The Symplectic Geometry of 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

Action 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

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

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 into a

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

2 sst

u w mod R if either

a u w M and w PSL C u

st

or

b u w M and the partitions of S corresp onding to u w coincide

cusp

The reader will verify that if u w M then u w mod R if and

nsst

only if w PSL C u

It is clear that R is an equivalence relation Set

Q M R Q M R Q M R Q M R

sst sst nsst nsst st st cusp cusp

each with the quotient top ology The elements of Q are uniquely deter

cusp

mined by their partitions Thus Q is a nite set It is clear that each

cusp

equivalence class in Q contains a unique PSL C orbit of nice semi

cusp

stable p oints whence the inclusion

M M

nsst sst

induces an isomorphism

Q M P S L C Q

nsst nsst sst

In case r r are rational then the quotient space Q can b e given a struc

1 n sst

ture of an algebraic variety by the techniques of geometric invariant theory

2 n

applied to certain equivariant pro jective emb edding of S see DM x

This concludes our review of DM x We now establish the connection with

the mo duli space M

r

We recall several basic denitions from symplectic geometry Supp ose

that N is a simplyconnected Kahler manifold with symplectic form and

c

G is a complex reductive Lie group acting holomorphically on N Let G b e

c

a maximal compact subgroup in G We may assume that G acts symplecti

cally Then the Lie algebra G of G maps naturally into the space of vector

elds on N Each element X G denes a Hamiltonian f N R so that

X



df Y X Y for every Y T N There exists a map N G

X

such that hz X i f z The map is called the momentum map for

X

1

the action of G The space G is called the symplectic quotient of N

by G to b e denoted by N G

2

Let v ol b e the SO invariant volume form on S normalized by

Z

v ol

2

S

2 n

Fix a vector r r r with p ositive entries We give S the symplec

1 n

tic form

n

X



r p v ol

j

j

j =1

 2 n 2

where p S S is the pro jection on the j th factor The maximal

j

compact subgroup SO PSL C acts symplectically on

2 n

S We let

2 n 3

S R

b e the asso ciated momentum map

Here we have identied the Lie algebra so of SO with the space

3 3 3 

R where is the usual crosspro duct and R R via the Euclidean

3 3

structure on R The identication R so is given by u ad

u

3

where ad v u v u v R

u

Lemma

u r u r u

1 1 n n

compare Lemma

2 3

Proof First note that in case n the momentum map S R for the

symplectic structure v ol is given by

u u

But the momentum map of a diagonal action on a pro duct is the sum of the

individual momentum maps 2

2 n

equipp ed with Thus the space M is the symplectic quotient of S

r

the symplectic structure dened ab ove by SO which is the sub quotient

1

S O

We obtain the following

Theorem The Gauss map is a homeomorphism from the moduli space

3

M of ngons in R with xed side lengths to the weighted symplectic quotient

r

2 n

of S by SO acting diagonal ly

We now prove that M is a complex analytic space Let M b e the

r r

subset of degenerate p olygons is a nite collection of p oints Then M

r

is the symplectic quotient of a Kahler manifold and is consequently a Kahler

manifold MFW Ch x It remains to give M a complex structure in

r

the neighb orho o d of a degenerate p olygon

To this end let P b e a degenerate ngon which has p forwardtracks

and q backtracks The following lemma is a sp ecial case of AGJ

Corollary except for the connection with the numb er of backtracks and

forwardtracks To establish this connection and for the sake of clarity we

prove the following

Lemma There is a neighborhood of P in M homeomorphic to the sym

r

n2

plectic quotient U S O where U is a neighborhood of in C SO

n2

acts by symplectic isometries of the paral lel symplectic form of C and is

n2

the Hamiltonian ow for the Hamiltonian h C R given by the formula

q p

X X

2 2

jw j jz j hz z w w

i i 1 p 1 q

i=1 i=1

3

Proof Let f g b e the standard basis of R We may assume that P is

1 2 3

contained in the xaxis and that the last edge e of P is given by e r

n n n 1

2 n

We lift P M to u S with

r

u

1 1 2 1 n1 1 n 1

Here fg there are p plus ones and q of minus ones

k n

and

n1

X

r r

k k n

k =1

2 n

Our goal is to investigate the symplectic quotient of S by SO near

SO b e the subgroup of SO xing Thus H is the u We let H

1

isotropy subgroup of P We will often write SO instead H in what follows

We b egin by constructing a slice S through u for the action of SO on

2 n 2 n

S Dene S f s s s S s g Then S is a smo oth

1 n n 1

2 n

submanifold of S of dimension n It is immediate that S satises

the slice axioms

hS S h H

If g S S g SO then g H

2 n

The natural map SO S S given by g s g s is

H

a dieomorphism

2 n

We transfer the symplectic form from S to X SO S It is

H

3

then immediate that the induced momentum map X R is given by

n

X

s g s g

i

i=1

We dene S S S by

0 1

n

X

2 n

r s R g S f s s s S

i i 1 1 1 n

i=1

n

X

2 n

r s g S f s s s S

i i 0 1 n

i=1

1

We note that SO S and consequently the map induces a

0

SO (2)

2 n

homomorphism S S O S S O We are done if we can prove

0

that there is a neighb orho o d V of u in S such that V S O is isomorphic

0

n2

to a neighb orho o d of in the symplectic quotient C S O describ ed

ab ove

P

n

3

We let f S R b e the map given by f s r s We want

i i

i=1

1

to investigate f near u Let f f f b e the comp onents of f Let

1 2 3

2 1 2 2

g f f whence g S R and g S Since dg T S T R

2 3 1 u u 0

is onto there is a neighb orho o d V of u which we may assume is SO

invariant such that V S is a smo oth manifold of dimension n Then

1

V S f s V S f s g

0 1 1

We observe that f is the Hamiltonian function for the SO action

1 1

on S Clearly SO carries S into itself

1

Let denote the isotropy representation of SO on T V S We

u 1

note that preserves the almost complex structure J on T V S given

u 1

by J u and preserves the parallel symplectic form on

u

T S After shrinking V and applying Darb ouxs Theorem we may assume

1

u

that the Riemannian exp onential map exp T S S induces a

1 1

u u

SO equivariant symplectic isomorphism from a neighb orho o d U of in

T S to V S Thus exp induces an isomorphism from U S O

1 1

u u

onto V S S O We have accordingly reduced the problem to the linear

1

case

We have SO equivariant inclusions of symplectic vector spaces

2 n

T S T S T S

1

u u u

where

T S f g

i 1 n

u

n

X

r s R g T S f

i i 1 1 i 1 n

u

i=1

But since is orthogonal to the xaxis for all i for T S we have

i 1

u

P

n

r and

i i

i=1

n

X

r g T S f

i i 1 i 1 n

u

i=1

The innitesimal linear isotropy representation d is given by the linear vec

tor eld

F d

1 1 1 n

We rst compute the Hamiltonian h for F on the larger space T S We

u

claim that

n1

X

2

r k k h

i i i

i=1

Indeed

n1

X

r dh

i i i i

i=1

for T S and

1 n u

n1

X

r

i i 1 1 i i F ( )

