Compactly supported Hamiltonian loops with non zero Calabi invariant

We give examples of compactly supported Hamiltonian loops with non zero Calabi invariant on certain open symplectic manifolds.


Introduction
Existence of such an example is indicated by McDuff in [3,Remark 3.10]. In addition, our interest to the problem was stimulated by [7]. An immediate corollary is that we get examples of open symplectic manifolds such that π 1 (Ham(M)) = 0.
Let us present one geometric consequence of the non-vanishing of Cal on For an element γ ∈ Π put and which also satisfies that it is homotopic to {f t } t∈ [0,1] in Ham(X).
with r 1 = r 2 , where r 1 , r 2 stand for the radii. This is a closed symplectic manifold with the symplectic form ω := area S 2 (r 1 ) ⊕ area S 2 (r 2 ) . Define the Hamiltonian function Proof: Near s we can find an S 1 -invariant neighbourhood which is S 1equivariantly symplectomorphic to an open neighborhood of 0 in C 2 with the Hamiltonian S 1 -action generated by: 1 1 More explicitly, if we look at S 2 × S 2 with cylindrical coordinates (θ 1 , α 3 , θ 2 , β 3 ) and on C 2 with polar coordinates (σ 1 , ρ 1 , σ 2 , ρ 2 ), then we can write the symplectomorphism explicitly where (z 1 , z 2 ) ∈ C 2 . We see that the Hamiltonian flow in C 2 is the loop of symplectic matrices e −2πit ⊕ e 2πit ∈ Sp(4), which has Maslov index 0. We get that s is a Maslov-zero fixed point. We can calculate This proves the claim and we see that all the requirements of Theorem 2.2 are satisfied.
Remark 2.4. We can extend Example 2.3 to higher dimensions.
Look at with the Hamiltonian where h i is the height function of the i-th copy of S 2 and r 1 = r 2 . Define s to be the fixed point s = ((0, 0, r 1 ), . . . , (0, 0, r 1 ), (0, 0, −r 2 )). Now s is a Maslov-zero fixed point, F (s) = 0 and F ω m = 0. We get again that all the requirements of Theorem 2.2 are satisfied.
Example 2.5. Let (X ′ , ω) be a closed symplectic manifold with an S 1 -action which has a Maslov-zero fixed point s, and is generated by a non-constant Hamiltonian function F . Under these conditions we can always construct an open symplectic manifold which admits a compactly supported Hamiltonian loop with a non-zero Calabi invariant. Normalize F so that F (s) = 0. Take another fixed point s ′ with F (s ′ ) = 0 (there are always at least two fixed points with different F -values because F achieves maximum and minimum on X ′ ). If X ′ F ω n = Vol(X ′ ) · F (s) then the requirements of Theorem 2.2 are satisfied right away. Assume that X ′ F ω n = 0. We first deal with the case where F (s ′ ) > 0. Choose a neighborhood U of s ′ so that F | U > 0. Perform an equivariant symplectic blow-up at s ′ (see [2, section 6.1]) with a small enough weight so that the symplectic ball B that we cut out in the blow-up will be inside U. Call the resulting symplectic manifold (X, ω), and the resulting Hamiltonian function F . The mean value of F will be less than zero because Hence, Note that in a neighborhood of s, we have F = F . We get that s is a Maslovzero fixed point of F and F (s) = 0. Hence, the requirements of Theorem 2.2 are satisfied.
If F (s ′ ) < 0 then we can define U to be a neighborhood of s ′ so that F | U < 0 and then continue as before.
Note that we can always perform an equivariant symplectic blow-up if the weight is small enough. From the equivariant Darboux theorem, we get that there is a small neighborhood of s ′ which is equivariantly symplectomorphic to a neighborhood of zero in C n with a linear symplectic S 1 -action inside U(n). Hence, we can choose a ball inside the neighborhood with radius λ and get that the action naturally extends to the blow-up with weight λ. In Remark 2.7 below we shall use the monotonicity in order to show that the non-contractible Hamiltonian loop that we have constructed in Ham(M) is also non-contractible in Ham(X).

