The Symplectic Displacement Energy Es(A)Ofa Non- Empty Subset a ⊂ M Is Deﬁned to Be

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In [5], a Hofer-like metric k·kHL was constructed on the group

Symp0(M,ω) of all symplectic diﬀeomorphisms of a compact symplectic manifold (M,ω) that are isotopic to the identity. It was proved recently by Buss and Leclercq [7] that the restriction of k·kHL to Ham(M,ω) is a metric equivalent to the Hofer metric. Let us now propose the following deﬁnition.

Deﬁnition 2. The symplectic displacement energy es(A)ofa non- empty subset A ⊂ M is deﬁned to be:

es(A) = inf{khkHL | h ∈ Symp0(M,ω), h(A) ∩ A = ∅} if some element of Symp0(M,ω) displaces A, and +∞ if no element of

Symp0(M,ω) displaces A.

Clearly, if A and B are non-empty subsets of M such that A ⊂ B, then es(A) ≤ es(B). The goal of this paper is to prove the following result.

Theorem 3. For any closed symplectic manifold (M,ω), the symplectic displacement energy of any subset A ⊂ M with non-empty interior satisﬁes es(A) > 0.

2. The Hofer norm k·kH and the Hofer-like norm k·kHL

2.1. Symp0(M,ω) and Ham(M,ω). Let Iso(M,ω) be the set of all compactly supported symplectic isotopies of a symplectic manifold (M,ω). A compactly supported symplectic isotopy Φ ∈ Iso(M,ω) is a smooth map Φ : M ×[0, 1] → M such that for all t ∈ [0, 1], if we denote by φt(x) = Φ(x, t), then φt ∈ Symp(M,ω) is a symplectic diﬀeomor- phism with compact support, and φ0 = id. We denote by Symp0(M,ω) the set of all time-1 maps of compactly supported symplectic isotopies.

Isotopies Φ = {φt} are in one-to-one correspondence with families of smooth vector ﬁelds {φ˙t} deﬁned by dφ φ˙ (x)= t (φ−1(x)). t dt t THE SYMPLECTIC DISPLACEMENT ENERGY 3

If Φ ∈ Iso(M,ω), then the one-form i(φ˙t)ω such that

i(φ˙t)ω(X)= ω(φ˙t,X) for all vector ﬁelds X is closed. If for all t the 1-form i(φ˙t)ω is exact, that is, there exists a smooth function F : M × [0, 1] → R, F (x, t) =

Ft(x), with compact supports such that i(φ˙t)ω = dFt, then the isotopy

Φ is called a Hamiltonian isotopy and will be denoted by ΦF . We deﬁne the group Ham(M,ω) of Hamiltonian diﬀeomorphisms as the set of time-one maps of Hamiltonian isotopies.

For each Φ = {φt} ∈ Iso(M,ω), the mapping

1 Φ 7→ (i(φ˙t)ω)dt , Z0 where [α] denotes the cohomology class of a closed form α, induces a ˜ well defined map S from the universal cover of Symp0(M,ω) to the first de Rham cohomology group H1(M, R). This map is called the Cal- abi invariant (or the flux). It is a surjective group homomorphism. Let Γ ⊂ H1(M, R) be the image by S˜ of the fundamental group of

Symp0(M,ω). We then get a surjective homomorphism 1 R S : Symp0(M,ω) → H (M, )/Γ. The kernel of this homomorphism is the group Ham(M,ω) [2, 3].

2.2. The Hofer norm. Hofer [14] deﬁned the length lH of a Hamil- tonian isotopy ΦF as 1 lH (ΦF )= (osc F (x, t)) dt, Z0 where the oscillation of a function f : M → R is

osc(f) = max(f) − min(f). x∈M x∈M For φ ∈ Ham(M,ω), the Hofer norm of φ is

kφkH = inf{lH (ΦF )}, where the inﬁmum is taken over all Hamiltonian isotopies ΦF with time-one map equal to φ, i.e. φF,1 = φ. 4 AUGUSTIN BANYAGA†, DAVID E. HURTUBISE, AND PETER SPAETH

The Hofer distance dH (φ, ψ) between two Hamiltonian diﬀeomor- phisms φ and ψ is

−1 dH (φ, ψ)= kφ ◦ ψ kH .

This distance is bi-invariant. This property was used in [8] to prove Theorem 1.

