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Suppose $A$ and $B$ are subsets of $\mathbb{R}^2$ homeomorphic (and thus $C^\infty$ diffeomorphic) to the open annulus (punctured $\mathbb{B}^2$) and let $G$ be the group of isometries of ${\mathbb R}^2$ that preserve both $A$ and $B$ $\big(\phi (A)=A$ and $\phi (B)=B$, for any $\phi\in G \big)$.

Question: is there always a smooth diffeomorphism $f:A\rightarrow B$ such that $$f\circ \phi=\phi \circ f$$ for any $\phi\in G$, i.e, a $G$-equivariant $C^\infty$ diffeomorphism between $A$ and $B$ ?

The same question for $A$ and $B$ homeomorphic to $\mathbb{B}^2$ with finitely many points removed.

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  • $\begingroup$ your interesting question is a motivation for the following related question: is the equivariant version of conformal classification of topological annuli, true?(see Riemann Surface,Farkas and Kra) Namely, assume that V is an open subset of the plane,with annuluar topology, which is invariant under the action of the isometry group of the plane. Does there exist an equivariant biholomorphic map between V and some annular reagin in the plane? If the answer to this question is affirmative, then your question has also an affirmative answer. $\endgroup$ Dec 2, 2013 at 12:38

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