Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six

Theorem 1.1: for $n\ge3$, $k\ge1$ and positive $R,R_1,\dots,R_k$, the closed balls $B^{2n}(R_i)$ embed disjointly and symplectically into $\mathrm{Int}\,B^{2n}(R)$ if and only if $\sum_iR_i^n<R^n$ and $R_i+R_j<R$ for $i\ne j$. The construction uses a positive-genus surface base with toric fibres and a Hamiltonian comparison lemma. Not shown: anything in dimension four, where more obstructions exist, or the stabilized packing problems Siegel and Yao also study.

Theorem 1.1: for $n\ge3$, $k\ge1$ and positive $R,R_1,\dots,R_k$, the closed balls $B^{2n}(R_i)$ embed disjointly and symplectically into $\mathrm{Int}\,B^{2n}(R)$ if and only if $\sum_iR_i^n<R^n$ and $R_i+R_j<R$ for $i\ne j$. The construction uses a positive-genus surface base with toric fibres and a Hamiltonian comparison lemma. Not shown: anything in dimension four, where more obstructions exist, or the stabilized packing problems Siegel and Yao also study.