2.3. The Hofer-like norm. Now let (M,ω) be a compact symplectic manifold without boundary, on which we ﬁx a Riemannian metric g.

For each Φ = {φt} ∈ Iso(M,ω), we consider the Hodge decomposition

[19] of the 1-form i(φ˙t)ω as

i(φ˙t)ω = Ht + dut, where Ht is a harmonic 1-form. The forms Ht and ut are unique and depend smoothly on t. For Φ ∈ Iso(M,ω), deﬁne

1 l0(Φ) = (|Ht| + osc(u(x, t))) dt, Z0 where |Ht| is a norm on the ﬁnite dimensional vector space of harmonic 1-forms. We let 1 l(φ)= (l (Φ) + l (Φ−1)), 2 0 0 −1 −1 where Φ = {φt }.

For each φ ∈ Symp0(M,ω), let

kφkHL = inf{l(φ)}, where the inﬁmum is taken over all symplectic isotopies Φ with φ1 = φ. The following result was proved in [5].

Theorem 4. For any closed symplectic manifold (M,ω), k·kHL is a norm on Symp0(M,ω).

Remark 5. The norm k·kHL depends on the choice of the Riemannian metric g on M and the choice of the norm |·| on the space of harmonic 1- forms. However, diﬀerent choices for g and |·| yield equivalent metrics. See Section 3 of [5] for more details. THE SYMPLECTIC DISPLACEMENT ENERGY 5

2.4. Some equivalence properties. Let (M,ω) be a compact symplectic manifold. Buss and Leclercq have proved:

Theorem 6. [7] The restriction of the Hofer-like norm k·kHL to

Ham(M,ω) is equivalent to the Hofer norm k·kH .

We now prove the following.

Theorem 7. Let φ ∈ Symp0(M,ω). The norm

−1 h 7→ kφ ◦ h ◦ φ kHL on Symp0(M,ω) is equivalent to the norm k·kHL.

Remark 8. We owe the statement of the above theorem to the referee of a previous version of this paper.

Proof. Let {ht} be an isotopy in Symp0(M,ω) from h to the identity, and let

i(h˙ t)ω = Ht + dut,

−1 be the Hodge decomposition. Then Ψ = {φ ◦ ht ◦ φ } is an isotopy −1 from φ ◦ h ◦ φ to the identity and Ψ˙ t = φ∗h˙ t. Therefore,

−1 ∗ ∗ −1 ∗ −1 ∗ −1 i(Ψ˙ t)ω =(φ ) (i(h˙ t)φ ω)=(φ ) (Ht +dut)=(φ ) Ht +d(ut ◦φ ).

−1 −1 Let {φs } be an isotopy from φ to the identity, and let LX = iX d + diX be the Lie derivative in the direction X. Then

d −1 ∗ −1 ∗ −1 ∗ −1 ((φ ) H )=(φ ) (L ˙−1 H )= d((φ ) i(φ˙ )H ), ds s t s φs t s s t ˙−1 d −1 where φt =( dt φt ) ◦ φt. Integrating from 0 to 1 we get

−1 ∗ (φ ) Ht − Ht = dαt where 1 −1 ∗ ˙−1 αt = ((φs ) i(φs )Ht) ds. Z0 Therefore, −1 i(Ψ˙ t)ω = Ht + d(ut ◦ φ + αt). 6 AUGUSTIN BANYAGA†, DAVID E. HURTUBISE, AND PETER SPAETH

Hence,

1 −1 l0(Ψ) = |Ht| + osc(ut ◦ φ + αt) dt Z 0 1 1 −1 ≤ |Ht| + osc(ut ◦ φ ) dt + osc(αt) dt Z Z 0 0 1 1 = (|Ht| + osc(ut)) dt + osc(αt) dt Z0 Z0

= l0({ht})+ K where 1 K = osc(αt) dt. Z0

−1 −1 −1 Let us now do the same calculation for Ψ = {φ ◦ ht ◦ φ }.

˙ −1 ˙ −1 −1 ˙ ˙ Since ht satisﬁes ht = −(ht )∗ht, the cohomology classes of i(ht)ω ˙ −1 and i(ht )ω are opposite. Since the Hodge decomposition is unique and the harmonic part of the ﬁrst form is Ht, the harmonic part of the second form is −Ht. Therefore, there is a smooth family of functions ˙ −1 vt such that the Hodge decomposition for i(ht )ω is

˙ −1 i(ht )ω = −Ht + dvt.

