@R.vanDobbendeBruyn Thank you for your comment. I added the additional assumption that the $V_i$ are assumed to be disjoint.

@RobertBryant For the setting I had in mind, the curve had boundary in a Lagrangian submanifold. I was hoping to control the "size" of the curve" (with respect to some, say J-compatible, background metric). However, the question as posed is clearly too general, as you pointed out.

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.

I don't know a full answer to this question, but observe for instance that my suggestion would clearly fail if the 1-form vanished in some subset of R4 whose complement is disconnected. It's important for my suggestion that the integral can always be made slightly bigger or smaller by slightly modifying the curve.

@YCor thanks for pointing out the typo. I edited the question again to make clear that the final assertion was false.

Well, if you consider the standard representation of $SL(2,\mathbb{C})$ as 2x2 matrices, then it means that this subgroup is an irreducible representation. i.e. it fixes a line.

Thanks a lot for your edit:) I apologize for the confusion!