Extension of volume preserving diffeomorphism

Assume $$\Omega$$ is a smoothly bounded open domain in $$\mathbb R^n$$ and $$B$$ is an open ball of equal volume. Let $$\Phi:\partial\Omega\to \partial B$$ be a diffeomorphism. Is it true that there exists a volume preserving diffeomorphism $$\tilde{\Phi}:\overline{\Omega}\to\overline{B}$$ such that $$\tilde{\Phi}\vert_{\partial\Omega}=\Phi$$?

• Thanks, yes $\tilde{\Phi}$ should be volume preserving. Aug 19, 2021 at 8:26
• I observe that $\Omega$ is not constrained to be connected, which can lead to an obstruction. Aug 19, 2021 at 16:29
• is there a special term for volume preserving diffeomorphism like how isometry is like metric preserving diffeomorphism or something?
– BCLC
Aug 19, 2021 at 19:02

Let $$\Omega$$ be a submanifold of $$\mathbb R^n$$ with boundary $$\partial\Omega$$ diffeomorphic to $$\partial B$$. As far as I know, it is not obvious that $$\overline\Omega$$ should be diffeomorphic to $$\overline B$$, and my guess would be that this is actually false in all generality. From what I understand, though, it is true if you assume $$\Omega$$ is contractible and $$n\geq5$$, see this question.
So there might be a topological obstruction. Once you get past this (for instance if you already assume $$\overline\Omega$$ is diffeomorphic to $$\overline B$$), then it reduces to the same question with $$\Omega=B$$. Now there might be another topological obstruction, which is that the diffeomorphism of the boundary may not extend to a diffeomorphism of the whole ball, see this answer.
Again, if you already know that $$\overline\Omega$$ is diffeomorphic to $$\overline B$$, then these obstructions cannot happen, more or less by definition. Then your question becomes: let $$\omega_1,\omega_2$$ be two smooth volume forms on $$B$$ with the same global volume, is there a diffeomorphism sending one to the other? This is a Moser-type theorem, and the version for manifolds with boundary is proved in Formes-volumes sur les variétés à bord (volume forms on manifolds with boundary) by A. Banyaga (1974).