i=1

n1

X

r

i i 1 i i 1

i=1

and the claim is established We note that h is a quadratic form of signature

p q recall that since e is a forwardtrack Also since SO

n n

0

preserves the complex structure J the quadratic form h satises hJ J

0

h ie h is a Hermitian form Now since SO carries T S into itself

u 1

F j is tangent to T S Hence the restriction of h to T S again

T (S ) u 1 u 1

u

1

denoted by h is the Hamiltonian for F j Thus we have only to compute

T (S )

u

1

the signature of this restriction Let W b e the orthogonal complement of

T S in T S for the quadratic form h It is immediate that W is spanned

u 1 u

by the two vectors

w

2 1 2 2 2 n1 2

and

w

3 1 3 2 3 n1 3

Indeed for k

n1 n1 n

X X X

hw r r r

k i i i k i i k i k i i

i=1 i=1 i=1

Thus T S T S W is a direct sum decomp osition which is orthogonal

u u 1

P

n1

r r k for h But hw w and hw w

i i n 2 3 k k

i=1

Hence hj is negative denite and hence hj is a Hermitian form of

W

T (S )

u

1

signature p q 2

We now give a complex structure to the symplectic quotient U S O

 p q

Let C act on C C by

1  p q

z w z w C z C w C

p q p q

Let C C denote the stable p oints and C C denote the nice

st nsst

semistable p oints Then

p q p q

C C fz w C C z and w g

st

p q p q

C C f g C C

nsst st

p q 

The Mumford quotient V of C C by C is by denition the ane variety

corresp onding to the ring of invariants



C

C z z w w

1 p 1 q

It is immediate that this ring is generated by the p olynomials f z w with

ij i j

relations generated by f f f f Thus V is a homogeneous quadratic

ij j i ii j j

pq

cone in C We observe that the top ological space V C underlying V is the

quotient space is the quotient

p q 

C C C

nsst

Note that we have an inclusion

n2

C S O V C

The following lemma gives the simplest example relating symplectic quotients

with Mumford quotients

Lemma The induced map is a dieomorphism

p q

Proof We will construct an inverse of Let z w C C Then

st

12 1 1

kz k kw k Let kz kkw k Then z w h since

1

kz k k w k 2

By transp ort of structure we obtain a complex analytic structure on

U S O This structure clearly agrees with the complex structure already

constructed on U S O fP g

We have proved the following

Theorem M is a complex analytic space It has isolated singularities

r

corresponding to the degenerate ngons in M These singularities are equiv

r

alent to homogeneous quadratic cones

We will now relate the symplectic quotient M to the space Q In

r sst

case the sidelengths r r are rational our theorem is a sp ecial case of

1 n

a fundamental theorem of Kirwan Kempf and Ness K KN N relating

symplectic quotients with quotients in the sense of Mumford of complex

pro jective varieties by complex reductive groups We note that

r u r u

1 1 n n

1

implies the semistability condition Therefore we have an inclusion

M and whence an induced map of quotients

nsst

1

M S O Q M P S L C

r sst nsst

Theorem The map is a complexanalytic equivalence

Proof In order to prove the theorem we will need some preliminary results

2

on the action of PSL C on measures on S

2

Denition A probability measure on S is cal led stable if the mass of

any atom is less than It is cal led semistable if the mass of any atom

is not greater than and nice semistable if it has exactly two atoms each

of the mass

The following is the basic example of semistable measure Take a vector

2

e M it denes a measure u r on S by the formula

r

n

X

r

j u

j

j =1

This measure has the total mass and is semistable

2 3

Let i S R b e the inclusion Then the center of mass B of a

2

measure on S is dened by the vectorintegral

Z

iud u B

2

S

We note that PSL C acts on measures by pushforward to b e denoted

2

by for PSL C and a measure on S



2

Lemma For each stable measure on S there exists PSL C

such that B The element is unique up to the postcomposition



g where g SO

3 3

Proof Denote by B the unit ball b ounded by S Douady and Earle in DE

3

dene the conformal center of mass C B for any stable probability

2

measure on S The assignment C has the following prop erties

a For any PSL C

C C



b B if and only if C

3

Note that in a the group PSL C acts on B as isometries of the

hyp erb olic space The lemma follows from the transitivity of this action

2

We can now prove that is an isomorphism of complex analytic spaces

By the previous lemma carries the nonsingular p oints of M continuously

r

to Q Also carries degenerate ngons to nice semistable p oints Thus

st

is a continuous bijection and consequently is a homeomorphism It is

easy to check that the complexlinear derivative d is invertible at all

non singular p oints Moreover it follows from the analysis of DM x

that is a complex analytic equivalence near the singular p oints of M

r

The theorem follows 2

We obtain the following

Corollary M has a natural complex hyperbolic cone structure see

r

Th for denitions

Proof It is proven in Th that M P S L C has a complex hyp erb olic

nsst

cone structure 2

Bending ows and p olygons

n3

In this section we will show that M admits actions of R by b ending along

r

1

n nonintersecting diagonals note that n dimM The orbits

r

2

0

are p erio dic and there is a dense op en subset M including the emb edded n

r

gons such that the action can b e renormalized to give a Hamiltonian action

1 n3 0

Thus M is almost a toric variety of the n torus S on M

r

r

In this paragraph it will b e more natural to work with the pro duct

n n

Y X

2

M f e S r e g

r j j

j =1 j =1

whence M M S O We need to determine the normalizations of the

r r

symplectic forms on factors of M in order that the zero level set of the

r

3

asso ciated momentum map M R is the set of closed ngons In what

r

Q

n

2

S r follows we will use S to b e the pro duct

j r

j =1

2

Let b e the form on S r given by

2 2

u v x u v r v ol u v x S r u v T S r

x x x

2

Here v ol denotes the Riemannian volume form on S r We dene the

symplectic form on M by

r

n n

X X

 

