area-preserving maps between regions of equal area

Suppose M, N are compact diffeomorphic subsets of the plane of equal area.

Does there exist an area-preserving diffeomorphism from M to N? If not, does it help to add additional assumptions on M and N (ie, that they are both simply-connected)?

Intuitively it seems like such a map should exist, and I imagine it's a standard theorem in differential geometry, but I haven't had much luck searching for it. The only paper I've found is the following, which (to my understanding) answers affirmatively in the simply-connected case: JSTOR: An Error Occurred Setting Your User Cookie

If yes, does there exist an area-preserving homotopy from M to N, ie, is it possible to smoothly deform M to N while preserving area?

What about in higher dimensions?