On Symplectic Packing Problems in Higher Dimensions

Let $B^{2n}(R)$ denote the closed $2n$-dimensional symplectic ball of area $R$, and let $\Sigma_g(L)$ be a closed symplectic surface of genus $g$ and area $L$. We prove that there is a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i) \times \Sigma_g (L) \overset{s}\hookrightarrow \operatorname{int}(B^4(R))\times \Sigma_g (L)$ if and only if there exists a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i) \overset{s}\hookrightarrow \mathrm{int}(B^4(R))$. This lies in contrast with the standard higher dimensional ball packing problem $\bigsqcup\limits_{i=1}^k B^{2 n}(R_i)\overset{s}\hookrightarrow \mathrm{int}(B^{2n}(R))$ for $n >2$, which we conjecture (based on index behavior for pseudoholomorphic curves) is controlled entirely by Gromov's two ball theorem and volume considerations. We also deduce analogous results for stabilized embeddings of concave toric domains into convex domains, and we establish a stabilized version of Gromov's two ball theorem which holds in any dimension. Our main tools are: (i) the symplectic blowup construction along symplectic submanifolds, (ii) an h-principle for symplectic surfaces in high dimensional symplectic manifolds, and (iii) a stabilization result for pseudoholomorphic holomorphic curves of genus zero.