of ω| W by λ.
For each N ∈ N we define a compatible almost complex structure J N on (X, ω) as in the proof of Theorem 2.1 such that the sequence J N only depends on N in the distinguished neigbhourhood of M . We choose J N to be cylindrical, resp., to ensure uniform C 0 -bounds on all holomorphic discs. This requires a modification of J N in a neighbourhood of ∂V \ M , resp., near the ends of V .
We assume that σ(L) is finite. Hence, there is a sequence of J N -holomorphic discs Moreover, we assume that w N (0) is contained in V \ [0, ε] × M . As in [31, p. 163] we choose a Riemannian metric on X of bounded geometry which is independent of N on X \ [−ε, ε] × M and is equal to a product metric on the neck [−N, N ] × M . An application of the mean value theorem to the path w N (t), t ∈ [0, 1], shows that there are no uniform gradient bounds on w N . In other words, after passing to a subsequence w ν , there exists a sequence z ν → z 0 inD such that R ν = |∇w ν (z ν )| −→ ∞. We call z 0 a bubbling off point.
We claim that there are only finitely many bubbling off points. In view of [31,Lemma 3.2] it is enough to show that there exists c > 0 such that for any (subsequence of a) bubbling off sequence z ν → z 0 and for any ̺ > 0 lim inf If a bubbling off point is contained in the interior of D the bubbling off argument on [31, p. 163-167] shows that there exists a finite energy plane v with Hofer-energy In the first two cases we get inf(α) ≤ E(v); in the third, invoking the compactness theorem [11,Theorem 10.5], E(v) is bounded from below by a uniform positive constant. If a bubbling off point is contained on the boundary ∂D we distinguish following [28,23] two cases: We view w ν as a J ν -holomorphic map on the upper half plane H + such that the bubbling off point equals 0. Using Hofer's Lemma [33, Lemma 6.4.5] we modify z ν = x ν + iy ν such that R ν y ν −→ r for some r ∈ [0, ∞], and that there exists a sequence ε ν ց 0 with ε ν R ν → ∞ and The first case is r = ∞. With the rescaling argument on [23, p. 560] we obtain a finite energy plane v in W ∪ [0, ∞) × M , R × M , or (−∞, 0] × M ∪ V , which has Hofer-energy E(v) uniformly bounded from below as in the above argument. It remains to consider the case r < ∞. Replace the sequence w ν by the rescaled sequence u ν (z) = w ν x ν + z/R ν . Set ζ ν = iR ν y ν , and observe that ζ ν → ir and |∇u ν (ζ ν )| = 1. Hence we get Denote the finite set of bubbling off points by Γ ⊂D. Recall that Γ = ∅. In the complement of any neighbourhood of Γ the sequence w ν admits uniform gradient bounds. Applying the mean value theorem we get C 0 -bounds such that a subsequence w ν converges in C ∞ loc (D \ Γ) to a punctured holomorphic disc w in W ∪ [0, ∞) × M with boundary in L. The Hofer-energy E(w) is strictly bounded from above by σ(L). We claim that w is not constant. Observe that for ̺ > 0 sufficiently small and z ∈ Γ we have a uniform bound Arguing by contradiction we see that all the circles, resp., chords w ν (∂D ̺ (z)) converge in C ∞ to a point in L. In both cases as on [40, p. 85-86] we can extent w ν (D ̺ (z)) smoothly to sphere maps into X. If ν ≫ 1 we can assume that the symplectic areas are positive. This contradicts our assumption that (X, ω) is symplectically aspherical. Therefore, w is a non-constant punctured holomorphic disc. All its boundary singular points can be removed by the above argument. We assume that all its removable interior punctures are removed as well. With This proves the Theorem 3.1.

3.2.
Proof of Theorem 1.2. In the two dimensional case a closed connected Lagrangian submanifold L is an embedded curve in the plane. The isoperimetric inequality implies that the enclosed bounded domain D has area ≤ 1 4π length(L) 2 . Notice, that L divides D * r L into two components of equal area. Precisely one component is mapped into D. Since a symplectomorphism preserves the area the area of D * r L is ≤ 1 2π length(L) 2 . Because the metric on L is a positive multiple of the metric induced by R/2πZ there exists ε > 0 such that the area of D * r L equals 4πεr and inf(g) = 2πε. It follows that Let 2n ≥ 4. Consider a symplectic embedding of the closure of D * r L into R 2n . Notice, that the Lagrangian submanifold L is displaceable. A theorem of Chekanov [14] implies that the Gromov width σ(L) is bounded by the displacement energy d(L) of L. In [47] Viterbo proved an isoperimetric inequality d(L) n ≤ ρ n vol(L) 2 for a positive constant ρ n . As explained in Remark 2.4 we have r inf(g) = inf(α) for the contact form α = λ can | T S * r L . Theorem 3.1 yields r inf(g) n ≤ ρ n vol(L) 2 .
This proves Theorem 1.2. With the inequality r inf(g) ≤ d(L) we get for the radius of a symplectically embedded codisc bundle taken w.r.t. the induced metric vol(L) 2 inf(g) n ≥ r n ρ n .
This inequality remains valid for all Riemannian metrics g induced by any Hamiltonian deformation of L. AsÁlvarez Paiva explained to the author a computation of the greatest value of r in the above inequality is related to questions in systolic geometry.

