The Riemann mapping theorem states that given any two simply connected open domains $A$ and $B$ of $\mathbb C$ that are neither empty nor equal to $\mathbb C$, there exists a unique (up to normalization) conformal mapping from one to the other. In higher dimensions of space Liouville's theorem states that the conformal mappings are only the Möbius transforms. One can loosen the requirement of being conformal but the exemple of Whitehead manifold shows that the topological properties shared by $A$ and $B$ would have to be extended beyond contractibility. These thoughts lead me to the following question

For $n\gt2$, is there a class of mapping $\mathscr M$ such that for any two homeomorphic simply connected open subsets $A$ and $B$ of $\mathbb R^n$ that are neither empty nor equal to $\mathbb R^n$ there is a

uniquemap $f\in\mathscr M$ verifying $f(A)=B$ ?

*Note*. This question actually came up while I was considering $A(f)=\left\{(x,y,z)\in\mathbb R^3, z\gt f(x,y)\right\}$ and $B=\{(x,y,z)\in\mathbb R^3, z\gt0\}$.