On the Gromov width of polygon spaces

Abstract

For generic $r=(r_1,\dots ,r_n)\in {\mathbb{R}}_{+}^n$ 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 $$2\pi \min \{2r_j,(\sum _{i\ne j}{r}_i)-r_j|j=1,\dots ,n\}$$ 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 $n\ge 4$.

39 pages, 14 figures, to appear on Transformation Groups