Suppose that we have open sets $A, B\subset \mathbb{R}^2$ such that $A$ and $B$ are simply connected and have smooth boundaries. Furthermore, let us assume that $m(A) = m(B) > 0$, where $m$ is the Lebesgue measure on $\mathbb{R}^2$. My question is as follows: when does there exist an area-preserving diffeomorphism between $A$ and $B$? In this context, an area-preserving diffeomorphism between $A$ and $B$ is a is a $C^{\infty}$-diffeomorphism $\phi : A\to B$ such that $m(S) = m(\phi(S))$ for all measurable $S\subseteq A$. If an area-preserving diffeomorphism doesn't always exist between sets $A$ and $B$ as above, are there any extra conditions we can impose on $A$ and $B$ such that there does exist an area-preserving diffeomorphism?
My thoughts thus far are that we always have a $C^{\infty}$-diffeomorphism between $A$ and $B$: this is a consequence of the Riemann mapping theorem. It's also obvious to me that if we can deform $A$ linearly to get $B$, then that deformation will be area-preserving. Furthermore, it seems that aside from this case, there's no reason why an area-preserving diffeomorphism should exist. However, I'm having a hard time proving that there necessarily isn't an area-preserving diffeomorphism between $A$ and $B$ if $B$ is not a linear deformation of $A$. Does anybody have any ideas?
See
Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, by R. E. Greene and K. Shiohama, Trans. Amer. Math. Soc. 255 (1979), 403-414.
They prove a version of Moser's theorem for noncompact manifolds of any dimension. In your setting, it yields a volume-preserving diffeomorphism.