3.3.
Non-embeddability of the cotangent bundles. Let (X, ω) be a symplectically aspherical symplectic manifold as described in Section 3.1. In [14] Chekanov proved for displaceable Lagrangian submanifolds L the inequality Proof. Let g be a metric on L. Arguing by contradiction we find for all positive r a symplectic embedding of the r-codisc bundle of L. With Theorem 3.1 and Chekanov's result [14] we find . Letting r tend to infinity yields a contradiction.
Remark 3.4. In the particular case the symplectic form ω = dλ on X is exact the restriction of λ to T L is a closed 1-form on L. Its cohomology class λ L , the so-called Liouville class, cf. [43], is independent of the choice of the primitive λ provided X is simply connected. Recall, that a norm on the space of cohomology 1-classes m can be defined via m = inf{sup L |µ| | µ ∈ m}, cf. [7]. If a neighbourhood of the closure of the λ L -codisc bundle of L embeds symplectically relative L the image L λ of the section into T * L representing −λ L is an exact Lagrangian submanifold of (X, dλ), see [3,Section 7]. This was pointed out to the author by Polterovich. With Chekanov's result [14] L λ is not displaceable. In particular, no subcritical Stein manifold contains a symplectically embedded cotangent bundle of a closed manifold, cf. [15]. Corollary 3.3 serves as a generalization to the symplectically aspherical case.
Remark 3.5. To give an example of non-embeddability of the cotangent bundles in the presence of holomorphic spheres we make the following remark. Barraud, Biran and Cornea [5,6,9,10] defined the relative Gromov radius c B (L) of a closed Lagrangian submanifold L in a symplectic manifold (V, ω) to be the supremum over all πr 2 such that there exists a symplectic embedding ϕ : B r → V with ϕ −1 (L) equal to B r ∩R n ⊂ C n . The relative Gromov radius of the zero section of T * L is not finite. Hence, if a cotangent bundle embeds symplectically into (V, ω) the relative Gromov radius of the image of the zero section must be infinite in (V, ω). Consider a closed Lagrangian submanifold L in T * Q × CP 1 , where Q is any closed manifold. Then c B (L) is bounded by the absolute Gromov radius c B (T * Q×CP 1 ) = π. Hence, no cotangent bundle does embed symplectically into T * Q × CP 1 .
Notice, that it is not known in general whether the relative Gromov radius of a closed Lagrangian submanifold L in R 2n is finite. Examples in the monotone case can be found in [10,13,18]. Its spherical variant will be discussed in Section 4.

3.4.
Proof of Theorem 1.3. We consider the torus R n /2πZ n with the metric induced from R n so that the shortest non-trivial closed geodesics have length 2π. The n-fold product of the maps (q, p) → 1 + 2p e iq , resp., 1/n + 2p e iq embed the 1/2-codisc, resp., the 1/2n-codisc bundle symplectically into R 2n . The images of the zero section are the Clifford tori T 1 and T 1/ √ n which equal the product of n circles in R 2 × · · · × R 2 of radius 1 and 1/ √ n, resp. The action on the corresponding cosphere bundles induced by the shortest non-trivial closed geodesics equal π and π/n, resp. This shows that c D (P ) ≥ π and c D (B) ≥ π n . Consider a metric g on S 1 . After a reparametrization there exist a positive constant ε such that g = ε 2 g 0 , where g 0 is the metric induced by R/2πZ. Therefore, inf(g) = 2πε. Assume that D * r (g)S 1 embeds into R 2 preserving the area such that S 1 is mapped into the open unit disc. This implies that 1/2 times the area of D * r (g)S 1 is < π. Because the area of D * r (g)S 1 is equal to 4πεr we get εr < 1/2. Hence, r inf(g) < π. This proves that c D is a capacity in dimension 2.
If 2n ≥ 4 we argue as follows. Non-trivial closed geodesics c on Riemannian manifolds (L, g) are in one-to-one correspondence with closed Reeb orbits γ on S * r (g)L for all positive r, see [22]. The correspondence assigns to a geodesic c which is assumed to have constant speed |ċ| = r the reparametrized speed curve γ = c • 1/r 2 . The action γ α, where α = λ can | T S * r (g)L , equals c λ can = r length(c). Therefore, r inf(g) = inf(α). On the other hand the action-area inequality in Theorem 3.1 and Chekanov's result [14] yield inf(α) < σ(L) ≤ d(L). The displacement energy d, which is known to be a special capacity, see [33], takes the value π on the open unit ball and the open unit symplectic cylinder. Hence, c D (Z) ≤ π. This proves Theorem 1.3. With Hermann's work [27] one obtains upper bounds for the codisc radius in terms of the Floer-Hofer resp. the Viterbo capacity, as well as with [25,24] in terms of the orbit capacity in dimensions ≥ 4. Motivated by the work of Cieliebak and Mohnke on the Lagrangian capacity [17,16] we conjecture that the link capacity on the unit ball equals π/n.