p v ol p

j j

2

r r

j

j

j =1 j =1

3

Lemma The momentum map S R for the diagonal action of

r

SO on S is given by

r

e e e

1 n

3

Proof It suces to treat the case n We replace r by r Let w R

1

2

so Then the induced vector eld w on S r is given by

w x w x

Let h b e the asso ciated Hamiltonian It suces to prove that h x w x

w w

2

To this end let v T S r Then dh v w v and

x w

v x w x v x w v x w v

w^ (x) x

2 2

r r

2

Remark The equation e is the closing condition for the ngons

3

in R with edges e e Thus the above normalization for is the correct

1 n

3

one However the map w w from R to the Lie algebra of vector leds

3

on R is an antihomomorphism of Lie algebras

We observe that M has an SO invariant Kahler structure The fol

r

lowing are the formulas for the Riemannian metric symplectic form

and almost complex structure J for u v T M

r

e

P

n

1

a u v u v

j j

j =1

r

j

P

e

n j

b u v u v

2

j j

j =1

r

j

e e

n

1

u u c J u

n 1

r r

n

1

Remark The normalization for is chosen in order that e wil l

be the closing condition for n gons The normalization for J is determined

2

by J I Consequently the normalization for is determined as

wel l

Now let P M and cho ose e M corresp onding to a closed ngon P

r r

in the congruence class P We may identify T M with the orthogonal

[P ] r

complement

hor

T M

r

e

of the tangent space to the orbit of SO passing through e The sub

hor 3 n

space T M consists of vectors R which satisfy the

r 1 n

e

following equations

i e

j j

P

n

ii

j

j =1

P

1

n

r e iii

j j

j

j =1

The rst equation corresp onds to the xed side lengths of our p olygon

the second is the innitesimal closing condition for the p olygon P The

last equation is the horizontality condition due to the following

Remark The equation iii is equivalent to the condition

P

1

n

3

iv r v e for al l v R

j j

j

j =1

3

We note that the vectors v e v e v R are the tangents to the

1 n

SO orbit through e in M

r

We obtain formulas for the pullback Riemannian metric symplectic

hor

form and almost complex structure J on T M by restricting formulas

r

e

a b c ab ove Note that formula iv ab ove may b e rewritten as

P

n

J v

j

j =1

hor

indicating that T M is J invariant

r

e

We now study certain Hamiltonian ows on M We will identify an

r

Q

n

2 n 2

SO invariant function on S or S r with the function it induces

i

i=1

on M without further comment We dene functions f f on S by

r 1 n3 r

2

f e e ke e k k n

k 1 n 1 k +1

Thus f corresp onds to the length squared of the k th diagonal of P drawn

k

from v to v Our goal is to nd an interpretation of the Hamiltonian

1 k +2

ow corresp onding to f in terms of the geometry of P

k

Lemma The Hamiltonian eld H associated to f is given by

f k

k

H e e e e

f 1 n k 1 k k +1

k

where e e is the k th diagonal of P

k 1 k +1

Proof Let e e e Since f do es not dep end on the last n k

1 n k

comp onents it suces to prove the lemma for the map

n 2

f S r R

n1 i

i=1

given by

2 2

f e ke e k kk

n1 1 n

where is the momentum map for the diagonal action of SO on

Q

n

2

S r By the equivariance of see the pro of of Lemma in K

i

i=1

the Hamiltonian eld H at e satises

f

n1

H e e

f

n1

Q

n

2

where v is the vector eld on S r corresp onding to v so But

i

i=1

v e v e and the lemma follows 2

Prop osition

ff f g

k l

for al l k l

Proof We may assume k l Then

k +1

X

e

i

e e ff f g H H

k i l i k l f f

k l

2

r

i

i=1

k +1

X

e

i k l k k l

i=1

2

t

We now study the Hamiltonian ow asso ciated to f Thus we must

k

k

solve the system of ordinary dierential equations

de

i

e i k

k i

dt

de

i

k i n

dt

3

We will use the following notation Recall that we have identied R

3

with the Lie algebra of SO and if u v R then we have

ad v u v

u

Accordingly we dene an element exp ad SO as the sum of the p ower

u

series

1

n

X

ad

u

exp ad

u

n

n=0

The following lemma is elementary and is left to the reader

3

Lemma Let be the oriented plane in R which is orthogonal to u

Then exp ad is the rotation in through an angle of kuk radians In

u

particular the curve exp tad has period kuk and angular velocity kuk

u

We can now solve the system

t

Prop osition Suppose P M has edges e e Then P t P

r 1 n

k

has edges e t e t given by

1 n

e t exp tad e i k

i i

k

e t e k i n

i i

Proof We will ignore the last n k edges since they are constants of

motion We make the change of unknown functions

e e e e e i k

