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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 $S^{7}$).

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 $S^{7}$).