Oriented Area as a Morse Function on Polygon Spaces

Abstract

We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces. We characterise critical points of the oriented area function in geometric terms and give a formula for the Morse indices. Thus we obtain a generalisation of isoperimetric theorems for polygons in the plane.

14 pages, 2 figures The difference from the previous version: minor changes in the structure of the document; the proof of Lemma 11 (which is now Lemma 4.7) is corrected; two pictures added