Singular fibers of the bending flows on the moduli space of 3D polygons

TL;DR

This paper characterizes the structure of singular fibers in the bending flows on the moduli space of polygons, showing they are isotropic submanifolds or orbispaces, extending to non-generic side lengths.

Contribution

It proves that singular fibers in the bending flows are isotropic (orbispaces) for generic and non-generic side lengths, generalizing previous results and suggesting similar analysis for related integrable systems.

Findings

Singular fibers are isotropic homogeneous submanifolds.

In non-generic cases, singular fibers are isotropic orbispaces.

Results extend to systems defined by disjoint diagonals.

Abstract

In this paper, we prove that in the system of bending flows on the moduli space of polygons with fixed side lengths introduced by Kapovich and Millson, the singular fibers are isotropic homogeneous submanifolds. The proof covers the case where the system is defined by any maximal family of disjoint diagonals. We also take in account the case where the fixed side lengths are not generic. In this case, the phase space is an orbispace, and our result holds in the sense that singular fibers are isotropic orbispaces. In a last part we provide leads in favor of a similar study of the integrable systems defined by Nohara and Ueda on the Grassmannian of 2-planes in $\mathbb{C}^n$.