The same calculation shows

˙ −1 −1 i(Ψt )ω = −Ht + d(vt ◦ φ − αt).

Hence, −1 −1 l0(Ψ ) ≤ l0({ht })+ K.

1 −1 We will now estimate K = 0 osc(αt) dt. Fix an isotopy {φs } from φ−1 to the identity. ConsiderR the continuous linear map

1 ∞ −1 L{φs } : H (M,g) → C (M) from the ﬁnite dimensional vector space of harmonic 1-forms given by

1 −1 ∗ −1 −1 ˙ L{φs }(θ)= ((φs ) i(φs )θ) ds. Z0 THE SYMPLECTIC DISPLACEMENT ENERGY 7

1 −1 Let ν ≥ 0 be the norm of L{φs } where the norm on H (M,g) is deﬁned by the metric g and C∞(M) is given the sup norm. Then

−1 −1 |L{φs }(θ)| ≤ ν|θ|. In our case αt = L{φs }(Ht). Therefore,

|αt| ≤ ν|Ht| and

osc(αt) ≤ 2|αt| ≤ 2ν|Ht|. This implies

osc(αt) ≤ 2ν (|Ht| + osc(ut)) and osc(αt) ≤ 2ν (|Ht| + osc(vt)) . Hence, 1 K = osc(αt) dt ≤ 2νl0({ht}), Z0 and 1 −1 K = osc(αt) dt ≤ 2νl0({ht }). Z0

Now recall that,

−1 −1 l0(Ψ) ≤ l0({ht})+ K and l0(Ψ ) ≤ l0({ht })+ K. Therefore, 1 l(Ψ) = l (Ψ) + l (Ψ−1) 2 0 0 1 ≤ l ({h })+2νl ({h })+ l ({h−1})+2νl ({h−1}) 2 0 t 0 t 0 t 0 t ≤ (2ν + 1)l({ht}). Taking the inﬁmum over the set I(h) of all symplectic isotopies from h to the identity we get

inf l(Ψ) ≤ (2ν + 1)khkHL, I(h) and since −1 kφ ◦ h ◦ φ kHL ≤ inf l(Ψ) I(h) we get −1 kφ ◦ h ◦ φ kHL ≤ kkhkHL with k =2ν + 1. 8 AUGUSTIN BANYAGA†, DAVID E. HURTUBISE, AND PETER SPAETH

We have shown that for every φ ∈ Symp0(M,ω) there is a k ≥ 1 (depending on an isotopy {φs} from φ to the identity) such that the preceding inequality holds for all h ∈ Symp0(M,ω). Applying this to φ−1 we see that there is an k′ ≥ 1 such that

−1 ′ kφ ◦ h ◦ φkHL ≤ k khkHL for all h ∈ Symp0(M,ω). Therefore, for any h ∈ Symp0(M,ω) we have −1 −1 ′ −1 khkHL = kφ ◦ (φ ◦ h ◦ φ ) ◦ φkHL ≤ k kφ ◦ h ◦ φ kHL. That is, 1 khk ≤kφ ◦ h ◦ φ−1k ≤ kkhk . k′ HL HL HL

Remark 9. If φ is Hamiltonian and contained in a Weinstein chart, then there is canonical isotopy {φs} with φ1 = φ. In this case k can be viewed as depending only on φ.

3. Proof of the main result We will closely follow the proof given by Polterovich of Theorem 2.4.A in [17] that e(A) > 0. We will use without any change Proposition 1.5.B. Proposition 1.5.B. [17] For any non-empty open subset A of M, there exists a pair of Hamiltonian diﬀeomorphisms φ and ψ that are supported in A and whose commutator [φ, ψ]= ψ−1 ◦ φ−1 ◦ ψ ◦ φ is not equal to the identity. For the sake of completeness we provide the following alternate proof of this proposition based on the transitivity lemmas in [3] (pages 29 and 109). (For a proof of k-fold transitivity for symplectomorphisms see [6].) Proof. Let U be an open subset of A such that U ⊂ A. Pick three distinct points a, b, c ∈ U. By the transitivity lemma of Ham(M,ω), there exist φ, ψ ∈ Ham(M,ω) such that φ(a) = b and ψ(b) = c. Moreover, we can choose φ and ψ so that supp (φ) and supp (ψ) are contained in small tubular neighborhoods V and W of distinct paths in U joining a to b and b to c respectively, and we can assume that c ∈ U\V . THE SYMPLECTIC DISPLACEMENT ENERGY 9

V U a b

Then (ψ−1φ−1ψφ)(a)=(ψ−1φ−1)(c)= ψ−1(c)= b. Hence [φ, ψ] =6 id.

