Unboundedness of the Lagrangian Hofer distance in the Euclidean ball

Let L denote the space of Lagrangians Hamiltonian isotopic to the standard Lagrangian in the unit ball in Euclidean space. We prove that the Lagrangian Hofer distance on L is unbounded.


Introduction
Let (M, ω) be a symplectic manifold and denote by C In [7], Khanevsky proved that, in dimension 2, the metric space (L, d) is unbounded. In higher dimensions, this seemingly basic question has remained open despite the fact that much progress has been made in the field of Lagrangian Hofer geometry; see for example [6,12,14]. Our main goal here is to prove the unboundedness of (L, d) in full generality. Let C ∞ c ([0, 1]) denote the space of smooth and compactly supported functions on Here is our main result.
In particular, the metric space (L, d) is unbounded.  [2,1,3,4,5]. A quasimorphism on a group G is a function µ : where D is a constant which is usually referred to as the defect of the quasimorphism µ. If µ(a k ) = kµ(a) for all a ∈ G and k ∈ Z, then µ is called a homogeneous quasimorphism.
We will next use the above theorem to prove Theorem 1. The fact that Theorem 1 follows from the existence of quasimorphisms with the above list of properties is implicitly present in Khanevsky's paper [7]. Various versions of the quasimorphisms employed in this note have appeared in the work of Entov, Polterovich and their collaborators; see for example [2,1,3,4,5]. This close connection to Entov and Polterovich's work allows us to prove the first two properties by standard techniques. The non-standard part is establishing Property (3), which in a sense states that, up to a bounded error, η δ (φ) is an invariant of the Lagrangian φ(L 0 ). In [7], Khanevsky proves this property using two dimensional methods. We will use a different approach which is applicable in all dimensions.
Proof of Theorem 1. We will prove the theorem for B 2n ( 1 √ π ) rather than B 2n . Of course, this is sufficient as B 2n ( 1 √ π ) and B 2n are conformally symplectomorphic. We will continue to denote by L 0 the restriction of Take φ ∈ Ham(B 2n ( 1 √ π )) and let ψ denote any other Hamiltonian diffeomorphism such that φ(L 0 ) = ψ(L 0 ). Using Properties (1) and (3) we obtain the following for all δ ∈ ( n n+1 , 1]: The above implies that ∀φ ∈ Ham(B 2n ( 1 √ π )) and ∀δ ∈ ( n n+1 , 1] we have , 1] and denote by C ∞ c (J) the set of functions whose support is compactly contained in the interior of J. We will construct a map Ψ : Observe thatf andg Poisson commute and hence φ −1 ). Therefore, the previous inequality is equivalent to Note thatf andg are constant on each of the tori T ( 1 δπ(n+1) ). Hence, using Property (2) from Theorem 2, we conclude that Picking δ so that f − g attains its maximum at n δ(n+1) yields : This completes our proof.
3.1. Spectral invariants on CP n . Let ω F S denote the Fubini-Study symplectic form on CP n normalized so that V ol(CP n ) = 1. Throughout the rest of this article we assume that CP n is equipped with the symplectic structure induced by ω F S .
It follows from the works of C. Viterbo, M. Schwarz, and Y.-G. Oh [13,10,9] that one can associate to each Hamiltonian H ∈ C ∞ ([0, 1] × CP n ) and each nonzero quantum homology class a ∈ QH * (CP n )) \ {0} a so called spectral invariant c(a, H) ∈ R. These invariants are constructed via the machinery of Hamiltonian Floer theory and, in fact, they can be defined on a very large class of symplectic manifolds, however, we will only be concerned with CP n . The only quantum homology class that we will be dealing with is the fundamental class [CP n ]; for brevity we will denote c(H) = c([CP n ], H).
For our purposes we must introduce the asymptotic version of the spectral invariant c. This is defined as follows: for H, Following Entov and Polterovich [2] we define the asymptotic spectral invariant of a Hamiltonian H by Our asymptotic spectral invariant, ζ, is defined for all time-dependent Hamiltonians. The restriction of ζ to the set of time-independent Hamiltonians, i.e. C ∞ (CP n ), is what Entov and Polterovich refer to as a symplectic quasi-state on CP n . In fact, the restriction of ζ to C ∞ (CP n ) is precisely the quasi-state constructed on CP n in [3]. According to Entov and Polterovich [4], a closed subset X ⊂ CP n is said to be superheavy with respect to ζ if The Clifford torus and RP n are two examples of superheavy subsets of CP n . Superheavyness of these two sets follows from Theorems 1.6 and 1.15 of [4], respectively. In [4], the notion of superheavyness is defined only for autonomous Hamiltonians. Using standard properties of spectral invariants one can easily show that if X ⊂ CP n is superheavy, then

