Let ${\Omega _1}$, ${\Omega _2}$ be two domains in $\mathbb R^n$ and $f$ be a diffeomorphism between them. For every convex compact $K \subset \Omega _1$, is there a diffeomorphism $g$ of $\mathbb R^n$ such that $g=f$ when restricted to $K$? This is a stronger statement of The extension of diffeomorphism which is proven true.
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What are your thoughts? Did you analyze the 1-dimensional case? What did you conclude? – Moishe Kohan Jan 01 '23 at 01:00
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@MoisheKohan The 1d case is obviously true as $K$ is contained in a closed interval $[a, b] \in U$ which can easily be extended by linear functions using derivatives at $a$ and $b$. – siyu Jan 01 '23 at 02:25
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Nope: Why do you think that $U$ is connected? – Moishe Kohan Jan 01 '23 at 02:27
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You're right. I've added that assumption. – siyu Jan 01 '23 at 02:30
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OK, then think about the case of an annulus $A$ in $R^2$ and $f: A\to A$. – Moishe Kohan Jan 01 '23 at 02:30
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More assumption added. – siyu Jan 01 '23 at 02:33
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Your question appears to be a moving target. Now, think of a spherical shell in $R^3$. – Moishe Kohan Jan 01 '23 at 02:35
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Sorry. My bad for didn't think this through. There shouldn't be any obvious counterexample now. – siyu Jan 01 '23 at 02:39
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Then it is true (at least in dimensions $\ne 4$) but non-trivial. – Moishe Kohan Jan 01 '23 at 03:44