1 1 k +1 k i i

It is immediate that e e satisfy the new system of equations

1 k +1

de

1

dt

de

i

e e i k

1 i

dt

Since e we nd that is invariant under the ow and by Lemma

1 k k

e t exp tad e i n

i i

k

It remains to nd e t Note that exp tad thus

1 k k

k

k +1 k +1

X X

exp tad e e t exp tad e t t

i i k 1 k

k k

i=2 i=2

exp tad e

1

k

2

t

Corollary The curve P is periodic with period where

k

k

ke e k

k 1 k +1

is the length of the k th diagonal of P

k

Remark If the k th diagonal has zero length thus v v then P

1 k +1

t

is a xed point of In this case the ow has innite period

k

t

P is the b ending ow describ ed in the intro duction It We see that

k

rotates one part of P around the k th diagonal with angular velo city equal

to the length of the k th diagonal and leaves the other part xed

0

We next let M M b e the subset of M consisting of those P for which

r r

r

0

no diagonal has zero length Then M is Zariski op en in M The functions

i r

r

0 2

and they Poisson commute Since f are smo oth on M

k 1 n3

r k

we have

df

k

d

k

k

and consequently

H H

f k

k k

Since is an invariant of motion the solution pro cedure in Prop osition

k

t

works for H as well Let b e the ow of H We obtain the following

k

k k

0 t

Prop osition Suppose P M has edges e e Then P t P

1 n

r k

has edges e t e t given by

1 n

e t exp tad e i k

i k i

k

e t e k i n

i i

t

Thus rotates a part of P around the k th diagonal with constant

k

t

angular velo city Hence P has p erio d and we have proved the

k

following

0

Theorem The space M of ngons such that no diagonal drawn from

r

the st vertex has zero length admits a free Hamiltonian action by a torus

1

0

dimM T of dimension n

r

2

Remark If n then M is a toric variety for generic r by Del

r

For n it suces to use the above choice of diagonals For n we

have to make dierent choice of diagonals v v v v v v Then if

1 3 3 5 5 1

0

r r for al l i j we conclude that M M Unfortunately for

j i r

r

0

any choice of nonintersecting diagonals leads to M M even for generic

r

r

values of r

t

Remark In what fol lows we wil l also denote by the normalized

d

bending in the diagonal d of the P

Actionangle co ordinates

In this section we use the geometry of P to intro duce global actionangle

0

co ordinates on the space M which was dened in the Intro duction

r

In x we will use the emb edding in M of the mo duli space N of planar

r r

2

p olygons with xed side lengths mo dulo the full group of isometries of E

This emb edding is constructed as follows Let b e a xed Euclidean plane

3 3

in E and b e the involution of E with as xedp oint set Then acts on

M We claim that the xedp oint set of on M consists of the p olygons

r r

3

that lie in up to isometry Indeed let P b e a ngon in E which is xed

by up to a prop er isometry Hence there exists a prop er Euclidean motion

3

g such that P g P But if the vertices of P span E we have g

a contradiction Hence P lies in a plane and can b e moved into by an

0

isometry The claim follows Let P b e a ngon in M and P b e a convex

r

2 0

ngon in R The diagonals d v v d v v of P are called disjoint

i j k s

0

or nonintersecting if the corresp onding diagonals of P do not intersect

0

in the interior of P

Fix a maximal collection of disjoint diagonals d d of P

1 n3

Lemma There exists a bending b of P in diagonals d d such that

1 n3

bP is a planar polygon

Proof The assertion is obvious for quadrilaterals The general case follows

by induction 2

Corollary The space M is connected

r

Proof The space N is connected by KM 2

r

0

Pick a p olygon P M The diagonals k n divide P

k

r

into n nondegerate triangles such that is a common

1 n2 k +1

side of and We orient in the direction v v Let b e the

k k +1 k k 1 k

element of R Z given by the dihedral angle measured from to

k k +1

k n see Intro duction So exp i rotates the plane of in the

k k

p ositive direction around into the plane of Recall that

k k +1 k k

Lemma

f g

i j ij

Proof From our description of the b ending ows we have

t

P P t mo d Z

i i ij

j

We obtain the lemma by dierentiating 2

Corollary

H H

i j

In order to prove that

f g

1 n3 1 n3

are actionangle co ordinates it suces to prove the following

Lemma

f g

i j

Proof Recall that N is the subspace of planar p olygons with xed side

r

lengths mo dulo the full group of planar Euclidean motions We have seen

that N is the xed submanifold of M under the involution We note that

r r



i n

i i

 

