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The Symplectic Geometry of Polygons in Euclidean Space

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The Symplectic Geometry of Polygons in Euclidean Space The Symplectic Geometry of Polygons in Euclidean Space Michael Kap ovich and John J Millson March Abstract We study the symplectic geometry of mo duli spaces M of p oly r gons with xed side lengths in Euclidean space We show that M r has a natural structure of a complex analytic space and is complex 2 n analytically isomorphic to the weighted quotient of S constructed by Deligne and Mostow We study the Hamiltonian ows on M ob r tained by b ending the p olygon along diagonals and show the group generated by such ows acts transitively on M We also relate these r ows to the twist ows of Goldman and JereyWeitsman Contents Intro duction Mo duli of p olygons and weighted quotients of conguration spaces of p oints on the sphere Bending ows and p olygons Actionangle co ordinates This research was partially supp orted by NSF grant DMS at University of Utah Kap ovich and NSF grant DMS the University of Maryland Millson The connection with gauge theory and the results of Gold man and JereyWeitsman Transitivity of b ending deformations Bending of quadrilaterals Deformations of ngons Intro duction Let P b e the space of all ngons with distinguished vertices in Euclidean n 3 space E An ngon P is determined by its vertices v v These vertices 1 n are joined in cyclic order by edges e e where e is the oriented line 1 n i segment from v to v Two p olygons P v v and Q w w i i+1 1 n 1 n are identied if and only if there exists an orientation preserving isometry g 3 of E which sends the vertices of P to the vertices of Q that is g v w i n i i Let r r r b e an ntuple of p ositive real numb ers Then M is 1 n r dened to b e the space of ngons with side lengths r r mo dulo isometries 1 n as ab ove The group R acts on P by scaling and we obtain an induced + n M for R Thus we lose nothing by assuming isomorphism M r + r n X r i i=1 We make this normalization to agree with DM x We observe that the map n P R n + which assigns the vector r of side lengths to an ngon P has image the p olyhedron D of KM x The mo duli spaces M then app ear as the b ers n r of and their top ology may b e obtained by the wall crossing arguments of KM and Wa We recall that the mo duli space M is a smo oth manifold r i M do esnt contain a degenerate p olygon We denote by the collection r of hyp erplanes in D describ ed in KM then M is singular i r n r This pap er is concerned with the symplectic geometry of the space M r 2 n We prove two main results The space S is given the symplectic structure n X r v ol i i=1 2 where v ol is the standard symplectic structure on S Our rst result gives 2 n a natural isomorphism from M to the weighted quotient of S for the r weights r r constructed by Deligne and Mostow in DM The construc 1 n tion go es as follows The closing condition e e 1 n for the edges of the p olygon P denes the zero level set for the momentum 2 n map for the diagonal action of SO on S Thus M is the weighted sym r 2 n plectic quotient of S by SO We give M the structure of a complex r analytic space with at worst quadratic singularities This is immediate at 2 n smo oth p oints of M since S is Kahler However it requires more eort r at the singular p oints We then extend the KirwanKempfNess theorem KN K N to show that the complex analytic quotient constructed by Deligne and Mostow is complex analytically isomorphic to the symplectic quotient We observe that the cusp p oints Q of DM corresp ond to the singular p oints of M which cusp r in turn corresp ond to degenerate p olygons ie p olygons that lie in a line Our second main result is the construction of b ending ows Supp ose P has edges e e We let 1 n e e k n k 1 k +1 b e the k th diagonal of P and dene functions f f 1 n3 on M by r 2 f P k k k k We check that the functions f Poisson commute and that the corresp onding k Hamiltonian ows have p erio dic orbits Since dimM n we see that r M is completely integrable and is almost a toric variety unfortunately we r cannot normalize the ows to have constant p erio ds if the functions f have k zeros on M The Hamiltonian ow for f has the following geometric de r k scription Construct a p olyhedral surface S b ounded by P by lling in the triangles 1 2 n2 where has edges e k n we have e k k k +1 k +1 n1 n Some of these triangles may b e degenerate The diagonal divides S into k two pieces Keep the second piece xed and rotate the rst piece around the diagonal with angular velo city equal to f P Thus the surface S is b ent k along the diagonal and we call our ows b ending ows If then P is k xed by the ow However the b ending ow do es