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 \(\bigsqcup \nolimits _{i=1}^k B^4(R_i) \times \Sigma _g (L) \overset{s}{\hookrightarrow }{\operatorname {int}}(B^4(R))\times \Sigma _g (L)\) if and only if there exists a symplectic embedding \(\bigsqcup \nolimits _{i=1}^k B^4(R_i) \overset{s}{\hookrightarrow }{\operatorname {int}}(B^4(R))\) . This lies in contrast with the standard higher dimensional ball packing problem \(\bigsqcup \nolimits _{i=1}^k B^{2 n}(R_i) \overset{s}{\hookrightarrow }{\operatorname {int}}(B^{2n}(R))\) 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 toric domains, and we establish a stabilized version of Gromov’s two ball theorem which holds in any dimension. The strategy is to show the holomorphic curves giving rise to (sharp) symplectic embedding obstructions in dimension four persist in higher dimensions; essential to this approach 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 curves of genus zero.
