Parametric Gromov width of Liouville domains

The classical Gromov width measures the largest symplectic ball embeddable into a symplectic manifold; inspired by the symplectic camel problem, we generalize this to ask how large a symplectic ball can be embedded as a family over a parameter space $N$. Given a smooth map $f:N\to \mathrm{\Omega }$, where $\mathrm{\Omega }$ is a symplectic manifold, we define the \emph{parametric Gromov width} $\mathrm{G}\mathrm{r}(f,\mathrm{\Omega })$ as the supremum of capacities $a>0$ for which there exists a family of balls, parametrized by $N$, of capacity $a$ whose centers trace out the map $f$. For Liouville domains $\mathrm{\Omega }$, we establish upper bounds on $\mathrm{G}\mathrm{r}(f,\mathrm{\Omega })$ using the Floer cohomology persistence module associated to $\mathrm{\Omega }$. Specializing to fiberwise starshaped domains in the cotangent bundle $T^{\ast }M$, we derive computable bounds via filtered string topology. Specific examples of $\mathrm{\Omega }$ – including disk cotangent bundles of thin ellipsoids, open books, and tori – demonstrate our bounds, and reveal constraints on parameterized symplectic embeddings beyond the classical Gromov width.