The relative spherical Gromov radius
The aim of this section is to prove Theorem 1.4.

4.1.
Proof of Theorem 1.4. Recall that a Lagrangian submanifold L in R 2n is monotone if there exists a positive real number η, the so-called monotonicity constant of L, such that the Liouville class λ L and the Maslov class µ L of L satisfy λ L = ηµ L . The proof of Theorem 1.4 below will show that monotone Lagrangian tori satisfy s(L) ≤ 4η. Notice that the minimal positive symplectic area of a smooth disc with boundary on L is equal to 2η. Moreover, it is attained by a holomorphic disc with Maslov number 2, see [13,18]. A theorem of Chekanov [14] implies that where d(L) denotes the displacement energy of L, see [33]. If in addition there is a metric of non-positive sectional curvature on L we get with [50,Theorem 2.5] for all k ∈ N and the k-th Ekeland-Hofer capacity [19].
Proof of Theorem 1.4. The proof of the theorem is an application of the relative neck stretching argument due to Abbas [2]. We consider a relative symplectic embedding ϕ of (S 2n−1 r , S n−1 r ) into (R 2n , L). For ε > 0 small enough we can assume that the neighbourhood on which ϕ is defined contains the spherical shell We consider the symplectic ellipsoid E =  [12,36,42] applied to the universal cover R n . We choose the radii r 1 , . . . , r n such that in addition there squares are rationally independent. Then all closed Reeb orbits on (M, α) correspond to intersections of ∂E with the complex coordinate axes and each Reeb chord of the Legendrian submanifold K is contained in a closed Reeb orbit. Moreover, the contact form α and the pair (α, K) are generic in the sense of [2, Chapter 3.2], i.e. the linearized Poincaré return map restricted to the contact structure ker(α) at any periodic point of the Reeb flow has no eigenvalue 1, and whenever the isotopic image K ′ of K under the Reeb flow intersects K itself, the contact structure ker(α) is spanned by the tangent spaces to K and K ′ at the intersection points. Therefore, the genericity assumptions of the compactness theorem in [2] are satisfied.
In order to define a sequence of almost complex structures on R 2n we describe a symplectic neighbourhood U ⊂ ϕ(U ε ) of M . Let Y be the Liouville vector field dual to λ. Following its flow near M in forward and backward time we obtain a symplectomorphic model of U for δ > 0, see [22], which we call the neck. Notice that Y , which is mapped to ∂ s , is tangent to L ∩ U so that the intersection of the Lagrangian submanifold L with U corresponds precisely to [−δ, δ] × K. In the same way we obtain for each N ∈ N a symplectomorphic copy of the neck We define a translation invariant almost complex structure on the N -neck as follows: On ker(α) it is required to restrict to a complex structure compatible with dα and on its complement it is required to map ∂ s to the Reeb vector field of α. This defines an almost complex structure J N on U for each N ∈ N. Near the boundary of U it is independent of N and therefore can be extended to R 2n in a uniform way, see [26]. Moreover, J N equals the complex structure of C n outside a fixed large ball. We consider the case where L ∩ W is simply connected. The case of L ∩ V being simply connected can not occur because otherwise there would be a Reeb chord on K with negative action by the analogue of the following argument: Let p be a point in the interior of L ∩ W . Because the Lagrangian torus L is monotone we can apply Damian's result [18,Theorem 1.5.(c)]. Therefore, we find for any N a J N -holomorphic disc u N : D → R 2n through p with boundary on L and Maslov index 2. In particular, the energy of u N is for all N , where η is the monotonicity constant of L. Notice, that the boundary curves u N (∂D) ⊂ L are not entirely contained in L ∩ W because these are not contractible in L.
By Abbas's compactness theorem [2] a subsequence of u N converges to a holomorphic building of total Hofer-energy equal to 2η. Its level structure corresponds to W ∪ [0, ∞) × M , several (or non) copies of R × M , and (−∞, 0] × M ∪ V . The boundary of the building lies in (L ∩ W ) ∪ [0, ∞) × K , the corresponding copies of R × K, and (−∞, 0] × K ∪ (L ∩ V ) resp., cf. [11,31]. Moreover, the building consists of punctured holomorphic spheres and discs with Lagrangian boundary conditions such that at least one disc is contained in each level. Over the interior punctures these are asymptotic to closed Reeb orbits of α and over boundary punctures to non-constant Reeb chords of (α, K), see [1,28]. In particular, in W ∪ [0, ∞) × M there exists a finite energy disc u with at least one boundary puncture. Its Hofer-energy satisfies chords over (the non-empty set of) boundary punctures of u is strictly less than 2η. Therefore, we get π 2 (r − ε) 2 < 2η, because the left hand side is precisely the shortest length of a Reeb chord in ∂E starting and ending on ∂E ∩R n . Letting ε tend to zero this proves the theorem.
Remark 4.1. Our proof requires the existence of a holomorphic discs D through any given point on a Lagrangian submanifold L with ∂D ⊂ L for any admissible almost complex structure such that the energy of D is uniformly bounded and ∂D is not contractible in L. Theorem 1.4 generalizes accordingly. By [18,Theorem 3.3.(b)] this is the case if L is a Lagrangian submanifold of a Liouville symplectic manifold (X, λ) convex at infinity such that any compact set in (X, dλ) is displaceable. L itself is required to be closed, oriented, and monotone such that the total singular Z 2 -homology of the universal cover L has finite dimension over Z 2 and the Z 2 -Euler characteristic of L does not vanish. The resulting discs in this situation all have Maslov index 2. Moreover, we used in the proof that any hypersurface of contact type symplectomorphic to the sphere separates X and that any smoothly embedded separating (n−1)-sphere in L bounds a simply connected domain in L. Notice that the 2-torus L has this property. Moreover, any manifold L such that any smoothly embedded (n − 1)-sphere in L bounds a homeomorphic n-disc satisfies this too. Examples can be obtained with the Jordan-Schoenflies theorem [12,36,42] if the universal cover is R n with n ≥ 3. Therefore, the inequality s(L) ≤ 2d(L) holds in the situation described.

