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Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?

Note: A possible approach could be the following: Is it true to say that the answer is affirmative because of Arnold conjecture (which is possibly true for the annulus region)? Is Arnold conjecture true (or discussable) for manifold with boundary at least in low dimensional case?

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    $\begingroup$ A theorem of Jürgen Moser says that all smooth volume forms on a compact, connected, smooth manifold are equivalent by a smooth diffeomorphism. If this theorem can be extended to manifolds with boundary (or just surfaces with boundary), that would show the Poincaré-Birkhoff theorem is true for an arbitrary smooth volume form on the closed annulus. $\endgroup$
    – Daniel Asimov
    Commented Sep 12, 2023 at 17:00
  • $\begingroup$ @DanielAsimov Is it realy the case? if it would be true so one may associate an intrinsic volum to every manifold M: the integral of arbitrary volum form over M. By theorem of change of variabble $\int_M f^* \alpha =\int M \alpha$. But I guess the theorem of Moser has a function multiplication too that is $\alpha =g f^* \beta$ for a positive function $g$. Do you agree? $\endgroup$
    – Ali Taghavi
    Commented Sep 12, 2023 at 20:50
  • $\begingroup$ @DanielAsimov I correct a typos: $\int_M f^* \alpha =\int_M \alpha$ $\endgroup$
    – Ali Taghavi
    Commented Sep 12, 2023 at 21:13
  • $\begingroup$ So if the multiplication version of the theorem is true we can not conclude that the Poincare Birkhoff theorem is true for arbitrary volum form. Am I mistaken? $\endgroup$
    – Ali Taghavi
    Commented Sep 12, 2023 at 21:15
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    $\begingroup$ I should have said that for any two positive volume forms on an oriented manfold, compatible with the orientation, there is a diffeomorphism taking one to the other up to a positive constant factor (math.tecnico.ulisboa.pt/~mabreu/GD/moser.pdf). $\endgroup$
    – Daniel Asimov
    Commented Sep 13, 2023 at 13:45

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