Parametric Gromov width of Liouville domains

TL;DR

The paper defines the parametric Gromov width $\text{Gr}(f,\Omega)$ for maps $f:N\to\Omega$ and develops Floer cohomology persistence modules to obtain upper bounds in Liouville domains, linking symplectic rigidity to parameterized embeddings. Specializing to fiberwise starshaped domains in $T^{*}M$, the authors derive computable, string-topology-based bounds via the BV-algebra structure, PSS isomorphisms, and a $\,\Theta$-morphism that ties string topology to Floer theory. They provide a catalog of sharp bounds in concrete examples (thin ellipsoids, open books, products with tori, and non-orientable surfaces), illustrating how parameterized phenomena constrain embeddings beyond the classical Gromov width. The work also develops a comprehensive framework unifying string topology and Floer cohomology through family Floer theory, with action filtrations and evaluation maps that yield quantitative obstructions to parametric ball embeddings, thereby enriching the toolbox for symplectic embedding problems.

Abstract

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 $f: N \to Ω$, where $Ω$ is a symplectic manifold, we define the \emph{parametric Gromov width} $\mathrm{Gr}(f,Ω)$ 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 $Ω$, we establish upper bounds on $\mathrm{Gr}(f,Ω)$ using the Floer cohomology persistence module associated to $Ω$. Specializing to fiberwise starshaped domains in the cotangent bundle $T^*M$, we derive computable bounds via filtered string topology. Specific examples of $Ω$ -- 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.

Figures (19)

• Figure 1: The cobordism between $M$ and $\mathbb{R}/\mathbb{Z}\times D(V)$ can be visualized as attaching a generalized handle to $M$. The simplest example is when $V=[0,1]$ where $M\simeq S^{2}$ and $\mathbb{R}/\mathbb{Z}\times D(V)\simeq T^{2}$.
• Figure 2: Compactness up-to-breaking of Floer differential cylinders at the punctures; a sequence of solutions $u_{n}$ defined on the pair-of-pants surface converging on compact subsets to a limit $u_{\infty}$, with the breaking of two solutions $v_{0},v_{1}$ of the $s$-independent equation at the punctures.
• Figure 3: The Floer differential
• Figure 4: Graph of $\rho:\mathbb{R}\to \mathbb{R}$
• Figure 5: A consequence of $\mathfrak{e}(\sum \gamma_{i})=1$ is the existence of a cylinder $(\eta,u)$ joining $\gamma_{i}$ to the center of the ball $B_{\eta}$. In the figure $Y,X$ are the Hamiltonian generators of $K,H$ where $(\mathfrak{a}=K_{\eta,s,t}\mathrm{d} s+H_{\eta,s,t}\mathrm{d} t,J_{\eta,s,t})$ is evaluation data as in §\ref{['sec:defin-evaluation-map']}.

Theorems & Definitions (133)

• Proposition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Corollary 9
• Proposition 10