Remark 2.7. In the previous examples we considered an open manifold
M which is a subset of a bigger closed manifold X. One could ask if the non-contractible loops that we have constructed in Ham(M) remain noncontractible in Ham(X). The constructed loop in Ham(M) is always homotopic to the original S 1 -action in Ham(X) so we get that it remains to check wether the original S 1 -action is non-contractible in Ham(X).
The Hamiltonian flow of F is the 2-turn rotation around the α 3 -axis times the 2-turn rotation around the β 3 -axis. It is a known fact that π 1 (Ham(S 2 )) = Z 2 , and that the generator is the 1-turn rotation (see [5, section 7.2]). We get that the 2-turn rotation is a contractible loop in Ham(S 2 ) and hence the flow of F is contractible in Ham(X).
The point s = ((0, 0, r 1 ), (0, 0, −r 2 )) is a Maslov-zero fixed point and X F ω n = Vol(X) · F (s). We will now show that the situation is different for monotone manifolds. For a closed monotone symplectic manifold X the Hamiltonian loops that we are considering will always be non-contractible in Ham(X). Proof. Consider the mixed action-Maslov invariant of the loop {f t } t∈[0,1] (see [6]). We calculate it via the constant loop {f t (s)} t∈[0,1] ⊂ X with the constant spanning disk s : D 2 → X. Normalize F so that its mean value will be zero, and calculate the symplectic action The Maslov index of the loop {d s f t } t∈[0,1] is zero so we get that the mixed action-Maslov invariant is Hence the loop {f t } t∈[0,1] is non-contractible in Ham(X).

Proof of Theorem 2.2
In order to prove Theorem 2.2 we will need the following lemma. and J is the standard linear complex structure on R 2n . In our situation H s,t =< x, Q s,t x > is a Hamiltonian function that generates the flow s → A s,t for Q s,t = 1 2 J ∂As,t ∂s A −1 s,t . We get that H is smooth, The latter assertion holds because This proves the claim.
Choose three balls B 1 ⊂ B 2 ⊂ B 3 ⊂ B such that for each t ∈ [0, 1], A t B 1 ⊂ B 2 . Choose a cut-off function a : R 2n → R such that a| B 2 = 1 and a| R 2n \B 3 = 0.
For a fixed t, define {g s,t } s∈[0,1] to be the Hamiltonian flow generated by the Hamiltonian function {a · H s,t } s∈[0,1] with the time variable s.
For a fixed s, {g s,t } t∈[0,1] is a loop with respect to t. That is true because H s,0 = H s,1 = 0 for every s, so g s,0 = g s,1 = Id.
Define {G s,t } t∈[0,1] to be the Hamiltonian function that generates g s,t for a fixed s with the time variable t, normalized so that G s,t (0) = 0. Denote g t := g 1,t . Note that for each t, because a| B 2 = 1 and A t B 1 ⊂ B 2 . Note also that g 0,t = Id because it is the time-0 map of the flow with the time variable s, so we get that Claim: For each ǫ > 0 there is δ > 0 such that if B 3 is with radius less than δ, then where ω 0 is the standard symplectic form on R 2n .
Proof: The Hamiltonian functions G s,t generates g s,t with time t, and a·H s,t generates g s,t with time s. We can use a known formula (see [5, section 6.1]), and get that Note that G 0,t = 0, so if we integrate over s we will get Note that G 1,t = 0 outside B 3 . Multiply with ω n 0 and integrate over R 2n to get For any two smooth functions F 1 , F 2 , the form {F 1 , F 2 }ω n 0 is exact, so from this we get that We know that ∂Hs,t ∂t is a smooth function so it is bounded on B × I × I. Hence for every t, s ∈ [0, 1] and for every δ such that B 3 ⊂ B, we have that ∂Hs,t ∂t < K for some K > 0. We get that This means that we can choose δ such that If we integrate over t we will get that This proves the claim.
We get that This proves the lemma.
Proof of Theorem 2.2. Assume that B 2 is an equivariant Darboux ball around z 0 (see [1], section 3.1). We choose the Darboux coordinates on B 2 such that z 0 is identified with 0 and the path f t | B 2 is identified with the path A t ∈ Symp(2n). From the fact that z 0 is a Maslov-zero fixed point, we know that the loop {A t } t∈[0,1] has Maslov index 0, and hence it is a contractible loop in Sp(2n). Normalize F so that F (z 0 ) = 0 and X F ω n = 0. Set ǫ = X F ω n 2 .
Use Lemma 3.1 to define the contractible loop {g t } t∈[0,1] and the ball B 1 such that for each t ∈ [0, 1], g t is supported in B 2 , g t | B 1 = A t | B 1 , and Define H t as the Hamiltonian function generating h t : Note that G t | B 1 = F | B 1 . Hence we get that H t | B 1 = 0. Choose a ball B ⊂ B 1 . The flow h t is defined on X\B and H t | B 1 \B = 0 so we get that H t is compactly supported on X\B. From the definition of the loop {g t } t∈[0, 1] we have that From this and from the fact that X F ω n = 0 we get that 1 0 X H t ω n dt = 0. However, This completes the proof.