We will say that a map h displaces A if h(A)∩A = ∅. Let us denote by D(A) the set of all h ∈ Symp0(M,ω) that displace A. We note the following fact.

Lemma 10. Let φ and ψ be as in Proposition 1.5.B, and let h ∈ D(A). Then the commutator

θ = [h, φ−1]= φ ◦ h−1 ◦ φ−1 ◦ h satisﬁes [φ, ψ] = [θ, ψ].

Proof. If x ∈ A then h(x) 6∈ A. Hence,

θ(x) = (φ ◦ h−1)(φ−1(h(x))) = φ(h−1(h(x))) since supp (φ−1) ⊂ A = φ(x),

−1 and we see that θ|A = φ|A. Similarly, for x ∈ A we have φ (x) ∈ A, and hence h(φ−1(x)) 6∈ A since h(A) ∩ A = ∅. Thus,

θ−1(x) = h−1(φ(h(φ−1(x)))) = h−1(h(φ−1(x))) since supp (φ) ⊂ A = φ−1(x),

−1 −1 −1 −1 and we see that θ |A = φ |A. Thus, (φ ◦ψ ◦φ)(x)=(θ ◦ψ ◦θ)(x) for all x ∈ A since supp (ψ) ⊂ A. 10 AUGUSTIN BANYAGA†, DAVID E. HURTUBISE, AND PETER SPAETH

Now, if x 6∈ A and θ(x) ∈ A we would have x = θ−1(θ(x)) = φ−1(θ(x)) ∈ A since supp (φ−1) ⊂ A, a contradiction. Hence, for x 6∈ A we have θ(x) 6∈ A and

(φ−1 ◦ ψ ◦ φ)(x)= x =(θ−1 ◦ ψ ◦ θ)(x) since both φ and ψ have support in A. Therefore, φ−1◦ψ◦φ = θ−1◦ψ◦θ, and we have [φ, ψ] = [θ, ψ].

Proof of Theorem 3 continued. Following the proof of Theorem 2.4.A in [17] we assume there exists h ∈ D(A) =6 ∅. Otherwise, we are done since es(A)=+∞. The commutator θ is contained in Ham(M,ω) because commutators are in the kernel of the Calabi invariant. Since both θ and ψ are in Ham(M,ω) and the Hofer norm is conjugation invariant, we have

−1 −1 k[θ, ψ]kH = kψ ◦ θ ◦ ψ ◦ θkH −1 −1 ≤ kψ ◦ θ ◦ ψkH + kθkH

= 2kθkH .

By Buss and Leclercq’s theorem [7] there is constant λ> 0 such that

kθkH ≤ λkθkHL.

Using the triangle inequality and the symmetry of k·kHL we have

−1 k[θ, ψ]kH ≤ 2λ kφ ◦ h ◦ φ kHL + khkHL ≤ 2λ (kkhkHL + khkHL) , where k > 0 is the constant given by Theorem 7. Therefore, k[θ, ψ]k 0 < H ≤khk . 2λ(k + 1) HL Since this inequality holds for all h ∈ D(A) we can take the inﬁmum over D(A) to get k[θ, ψ]k 0 < H ≤ e (A). 2λ(k + 1) s This completes the proof of Theorem 3. THE SYMPLECTIC DISPLACEMENT ENERGY 11

Remark 11. The proof of Theorem 1 relied on the bi-invariance of the distance dH , whereas the proof of Theorem 3 relied on the equivalence −1 of the norms h 7→ kφ ◦ h ◦ φ kHL and k·kHL, i.e. the invariance of dHL up to a constant.

4. Examples

A harmonic 1-parameter group is an isotopy Φ = {φt} generated by the vector field VH defined by i(VH)ω = H, where H is a harmonic 1-form. It is immediate from the definitions that

−1 l0(Φ) = l0(Φ )= |H| where |·| is a norm on the space of harmonic 1-forms. Hence l(Φ) = |H|.

Therefore, if φ1 is the time one map of Φ we have

kφ1kHL ≤ |H|.

