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I recently heard the it is an open question whether or not an unknotted $S^2$ in $S^4$ bounds a unique $B^3$ in $S^4$, where by unique, I mean up to isotopy rel boundary in $S^4$. The rel boundary condition is what makes this interesting.

I am interested in hearing what is known about this other dimensions/codimensions. Namely, given a ball $B^{m+1}$ embedded in $S^n$, are there other balls bounding $S^m = \partial B^{m+1}$ in $S^n$ that are not isotopic to the original ball.

For what it's worth, I like the smooth category - smooth embeddings, smooth isotopies - although I am happy to hear about other stuff.

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    $\begingroup$ R. Palais proved in theorem B of "Extending diffeomorphisms", Proc. Amer. Math. Soc. 11 (1960), 274-277 that any two smoothly embedded $k$-disks in a closed $n$-manifold with $k<n$ can be moved to each other by a diffeomorphism that is isotopic to identity. Perhaps, one can modify the proof to make the isotopy rel boundary. $\endgroup$
    – Igor Belegradek
    Commented Sep 10, 2019 at 22:06
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    $\begingroup$ @IgorBelegradek I was told that if all the 3-balls in $S^4$ are isotopic rel boundary, then smooth Schoenflies follows. So the modification would presumably need to be tricky. $\endgroup$
    – user101010
    Commented Sep 11, 2019 at 9:37

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