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Before that I mention my question explicitly, I start with my motivation:

Look at $\mathbb{D} \times \mathbb{C}=\{(x_{1},x_{2},y_{1},y_{2})\mid x_{1}^{2}+x_{2}^{2}< 1\}$.This can be identified with the cotangent bundle of $M:=\mathbb{D}$, that is $T^{*} M$

However the capacity of this symplectic manifold $T^{*}M$ is not finite with respect to $\omega=dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2}$, but the capacity is finite with respect to

$\omega=dx_{1}\wedge dx_{2}+dy_{1}\wedge dy_{2}$.( According to Gromov Squeezing theorem)

Now my question is about the globalization of this second 2-form. In fact my question is:

Is there a symplectic structure on $T^{*} M$ such that a relevant capacity is finite, when $M$ is a compact manifold? If the answer is affirmative, does this capacity depend on differential structure of $M$(for example when $M$ is $S^{7}$).

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    $\begingroup$ The standard structures satisfies the important property that every diffeomorphism of $M$ lifts to a symplectomorphism of its cotangent bundle. Do you want to keep this property? $\endgroup$
    – alvarezpaiva
    Commented Jan 8, 2014 at 17:26
  • $\begingroup$ To start, we do not require this property. (can the property which you mentioned be found in the MCDuff books?) $\endgroup$
    – Ali Taghavi
    Commented Jan 8, 2014 at 17:32
  • $\begingroup$ Do you even know of any example of symplectic capacities which is defined for general symplectic manifolds and general symplectomorphism !!! $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jan 8, 2014 at 17:45
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    $\begingroup$ Sure, the Gromov width makes sense for any symplectic manifold. $\endgroup$
    – John Pardon
    Commented Jan 8, 2014 at 23:22

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