Hence d d and since we have

i i

H H i n



i i

Hence if P is a planar p olygon we have

H P T N i n

P r

i

Since N is Lagrangian we have for P N

r r

H P H P i j n

P

i j

0

Now let P b e a general element of M There exists b T such that bP N

r

r

see Lemma Since the H and H commute by Corollary the

b

i j

Hamiltonian elds H i n are invariant under b ending and

i

consequently

H bP dbH P i n

i i

Since is invarant under b we have

H P H P dbH P dbH P

P bP

i j i j

The lemma follows 2

We have proved the following

Theorem

f g

1 n3 1 n3

0

are actionangle coordinates on M

r

The connection with gauge theory and the

results of Goldman and JereyWeitsman

In this section we rst review the description of M given in KM in terms

r

of relative deformations of at principal E bundles over the n times

punctured sphere here E denotes the group of orientation preserving

3

isometries of R We then show that the Lie algebra e of E admits an

invariant non degenerate symmetric bilinear form b not the Killing form of

3

course This form is closely related to the scalar triple pro duct in R We use

the form b together with wedge pro duct to give a gaugetheoretic description

of the symplectic structure on M This description is the analogue of the

r

usual one in the semisimple case the form b replaces the Killing form It is

then clear how our results on b ending are analogues for E and relative

deformations of those of Go JW and W

We b egin by briey reviewing our pap er KM on relative deformation

theory It is more convenient to use relative deformations of representations

here for the details of the corresp ondence with at connections see KM

Let b e a nitelygenerated group R f g a collection of sub

1 r

groups of G b e the set of real p oints of an algebraic group dened over

R and G a representation In KM we intro duce the relative

0

representation variety H om R G Real p oints of this variety consist of

representations G such that j is a representation in the closure

j

of the conjugacy class of j For any linkage with n vertices in the Eu

0

j

m

clidean space E we constructed an isomorphism of ane algebraic varieties

C H om R E m

n

Here C is the conguration space of the linkage we do not divide out

by the action of E m The group is the free pro duct of n copies of

n

Z R is a collection of dihedral subgroups Z Z of determined

n

by the edges of the linkage and E m is the full group of isometries of the

Euclidean space

3

We assume henceforth that the linkage is an ngon in E with side

length r r r and as ab ove M denotes the mo duli space M

1 n r

C E We have an induced isomorphism

H om R E mE m M

n

for any linkage

2

Let S fp p g denote the sphere punctured at fp p g let

1 n 1 n

U U denote disjoint disc neighb orho o ds of p p and U U U

1 n 1 n 1 n

The subgroup consisting of words of even length in generators

n n

is isomorphic to see KM Lemma Indeed put

1 n 1 i

i n Then Let H om R E m Then

i i+1 1 n n

induces a representation E and a at principal E bundle P

n

over We let adP b e the asso ciated at Lie algebra bundle In our case

we can use the restriction map to replace the ab ove relative representation

variety with one that makes the connection with M transparent Let T b e

r

the set of conjugacy classes in given by

n

T fC C g

1 n

Here C denotes the conjugacy class of Then it is immediate that we

have an induced isomorphism

H om T E E M

n r

Indeed each ngon corresp onds to the n translations in the direction of

its edges e e The relation corresp onds to the closing

1 n 1 n

condition

e e

1 n

Note that r is the translational length of We will henceforth abbre

i 0 i

viate H om T E E to X

n nr

Remark If is an automorphism of that preserves each class

n

C C

1 n

then acts trivial ly on H om T E Indeed since is contained

n n

in the translation subgroup of E xing the conjugacy class of amounts

to xing Thus quantizing M wil l produce only trivial representations

r

of the pure braid group

Let X We dene the parab olic cohomology

nr

1

H adP

par

1

to b e the subspace of the de Rham cohomology classes in H adP whose

restrictions to each U are trivial By KM we may calculate the relative

i

1

deformations of and consequently H adP by using the dierential

par



graded Lie algebra B U adP This algebra is the subalgebra of the de

Rham algebra consisting of sections of adP which are constant on U in degree

zero and adP valued forms which vanish on U in degrees and

We now give our gaugetheoretic description of the symplectic form on

M Since the Lie algebra e of E is not semisimple the Killing form

r

of e is degenerate and we can not give the usual ie for G semi simple

description of the symplectic form However it is a remarkable fact that there

is another E invariant symmetric form b on e which we now describ e

2 3

We recall that we may identify R with so by asso ciating to u v the

3

element of E ndR also denoted by u v given by

u v w u w v v w u

We dene a bilinear form

3

a so R R

by

au v w u v w

3

We split e according to e so R and dene the split form

b e e R by

bu v w u v w au v w au v w

1 1 1 2 2 2 1 1 2 2 2 1

The following prop osition is immediate

Prop osition The form b is symmetric nondegenerate and invariant

under E

We combine the form b on e with the wedgepro duct to obtain a skew

symmetric bilinear form

1 1 2

B H adP H adP H U R

par par

we then evaluate on the relative fundamental class of M to obtain a skew

symmetric form

1 1

A H adP H adP R

par par

It follows from Poincare duality that A is nondegenerate and we obtain a

symplectic structure on X In order to relate A to the symplectic form

nr

of Section we need to make explicit the induced isomorphism

d T X T M

p nr r

To do this we need to pass through the group cohomology description of

1

H adP In the following discusssion we let G b e any Lie group We

par

denote by G the Lie algebra of G Recall that we may identify the universal

2

cover of with the hyp erb olic plane H we will make this explicit later

 

Let p b e the covering pro jection We will identify A p adP with

the G valued dierential forms on via parallel translation from a p oint v

1

1 

Given H adP cho ose a representing closed form A adP



Let p and f G b e the unique function satisfying

f v

1

df

1

We dene a co chain h Z G with co ecients in G by

1

h f x Ad f x

We note that the righthand side do esnt dep end on x We dene to b e

1

the class of h in H G It is easily checked see GM x that is an

1

isomorphism We note that H M adP if and only if the restriction

par

of h to the cyclic groups generated by are exact for all i We denote

i

1 1

the set of all co cycles in Z G satisfying this prop erty by Z G

par

We now return to the case G E Let H om T E Since

n

is a translation for all it follows that if

0 n

c T H om T E

n

then c is an innitesimal translation for all and consequently we

3

may identify c with an element of H om R The condition that c is a

n

parab olic co cycle is equivalent to c e where is a translation by

i i i

e We leave the pro of of the next lemma to the reader

i

Lemma

d c T M

1 n r

e

where c i n

i i

Remark Here we think of T M as the quotient of T M by SO

r r

e e

see x for the denitions

We can now state the main result of the next section Recall that the

symplectic structure on M was describ ed in x

r

Theorem With the above identication we have A

Proof We will work in the more general framework where E is replaced

by a Lie group G admitting an invariant symmetric bilinear form on its Lie

3

algebra G We have in mind an eventual application to ngon linkages in S

2

We construct a fundamental domain D for op erating in H as follows

n

Cho ose a p oint x on and make cuts along geo desics from x to the cusps

0 0

The resulting fundamental domain D is a geo desic ngon with n interior

1 1

vertices v v and n cusps v v These o ccur alternately so that

1 n

1 n

1 1 1

pro ceeding counterclo ckwise around D we see v v v v v v The

1 2 n

1 2 n

1

generator xes v and satises v v We take v as our base p oint

i i i i+1 1

i

2

x in H

0

Remark We have changed our original generators of to their in

n

verses

0 1

Now let H om T GG and c c H G b e tangent vectors

n n

par

0

at Let and b e the corresp onding elements of the de Rham cohomology

0 1

G i n adP By assumption there exist vectors w w group H

i

i par

such that

w Ad w c

i i i i

0 0 0

w Ad w c

i i

i i

We let B b e the bar resolution of MacL Thus B is the free

 k

Zmo dule on the symb ols j jj with

1 2 k

j jj jj

1 2 k 1 2 k

k 1

X

k

jj jj j jj

1 i i+1 k 1 2 k 1

i=1

We let C B Z where Z acts on Z via the homomorphism

k k Z[]