4.2.
A remark on dimension 4. In the case of a Stein surface X and a monotone Lagrangian 2-torus L both quantities c B (L) and s(L) coincide: Consider a symplectic embedding ϕ of S 3 r into X relative L. Then ϕ(S 3 r ) cuts an exact symplectic filling of out X. By a theorem of Gromov [26] ϕ extends (after restriction to a smaller neighbourhood of S 3 r ) to the ball B 4 r , see [40,Theorem 9.4.2]. It follows from the proof of Theorem 1.4 that the intersection of ϕ(B 4 r ) with L is a 2-disc. Therefore, ϕ −1 (L) is a local Lagrangian knot inside the ball B 4 r in the sense of [20]. As Polterovich pointed out to the author with [20, Theorem 1.1.A, Proposition 5.1.A.2)] one can assume that the local Lagrangian knot is isotopic to R 2 through local Lagrangian knots whose trace of the non-flat regions stay inside B 4 r . Hence, with Hamiltonian isotopy extension, cf. [43, p. 43] and [39, p. 96], ϕ can be Hamiltonianly isotoped to ψ inside B 4 r such that ψ coincides with ϕ near ∂B 4 r and maps R 2 ∩ B 4 r to L. Hence, ϕ extends to a symplectic embedding of B 4 r relative L.

4.3.
A non-monotone example. We consider a closed Lagrangian torus L = L ′ × S 1 ̺ in R 2n−2 × R 2 such that the Lagrangian torus L ′ ⊂ R 2n−2 is rational. This means that the minimal positive symplectic area inf(L ′ ) of a disc with boundary on L ′ is positive. We choose the radius ̺ of the circle S 1 ̺ = ∂D ̺ such that inf(L ′ ) ̺ is a natural number. This implies that L itself is rational with inf(L) = ̺.
Notice that for the complex structure of C n the family of holomorphic discs defines a smooth filling. With transversallity as in [40,49] and Gromov compactness, see [21,26,35], we get that for all tamed almost complex structures J standard at infinity and any point p on L there exists a J-holomorphic map u : (D, ∂D, 1) −→ (R 2n , L, p) with symplectic area