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Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist a diffeomorphism $\phi$ such that $\phi \circ g_0 = g_1$ ?

More generally given 2n symplectic embeddings $(g_1, ......, g_n)$ and $(g_1^\prime,......, g_n^\prime)$ of a 4-ball of radius $r$ does there exist a diffeomorphism $\phi$ such that $\phi \circ g_i = g_i^\prime ~ ~\forall i \in \{1,....n\}$.

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    $\begingroup$ Yes, "Palais-Cerf lemma", see eg here. The only reason the embeddings being symplectic is relevant is that they are all consistently oriented. $\endgroup$
    – mme
    Commented Feb 27, 2019 at 23:02
  • $\begingroup$ But did you want the diffeomorphism to preserve the symplectic structure? $\endgroup$
    – Tom Goodwillie
    Commented Feb 27, 2019 at 23:15
  • $\begingroup$ @Tom Goodwillie No not necessarily. $\endgroup$
    – cr1t1cal
    Commented Feb 27, 2019 at 23:22

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