dened by

m m

X X

a a

i i i

i=1 i=1

Thus C is the free ab elian group on the symb ols

k

jj j jj and

1 k 1 2 k

jj jj

1 k 2 k

k 1

X

k

jj jj j jj

1 i i+1 k 1 2 k 1

i=1

We dene a relative fundamental class F C by the prop erty

2

n

X

F

i

i=1

Let C b e the chain

2

n

X

j

i i1 i2 1

i=2

The reader will easily verify the following lemma which was p ointed out to

us by Valentino Zo cca

Lemma is a relative fundamental class

We abuse notations and use B to denote the ab ove wedge pro duct of

the de Rham cohomology classes and the cuppro duct of Eilenb ergMacLane

co chains using the form b on the co ecients

Prop osition

Z

n

X

0 0 0

hB c w i B hB c c i

i

i



i=1

Proof The reader will verify that the righthand side of this formula do esnt

0

dep end on the choices of w i n and of a relative fundamental class

i

1

In the following we let e b e the oriented edge of D joining v to v

i i

i

1

and e b e the oriented edge joining v to v Then e e We remind

i i+1 i i i

i

0

the reader that the forms and vanish in neighb orho o ds of the cusps

1

v i n Prop osition will b e a consequence of the following three

i

lemmas

Lemma

Z Z

0 0 1 0

B f bc f v bc v B f

i i+1 i

i

e e

i i

Proof

Z Z Z Z

0 0 0 0

B f B f B f B f

1

e e e e

i i i i

i

Z

0 1  0

B f B f

i

e

i

Z

0 1  1  0

B f B f

i i

e

i

Z

0 1  1  0

B f B Ad f Ad

i i

i i

e

i

Z

1  0

B f Ad f

i

i

e

i

Z

0 0 0 1

B c B c f v B c f v

i i i+1 i

i

e

i

2

We obtain

Z

n n

X X

0 1 0 0

bc f v bc f v B

i i i+1

i



i=1 i=1

To evaluate the second sum we need

Lemma

0 1 0

f v c i n

i i

i

were is the EilenbergMacLane coboundary

2

Proof By denition for any x H we have

0 0 0 1

x c f x Ad f

i i

i

0 1

Since f is a covariant constant near the cusps we may allow x to tend to v

i

1

1 1

v v we obtain in the ab ove formula Since

i

i i

0 1 0 0 1

Ad f v c f v

i i

i i

2

We now evaluate that sum over the interior vertices

Lemma

n

X

0 0

bc f v hB c c i

i i+1

i=1

n n

X X

0 0 1

hB c w i bc f v

i i

i i

i=1 i=1

2

Proof By denition for any x H we have

0 0 0 1

c f x Ad f x

1

Substituting x v and using v v we obtain

i+1 i i+1 i

i

0 0 0

c f v Ad f v

i i+1 i i

0

Using f v we conclude that

1

0

f v c

i i1 i2 1

Hence

n n

X X

0 0

bc c bc f v

i i i1 1 i i+1

i=1 i=1

n n

X X

0 0

bc Ad c bc c

i i i1 1 i i

i=1 i=1

n

X

0 0

bc c hB c c i

i i

i=1

0 0 1 0 1

We substitute c f v Ad f v and use the formula

i i

i i

0 0

hB c w i bc Ad w

i i i

i i

to obtain the lemma 2

We have proved Prop osition and now sp ecialize to the case at hand

0

namely G E In this case c and c take values in the Lie subalgebra of

innitesimal translations Since this is a totallyisotropic subspace for b we

obtain

0

hB c c i

It remains to evaluate the sum over the cusps

Lemma

e

i

0 0

hB c w i

i i

i i

2

r

i

0 0

where c c i n

i i i

i

Proof We rst note that

0 0 1 0 0

bc w c w bAd i bc Ad w hB c w

i i i i i

i i i i i

The last equality holds b ecause is a translation and c is an innites

i i

imal translation A direct computation in the Lie algebra e shows that

e e

i i

0 0 0

Ad

i

i i i

2 2

r r

i i

e

0 0

i

i n and the lemma follows from Hence we may cho ose w

2

i i

r

i

the denition of b 2

We have accordingly

Z

n

X

e

i

0 0 0

A B

i

i

2

r



i

i=1

Comparing with our formula for the symplectic structure in x we obtain

Theorem 2

We can now relate our results on the b ending of ngon linkages with the

work of Goldman JereyWeitsman and Weitsman There are two inde

p endent class functions on E translation length and the trace of the

2

rotation part t We replace by f to get a p olynomial invariant

Given we dene

n

f H om T E E R

n

by f f In the case i n it is easily seen

i 1 2 i

that the Hamiltonian ow of f corresp onds to the unnormalized b ending

ow in the ith diagonal The decomp osition of the p olygon P by diagonals

drawn from a common vertex corresp onds to a decomp osition of into pairs

of pants using the curves Thus our real p olarization of M ie

2 n2 r

singular Lagrangian foliation obtained by b ending in the ab ove diagonals

corresp onds to that of JW and W obtained from twists with resp ect to

2 n2

Transitivity of b ending deformations

Denition An embedded polygon P M is cal led a pseudotriangle

r

3

if the union of edges of P is a in R The vertices of this triangle

are cal led the pseudovertices of the pseudotriangle P

It is easy to see that M contains only a nite numb er of pseudo

r

3

triangles T T where N n The main result of this paragraph is

1 N

the following

Theorem a For each nonsingular moduli space M there exists a num

r

ber r such that each polygon P M can be deformed to a pseudotri

r

angle via not more than bendings

b The function r is bounded on compacts in D

n

c The subset of polygons P M which can be deformed to a pseudo

r

triangle T by at most bendings is closed in M

r

d Each pair of polygons P Q M can be deformed to each other by a

r

3

sequence of not more than n bendings

Bending of quadrilaterals

We rst consider a sp ecial case of b ending assuming that

a P is a planar with a nonsingular mo duli space M

r

b we allow sequences of b endings in b oth diagonals of P

c b ending are always so the b ending of a planar p olygon is

again a planar p olygon Such a b ending will b e called a b ending

We have an action of the group Z Z on the mo duli space N of planar

2 2 r

quadrilaterals given by b endings along the diagonals

Denote the b ending in the diagonal v v by and the b ending in

1 3

the diagonal v v by We assume that xes the vertex v and xes

2 4 2

the vertex v We shall normalize gons P so that e r u u C

1 1 1 1 1

Then

r r u r r u

1 2 2 1 2 2

1 1

u u u u u u u u

1 2 3 4 1 2

3 4

1 1

r r u r r u

1 2 1 2

2 2

r r u r r u

1 2 2 1 2 2

1 1

u u u u u u u u

1 2 3 4 1 4

2 3

1 1

r r u r r u

1 2 1 2

2 2

4

Both maps are birational transformations of C The mo duli space N

r

1

S is the quotient of the curve

4

X

4

r u g E fu u u u C u ju j j

j j 1 2 3 4 1 j

j =1

by action of the involution u u u u u u u u

1 2 3 4 1 2 3 4

c

Lemma The complexication E of the curve E is a nonsingular con

nected el liptic curve The composition extends to an automorphism

c c

of E which is xedpoint free

Proof The rst statement of Lemma was proven in G GN The biholo

c c

morphic extension exists since E is Zariski dense in E The transfor

mation E E has no xed p oints It follows from the classication of

c

automorphisms of elliptic curves that is also xedp oint free 2

In particular the selfmap of E preserves the metric on E given by the

c

restriction of a at metric on E Therefore if we identify E h i with the

1 1

unit circle then S S is a rotation Denote by a b the parameter

s t

families of b endings in the diagonals v v v v so that a b

1 3 2 4

1

Lemma An arc x x between x x on S is contained in the

orbit b a x where s t R Z

t(s) s

Proof For each p oint a x we take b to b e one of two b endings which

s t(s)

makes a x planar We cho ose b so that it dep ends continuously on s

s

t(s)

and b b id b b It is clear that for s the p olygon

t(0) 0 t( )

b a x is equal to x This proves the Lemma 2

s

t(s)

