Let $\Omega$ be a star-shaped open set of $\mathbb{R}^3$. Under which conditions is $\Omega$ analytically diffeomorphic to $\mathbb{R}^3$?

Question

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A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$

I found a proof that $\Omega$ is always diffeomorphic to $\mathbb{R}^3$. In which cases can such a diffeomorphism be analytical?

Answer 1

There is a general theorem (Morrey and Grauert) stating that if two real-analytic manifolds are diffeomorphic then they are real-analytically isomorphic.

H. Grauert, On Levi’s problem and the imbedding of real analytic manifolds, Ann. of Math., 68 (1958), 460-472.

C. B. Morrey, The analytic embedding of abstract real analytic manifolds, Ann. of Math., 68 (1958), 159-201.