2n For instance, take the torus T with coordinates (θ1,...,θ2n) and the ﬂat Riemannian metric. Then all the 1-forms dθi are harmonic. 2n Given v = (a1,...,an, b1,...,bn) ∈ R , the translation x 7→ x + v on 2n 2n R induces a rotation ρv on T , which is a symplectic diﬀeomorphism. Moreover, x 7→ x + tv on R2n induces a harmonic 1-parameter group t 2n {ρv} on T . Taking the 1-forms dθi for i = 1,..., 2n as basis for the space of harmonic 1-forms and using the standard symplectic form n

ω = dθj ∧ dθj+n Xj=1 on T 2n we have n ˙t i(ρv)ω = (ajdθj+n − bjdθj) . Xj=1 Thus, t l({ρv})= |(−b1,..., −bn, a1,...,an)| where |·| is a norm on the space of harmonic 1-forms, and we see that

kρvkHL ≤|v| 12 AUGUSTIN BANYAGA†, DAVID E. HURTUBISE, AND PETER SPAETH if we use |v| = |a1| + ··· + |an| + |b1| + ··· + |bn| as the norm on both R2n and the space of harmonic 1-forms. Consider the torus T 2 as the square:

{(p, q) | 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1} ⊂ R2

1 with opposite sides identiﬁed. For any r < 2 let A˜(r)= {(x, y) | 0 ≤ x

2 and let A(r) be the corresponding subset in T . If v = (a1, 0) with r ≤ a1 ≤ 1−r, then the rotation ρv induced by the translation (p, q) 7→

(p + a1, q) displaces A(r). Therefore, using the norm |v| = |a1| + |b1| we have t kρakHL ≤ l({ρa})= a1.

Since this holds for all r ≤ a1 we have,

es(A(r)) ≤ r.

Remark 12. Note that in the above example the symplectic displacement energy is finite, whereas the Hamiltonian displacement energy e(A(r)) is infinite. This follows from a result proved by Gromov [12]: If (M,ω) is a symplectic manifold without boundary that is convex at infinity and L ⊂ M is a compact Lagrangian submanifold such that [ω] vanishes on π2(M, L), then for any Hamiltonian symplectomorphism φ : M → M the intersection φ(L) ∩ L =6 ∅. Stronger versions of this result can be found in [9], [10], and [11]. See also Section 9.2 of [15].

5. Application The following result is an immediate consequence of the positivity of the symplectic displacement energy of non-empty open sets. For two isotopies Φ and Ψ denote by Φ−1 ◦ Ψ the isotopy given at time t by −1 −1 (Φ ◦ Ψ)t = φt ◦ ψt.

Theorem 13. Let Φn be a sequence of symplectic isotopies and let Ψ be another symplectic isotopy. Suppose that the sequence of time-one maps φn,1 of the isotopies Φn converges uniformly to a homeomorphism −1 φ, and l(Φn ◦ Ψ) → 0 as n →∞, then φ = ψ1. THE SYMPLECTIC DISPLACEMENT ENERGY 13

This theorem can be viewed as a justiﬁcation for the following, which appeared in [1] and [4].

Deﬁnition 14. A homeomorphism h of a compact symplectic manifold is called a strong symplectic homeomorphism if there exist a sequence Φn of symplectic isotopies such that φn,1 converges uniformly to h, and l(Φn) is a Cauchy sequence.

−1 Proof of Theorem 13. Suppose φ =6 ψ1, i.e. φ ◦ ψ1 =6 id. Then there −1 exists a small open ball B such that (φ ◦ ψ1)(B) ∩ B = ∅. Since φn,1 −1 converges uniformly to φ, ((φn,1) ◦ψ1)(B)∩B = ∅ for n large enough.

Therefore, the symplectic energy es(B) of B satisﬁes

−1 −1 es(B) ≤k(φn,1) ◦ ψ1kHL ≤ l(Φn ◦ Ψ).

The later tends to zero, which contradicts the positivity of es(B).

Remark 15. This theorem was ﬁrst proved by Hofer and Zehnder for M = R2n [13], and then by Oh-M¨uller in [16] for Hamiltonian isotopies using the same lines as above, and very recently by Tchuiaga [18], using the L∞ version of the Hofer-like norm.

Acknowledgments We would like to thank the referee for carefully reading an earlier version of this paper and providing the statement of Theorem 7.

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Department of Mathematics, Penn State University, University Park, PA 16802 E-mail address: [email protected]

Department of Mathematics and Statistics, Penn State Altoona, Altoona, PA 16601-3760 E-mail address: [email protected]