Corollary Suppose that r doesnt belong to a face of the polyhedron D

4

Let be the rotation angle of the element m

Then for each two points x y in the space of quadrilaterals M there exists a

r

composition of at most m bendings which transforms x to y

Proof Our assumptions imply that the mo duli space M is not a single p oint

r

Thus the angle is dierent from zero We rst apply a single b ending to each

0 0

x y to make them planar p olygons x y There exists a comp osition of at

0 00 0 0

most m b endings which sends x to a p oint x on the arc y y

00 0

Then we apply Lemma to transform x to y 2

See Figure for the deformation of a square to a parallelogramm via two

b endings

Remark The rotation angle depends continuously on the parameter

r The angle can be arbitrary close to zero as r approaches the wal ls or

the boundary of the polyhedron D This corresponds to a degeneration of the

4

el liptic curve In the limit the birational transformation wil l have isolated

xed points singular points of the curve E

Deformations of ngons

Lemma Suppose that Q M is a nondegenerate ngon Then there

r

exists a diagonal d of Q such that the bending in d changes the distance

between at least two vertices A B

Proof Supp ose that Q M is a ngon and M is a nonsingular mo duli

r r

space Our problem is to deform Q via b ending so that the distance b etween

two distinct vertices A v B v of the p olygon Q is changing assuming

1 s

that js j If the distance jAB j do esnt change under b ending in a

diagonal v v then either A or B b elong to the line v v

k s+i k s+i

Supp ose that the distance jAB j do esnt change for any choice of s k

ab ove Then either A or B b elong to the line v v for all k s

k s+i

Figure

i n s This means that either all the vertices of Q except A b elong to a

single line v B through B or all the vertices of Q except of B b elong to

2

a single line v A The p olygon Q must b e a pseudotriangle Supp ose

2

that the former case takes place Then instead of vertices A B we cho ose

say v v and applying b ending along the diagonal AB we can change the

2 n

distance jv v j 2

2 n

1

Supp ose that f S R is a continuous function Then we dene

1

f jmax f min f j v ar

t t

t2S

1

to b e the variation of the function f on the circle S

t

We recall that denotes the normalized b ending in the diagonal d of a

d

p olygon P M jdj denotes the length of this diagonal

r

Denition Suppose that P P and P We dene the fol lowing

n

function

t t

1

j A B j A B are vertices of P P maxfjdjv ar

t2S

d d

d is a diagonal of P g

A pair of diagonals A B d providing this maximum wil l be cal led a max

imal pair

It is clear that the function is continuous Thus by Lemma for any

compact K D there is a numb er such that P for

n K K

1

each P K

Pro of of Theorem

We have already proved Theorem for quadrilaterals thus we can assume

that the numb er n of vertices is at least Arguing by induction we can

assume that Theorem is valid for all spaces of k gons where k n

Step Take a ngon P M The space M is nonsingular thus

r r

P where dep ends only on r Cho ose a pair of vertices A B and

diagonal d of P which maximize the function P The maximal variation

t t

1

j A B j is at least since the length of the diagonal d is at v ar

t2S

d d

0 00

most Split P along the diagonal A B in two p olygons P P treating

A B as a new side with xed length There is at most n of bad values

0 00

00 0

of P P are singular We M of jAB j such that the mo duli spaces M

r r

also include zero in the list of bad values Denote the numb er of vertices

0 0 00 00

of P by n and the numb er of vertices of P b e n

Then we use a b ending in the diagonal d to deform P to a p olygon P



t t

so that the distance from j A B j to each of these bad values is

d d

maximal which is at least n

The pro of of the following prop osition is obvious and is left to the reader

Prop osition Let C D be a compact Then there are two com

n

1 00 0

00 0

such that for any P C we have C D pacts C D

n n

0 0 00 00

P C P C

 

0 00

Thus the p olygons P P satisfy the prop erty that the function on their

 

1 0 1 0

00 0

P is b ounded from b elow P M mo duli spaces M

r r

by some p ositive numb er

1

1 2 2 1

Dene a relation R in P f ng P as the

n n



set of tuples P i j k s P where the diagonals A B v v d

i j

v v form a maximal pair and the p olygon P is obtained from P via

k s 

b ending in the diagonal d as ab ove

Prop osition The relation R is closed

1

Proof It is enough to prove that if lim P P in the space P

s!1 s n



R and suciently large s the pairs then for any P A B d P

s s s s

s

A B d are maximal for the limiting p olygon P as well

s s s

Pick a subsequence with constant A B d By continuity of the

s s s

k k k

function it is enough to check that the jd P j for the limiting p oly

s

k

gon P The nonvanishing of jd P j follows from the inequalities

s

k

P jd P j 2

s s s

k k k

Step

Lemma Let r D Then for each i the moduli space M contains

k r

a pseudotriangle T v v such that v is a pseudovertex of T

i 1 k i i

0

Proof Assume that the assertion is valid for all k k By renumeration

of vertices it is enough to construct the pseudotriangle Q Since the p erime

1

ter of the p olygon is normalized to b e equal to there is a pair r r

j j +1

j Z dierent from r r such that r r Hence we can apply

k k 1 j j +1

the same induction argument as Lemma in KM to nd a p olygon in M

r

where e e forms an edge As the result of this pro cedure we construct

j j +1

the required pseudotriangle 2

Remark The case when T T happens exactly when there is a num

1 2

ber s k such that r r and r r r

1 s s+1 k 1

Recall that k n and by the induction hyp othesis we have a function

r

2

Dene the relation on P to b e the set of pairs P T where

 k

i the pseudotriangle T b elongs to the mo duli space of P and ii P can b e

deformed to T via at most r b endings

Prop osition The relation is closed in P P and its pro

T k k

jection to the rst factor is onto

Proof Since we consider only parameters r the equality r r in

j j +1

the pro of of Lemma is imp ossible Thus the relation on P P

k k

given by the condition i is closed The statement of the Prop osition follows

from the induction hyp othesis in Theorem 2

This lemma together with Lemma implies that arguing by induction

0 00 0 00

we can deform via b ending each of p olygons P P to pseudotriangles T T

 