not preserve the complex structure of M r 0 Let M denote the dense op en subset of M consisting of those p olygons r r 0 P such that none of the ab ove diagonals have zero length Thus M contains r p all the emb edded p olygons Then the functions f i n i i 0 are smo oth on M The resulting ows are similar to those ab ove except they r have constant p erio ds We obtain a Hamiltonian action of an n torus T 0 by b ending as ab ove If we further restrict to the dense op en subset on M r 0 0 M M consisting of those p olygons so that and e are not collinear i i+1 r r 0 i n then we can intro duce actionangle co ordinates on M Note r that under the ab ove hyp othesis none of the triangles is degenerate We i let R Z b e the oriented dihedral angle b etween and In x i i i+1 we prove that 1 1 n3 1 n3 are actionangle variables In Section we relate our results on b ending to the twist deformations of Go JW and W Our main new contribution here is the discovery of an invariant nondegenerate symmetric bilinear form on the Lie algebra of the group of Euclidean motions Also in Prop osition we give a general formula for the symplectic structure on the space of relative deformations of a at Gbundle over an n times punctured sphere Here we assume that the Lie algebra G of G admits a nondegenerate Ginvariant symmetric bilinear form In x we show that the subgroup of the symplectic dieomorphisms of M generated by b endings on the diagonals of P acts transitively on M In r r Figure we show how to b end a square into a parallelogram It is a remarkable fact that most of the results and arguments of this 1 pap er generalize to the space of smo oth isometric maps from S with a xed 3 Riemannian metric to E mo dulo prop er Euclidean motions ie to regular gons These results will app ear in MZ and MZ After this pap er was submitted for publication we received the pap er Kl by A Klyachko Klyachko also discovered a Kahler structure on M and that r 2 n it is biholomorphically equivalent to the conguration space S P S L C however he didnt use the conformal center of mass construction and didnt give a pro of of this equivalence Otherwise the main emphasis of Kl is on construction of a celldecomp osition of M and calculation of cohomological r invariants of this space Kl do esnt contain geometric interpretation and transitivity of b ending ows actionangle co ordinates and connections with gauge theory Acknowledgements We would like to thank Mark Green for showing us how to prove Lemma b efore GN came to our attention We would also like to thank Janos Kollar Yi Hu Valentino Zo cca and David Rohrlich for helpful conversations Mo duli of p olygons and weighted quotients of conguration spaces of p oints on the sphere Our goal in this section is to give M the structure of a complex analytic r space and to construct a natural complex analytic equivalence from M to r 2 Q the weighted quotient of the conguration space of n p oints on S by sst 2 n PSL C constructed in DM x We dene a subspace M S by r n X 2 n r u g M fu S j j r j =1 Each p olygon P in the mo duli space M corresp onds up to translation to r the collection of vectors 3 n e e R fg 1 n 2 The normalized vectors u e r b elong to the sphere S The p olygon P j j j is dened up to a Euclidean isometry therefore the vector u u u 1 n is dened up to rotation around zero Since the p olygon P is closed we conclude that n X r u j j j =1 Thus there is a natural homeomorphism M M M S O r r r We will call the Gauss map We now review the denition of the weighted quotient Q of the cong sst 2 2 n uration space of n p oints on S following DM x Let M S b e the set of ntuples of distinct p oints Then Q M P S L C is a Hausdor complex manifold 2 n Denition A point u S is cal led r stable resp semistable if X r resp j u =v j 2 for al l v S The sets of stable and semistable points wil l be denoted by 2 n M and M respectively A semistable point u S is said to be a nice st sst semistable point if it is either stable or the orbit PSL C u is closed in M sst We denote the space of nice semistable p oints by M We have the nsst inclusions M M M st nsst sst Let M M M We obtain the p oints in M in the following cusp sst st cusp way Partition S f ng into two disjoint sets S S S with S 1 2 1 fi i g S f g in such a way that r r whence 1 k 2 1 nk i i 1 k r r Then u is in M if either u u or j j cusp i i 1 1 nk k u u The reader will verify that u M is a nice semistable j j cusp 1 nk p oint if and only if b oth sets of equations ab ove hold All p oints in M are cusp obtained in this way Clearly the nice semistable p oints corresp ond under 1 to the degenerate p olygons with S determined by the forwardtracks and 1 S by the backtracks On M we dene a relation R via
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