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Apr 7, 2018 at 0:57 comment added user142700 I think this is a good idea and I agree that the existence of such an embedding would imply the claim. However, I think that the existence of a Lagrangian cylinder connecting two curves of equal area is in general not true (although I don't know an easy counterexample). I think it's also important to observe that it's not hard to construct isotopies of curves of constant area which don't trace out a Lagrangian.
Apr 6, 2018 at 19:20 history edited Piotr Hajlasz CC BY-SA 3.0
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Apr 6, 2018 at 19:09 history edited Piotr Hajlasz CC BY-SA 3.0
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Apr 6, 2018 at 18:27 history answered Piotr Hajlasz CC BY-SA 3.0