Entov and Polterovich's Calabi quasi-morphism.
In [2], Entov and Polterovich construct a quasimorphism µ : Ham(CP n ) → R which is defined by the following expression: In [2], it is proven that µ does not depend on the choice of the generating Hamiltonian F and hence it is a well-defined map from Ham(CP n ) to R. Furthermore, µ is a homogeneous quasimorphism. The relationship between this quasimorphism and the aforementioned quasi-state ζ is discussed in detail in [3]. For example, the above formula relating ζ and µ can be found in Section 6 of [3].
Recall that a subset U of a symplectic manifold is said to be displaceable if there exists a Hamiltonian diffeomorphism ψ such that U ∩ ψ(U ) = ∅. If the support of a Hamiltonian F is displaceable, then (4) ζ(F ) = 0, This is referred to as the Calabi property of µ and for this reason µ is often referred to as a Calabi quasimorphism; for further details see [2,3].

3.3.
Constructing η δ . In this section we closely follow Section 4 of [1]. For 0 < δ 1, define embeddings θ δ : B 2n ( 1 √ π ) → CP n by the formula: where z ′ i s denote the standard complex coordinates on C n . The maps θ δ pull ω F S back to δ ω 0 and so these embeddings are all conformally symplectic. Clearly, θ 1 is a genuine symplectic embedding.
For δ > n n+1 , we define η δ : Ham(B 2n ( 1 √ π )) → R as follows: . We must show that η δ does not depend on the choice of the generating Hamiltonian F . One can easily check that the time-1 map of the Hamiltonian δF • θ −1 δ : [0, 1] × CP n → R is the Hamiltonian diffeomorphism θ δ φθ −1 δ . This combined with Equation (3) yields: A simple computation shows that where Cal : Ham(B 2n ( 1 √ π )) → R denotes the Calabi homomorphism: Hence, we have obtained the following alternative definition for η δ : . It is clear from the above formula that η δ is well-defined and does not depend on the choice of the generating Hamiltonian F . Furthermore, η δ is a quasimorphism as it is a linear combination of quasimorphisms. Results from [11] on descent of asymptotic spectral invariants provide an alternative method for proving that η δ is a well-defined quasimorphism. Additionally, one can use Formula (6) to show that the quasimorphisms η δ have bounded defect. Indeed, Formula (6), combined with the fact the Cal is a homomorphism, implies that . From this we obtain that where Def (µ) denotes the defect of µ. The last inequality holds because δ ∈ ( n n+1 , 1]. 3.4. Proof of Theorem 2. Part (1) of the theorem follows from Formula (6) and the fact that both µ and Cal are Lipschitz continuous with respect to Hofer's norm. Part (2) follows from Formula (5): Indeed, if F c on T ( n πδ(n+1) ), then δF • θ −1 δ δc on the Clifford torus. Since the Clifford torus is superheavy for ζ we see that n Def (µ). Inequality (7) implies that |η δ (φ −1 ψ) + η δ (φ) − η δ (ψ)| D. The Hamiltonian diffeomorphism φ −1 ψ preserves L 0 and hence, it is sufficient to prove that η δ vanishes on the set of Hamiltonian diffeomorphisms which preserve L 0 .