00

keeping the length jAB j xed so that A is a pseudovertex of T and B is a

0

pseudovertex of T The numb er of b endings which we have to use here is

0 00

b ounded from ab ove by a function which dep ends on r r only In the case

0 00

when b oth A B are pseudovertices of T and T these pseudotriangles form

a quadrilateral and we can go directly to the Step see Figure Assume

00 0

that B is not a pseudovertex of T and A is a not a pseudo vertex of T see

0 00

Figure The triangles T T form a split it along the diagonal

0  0 00 0

d X Y into two quadrilaterals S S where d is a side of xed length

0 00

Then the triangle inequalities imply that b oth S S can b e deformed to

0 00 

pseudotriangles L L where X Y are pseudovertices This again gives us

a quadrilateral Thus we can go to the Step

0

In the remaining case when A is not a pseudovertex of T and B is a

0 00 0 00

pseudovertex in b oth T T we split the p olygon formed by T T along the



diagonal X B see Figure and deform the quadrilateral X B X A to



a pseudotriangle with the vertices X B X keeping the triangle X Y B

xed Then we again can go to the Step

Step As the result of Step we deform the p olygon P to a p olygon Q

0 00

which is the union of two pseudotriangles minus the diagonal A B

Remark As before the relation which consists of the pairs P Q

above is closed in P P

n n

The mo duli space of the quadrilateral Q is nonsingular since r We

again apply the induction to deform Q so that it b ecomes a pseudotriangle

T It follows from the induction hyp othesis that the relation which consists

of pairs Q T as ab ove is closed in P P

4 4

Thus we have proved the assertion a of Theorem for ngons Namely

for each p olygon P we have constructed a piecewisesmo oth bending curve

P M which connects the p olygon P with a pseudotriangle T Each

r

smo oth arc of this curve is given by b ending in one of diagonals However

the curve is not necessarily unique The fact that the function r the

numb er of b endings is b ounded on compacts follows from the induction

hyp othesis via Prop osition This implies the assertion b of Theorem

For a pseudotriangle T M denote by Y T the subset in M consist

r r

ing of those p olygons P such that at least one of the b ending curves P

terminates at T Thus the relation fP T P Y T g is the comp osition

of closed relations R This implies that each Y T is closed and the

assertion c follows

It remains to prove the assertion d The closed subsets Y T can in

tersect We say that two pseudo triangles T T are equivalent if Y T

1 2 1

Y T This generates an equivalence relation on the nite set of pseudo

2

triangles in M The space M is connected and all Y T are closed sets thus

r r

all pseudotriangles are mutually equivalent Hence any p olygon P can b e

deformed to a pseudotriangle via at most r b endings by the assertion

a and any two pseudotriangles can b e deformed to each other via at most

3

n r b endings This nishes the pro of 2

References

AGJ J Arms M Gotay G Jennings Geometric and algebraic reduction

for singular momentum maps Advances in Math

Del T Delzant Hamiltoniens periodiques et images convexes de

lapplication moment Bull So c Math France N

DE A Douady C Earle Conformal ly natural extension of homeomor

phisms of the circle Acta Math

DM P Deligne G Mostow Monodromy of hypergeometric functions and

nonlattice integral monodromy Publications of IHES Vol

p

G ML Green private communication

GN CG Gibson PE Newstead On the geometry of the planar bar

mechanism Acta Applicandae Mathematicae Vol

Go WM Goldman Invariant functions on Lie groups and Hamiltonian

ows of surface group representations Invent Math p

GM W Goldman JJ Millson The deformation theory of representa

tions of fundamental groups of compact Kahler manifolds Publ

Math of IHES vol

JW L Jerey J Weitsman Torus actions moment maps and the moduli

space of at connections on a twomanifold Preprint

KM M Kap ovich JJ Millson On the moduli space of polygons in the

Euclidean plane Journal of Di Geom N

KM M Kap ovich JJ Millson The relative deformation theory of repre

sentations of at connections and deformations of linkages in con

stant curvature spaces Comp ositio Math to app ear

KN G Kempf L Ness The length of vectors in representation spaces

In Algebraic Geometry Pro ceedings Cop enhagen Springer

Lecture Notes in Math

K F Kirwan Cohomology of Quotients in Symplectic and Algebraic

Geometry Mathematical Notes Princeton University Press

Kl A Klyachko Spatial polygons and stable congurations of points

on the projective line In Algebraic geometry and its applications

Yaroslavl p

MacL S MacLane Homology Die Grundelheren der Mathematischen

Wissenschaften SpringerVerlag

MZ JJ Millson B Zombro A Kahler structure on the moduli space

of isometric maps from the circle into Euclidean space Inventiones

Math to app ear

MZ JJ Millson B Zombro in preparation

MFW D Mumford J Fogarty F Kirwan Geometric Invariant Theory

Third Enlarged Edition Ergebnisse der Mathematik und Ihrer Gren

zgebiete SpringerVerlag

N L Ness A stratication of the nul l cone via the moment map Amer

Journal of Math vol

Th W Thurston Shapes of Polyhedra Preprint

Wa K Walker Conguration spaces of linkages Undergraduate thesis

Princeton Univ

W J Weitsman Real polarization of the moduli space of at connections

on a Riemann surface Comm Math Phys p

Michael Kap ovich John Millson

Department of Mathematics Department of Mathematics

University of Utah University of Maryland

Salt Lake City College Park

UT USA MD USA

kap ovichmathutahedu jjmjuliaumdedu X

A B

Y

X* Y

B

* A B Y X X Y*

A

X* X*

Y Y

A X X B B

A

X*

X*

Figure