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Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover that any $J$-holomorphic disk in $M_1$ is send to a $J$-holomorphic disk in $M_2$ with the same symplectic area. Is this diffeomorphism necessarily symplectomorphism?

Would there be a generalization of this question without a condition of preserving Lagrangian submanifolds or maybe without fixed $J$ structures? Or maybe we should think about Lagrangian immersions instead of submanifolds?

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  • $\begingroup$ How such a rescaling would preserve unit disk area and Cauchy-Riemann equations (j-holo condition) at the same time? $\endgroup$
    – OSBM
    Commented Mar 26, 2022 at 20:19
  • $\begingroup$ Above comment was response to a comment (which disappeared) saying that "rescaling of a plane is a trivial counter example". $\endgroup$
    – OSBM
    Commented Mar 27, 2022 at 20:49

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