On the Gromov width of polygon spaces

TL;DR

This paper determines the Gromov width of generic polygon spaces in three dimensions, providing explicit formulas for 5- and 6-gons and conditions under which the width equals a specific value, linking geometry and symplectic topology.

Contribution

It derives explicit formulas for the Gromov width of polygon spaces, extending understanding of their symplectic geometry and identifying cases where the width matches a known geometric quantity.

Findings

Gromov width formula for 5- and 6-gon spaces

Lower bounds for all 6-gon spaces

Exact Gromov width when space is symplectomorphic to complex projective space

Abstract

For generic $r=(r_1,\ldots,r_n) \in \mathbb{R}^n_+$ the space $\mathcal{M}(r)$ of $n$--gons in $\mathbb{R}^3$ with edges of lengths $r$ is a smooth, symplectic manifold. We investigate its Gromov width and prove that the expression $$2\pi \min \{2 r_j, (\sum_{i \neq j} r_i) - r_j\,\,|\, j=1,\ldots,n\}$$ is the Gromov width of all (smooth) $5$--gon spaces and of $6$--gon spaces, under some condition on $r \in \mathbb{R}^6_+$. The same formula constitutes a lower bound for all (smooth) spaces of $6$--gons. Moreover, we prove that the Gromov width of $\mathcal{M}(r)$ is given by the above expression when $\mathcal{M}(r)$ is symplectomorphic to $\mathbb{C}\mathbb{P}^{n-3}$, for any $n \geq 4$.