Symplectically convex and symplectically star-shaped curves -- a variational problem
SSymplectically convex and symplectically star-shaped curves–a variational problem
Peter Albers ∗ Serge Tabachnikov † January 6, 2021
Abstract
In this article we propose a generalization of the 2-dimensional notions of convexityresp. being star-shaped to symplectic vector spaces. We call such curves symplecticallyconvex resp. symplectically star-shaped. After presenting some basic results we study afamily of variational problems for symplectically convex and symplectically star-shapedcurves which is motivated by the affine isoperimetric inequality. These variational prob-lems can be reduced back to two dimensions. For a range of the family parameter extremalpoints of the variational problem are rigid: they are multiply traversed conics. For allfamily parameters we determine when non-trivial first and second order deformationsof conics exist. In the last section we present some conjectures and questions and twogalleries created with the help of a Mathematica applet by Gil Bor.
In the paper we make a step toward expanding some notions and results of equiaffine differ-ential geometry of the plane to symplectic spaces.Let γ be a smooth, closed, strictly convex, positively oriented plane curve. One can givethe curve a parameterization γ ( t ) such that [ γ (cid:48) ( t ) , γ (cid:48)(cid:48) ( t )] = 1 for all t , where the bracketdenoted the determinant made by two vectors. This is called an equiaffine parameterizationand, accordingly, one defines the equiaffine length of the curve.The affine isoperimetric inequality between the equiaffine length L and the enclosed area A assert L ≤ π A, with equality if and only if γ is an ellipse, see, e.g., [6, 11]. Note that the inequality goes inthe “wrong” direction, compared to the usual isoperimetric inequality!Assume that the origin is inside the curve, then γ is also star-shaped, in addition tobeing convex. For any parameterization γ ( s ), one has a well-defined (i.e., independent of theparameterization) differential 1-form and a cubic form[ γ ( s ) , γ (cid:48) ( s )] ds, [ γ (cid:48) ( s ) , γ (cid:48)(cid:48) ( s )] ds , ∗ Mathematisches Institut, Universit¨at Heidelberg, 69120 Heidelberg, Germany; [email protected] † Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA;[email protected] a r X i v : . [ m a t h . S G ] J a n uch that A = 12 (cid:90) [ γ ( s ) , γ (cid:48) ( s )] ds, L = (cid:90) (cid:112) [ γ (cid:48) ( s ) , γ (cid:48)(cid:48) ( s )] ds. Thus the affine isoperimetric inequality relates the integrals of these two 1-forms along aconvex closed curve.Let γ ( t ) be a smooth closed curve in the standard symplectic vector space ( R n , ω ).Call γ symplectically star-shaped if ω ( γ ( t ) , γ (cid:48) ( t )) > t , and symplectically convex if ω ( γ (cid:48) ( t ) , γ (cid:48)(cid:48) ( t )) > t . Remark 1.1
We point out that an alternative definition of symplectically star-shaped resp. con-vex is to require (cid:54) = instead of > above. That this is actually more general is explained inSection 2 where we construct examples of curves with all possible sign combinations.Similarly to the plane case, we may define two differential 1-forms along a symplecticallystar-shaped and symplectically convex curve γ by ω ( γ, γ (cid:48) ) dt, (cid:112) ω ( γ (cid:48) , γ (cid:48)(cid:48) ) dt. Both forms are well defined and have no zeroes. Inspired by the affine isoperimetric inequality,we are interested in the relative extrema of (cid:82) γ (cid:112) ω ( γ (cid:48) , γ (cid:48)(cid:48) ) dt constrained by (cid:82) γ ω ( γ, γ (cid:48) ) dt . Infact, we consider a more general variational problem: describe the curves γ that are therelative extrema of (cid:82) γ ω ( γ (cid:48) , γ (cid:48)(cid:48) ) a dt constrained by (cid:82) γ ω ( γ, γ (cid:48) ) dt where a is a real exponent.We point out that the case a = corresponds to the affine isoperimetric inequality.Before we come to our main results we point out that the corresponding metric problem,extremizing the L -norm of the (metric) curvature on a class of plane curves, is a widelystudied topic going back to Bernoulli and Euler and goes under the name of elastica , see [12].In addition, we mention the recent article [13] in which the L p -norms of the curvature arestudied and [7] where the corresponding gradient flows are developed. The latter is a naturalnext step also for the affine context from this article. See [11] for an affine analog of theEuclidean curve shortening flow. Main results
We prove that such extremal curves of this variational problem lie in sym-plectic affine 2-planes, and therefore the problem reduces to a 2-dimensional one (Proposition3.4).We then fix the constraint by giving the curve the centroaffine parameterization, i.e., weassume [ γ ( t ) , γ (cid:48) ( t )] = 1. Then Hill’s equation γ (cid:48)(cid:48) ( t ) = − p ( t ) γ ( t ) holds, and we may considerconsider the functional B a ( γ ) := (cid:82) [ γ ( t ) , γ (cid:48) ( t )] a dt .For a ∈ [ , a = , (Theorem 3).In Theorem 1 we describe non-trivial infinitesimal deformations of multiply traversedconics in the class of extremal curves: if a = , then the n -fold ellipse is infinitesimally rigid;otherwise, a non-trivial infinitesimal deformation of the n -fold ellipse exists if and only if a = k − n k − n for some positive integer k (cid:54) = n . 2heorem 2 concerns second order deformations of conics: for a <
0, the circle γ is a localminimum of B a ; for a ∈ (0 , ), it is a local maximum; for a > , it is a local minimum; andin other cases the Hessian is not sign-definite. The Hessian is degenerate (with 1-dimensionalkernel) if and only if a = k − k − k .In Section 7 we present examples of extremal curves and formulate some conjectures aboutthem.This introduction would not be complete if we failed to mention another reason for ourinterest in centroaffine differential geometry, namely its close relation with the Korteweg-deVries equation, discovered by U. Pinkall [9] and studied by a number of authors since then.When the exponent a equals 2, the extremal curves are periodic solutions to Lam´e’s equationthoroughly studied in this context in a recent paper [4]. Acknowledgements : We are very grateful to Gil Bor for writing a Mathematica pro-gram that made it possible to experiment with extremal curves. ST was supported by NSFgrant DMS-2005444. PA was supported by Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) through Germany’s Excellence Strategy EXC-2181/1 - 390900948 (theHeidelberg STRUCTURES Excellence Cluster), the Transregional Colloborative ResearchCenter CRC/TRR 191 (281071066) and the Research Training Group RTG 2229 (281869850).
In this section we construct curves γ with all possible sign combinations of the quantities ω ( γ, γ (cid:48) ) (cid:54) = 0 and ω ( γ (cid:48) , γ (cid:48)(cid:48) ) (cid:54) = 0. In particular, we assume that all curves are immersed. Westart with a remark concerning symplectically star-shaped curves. The sphere S n − ⊂ R n carries a contact structure defined by the symplectic orthogonal complement to the positionvector. A symplectically star-shaped curve projects to a transverse curve in S n − . A similarremark applies to the contact RP n − , the projectivization of R n . If ω ( γ, γ (cid:48) ) > γ is positively transverse and < corresponds to negatively transverse. A somewhat similarinterpretation for the condition ω ( γ (cid:48) , γ (cid:48)(cid:48) ) (cid:54) = 0 is derived in Lemma 2.2 in case of R .We consider the unit sphere S ⊂ C = R with its standard contact structure. Thestandard contact form at a point q ∈ S is ω ( q, · ) | T q S . Let γ ( t ) be a smooth closed Legendriancurve in S , i.e., ω ( γ, γ (cid:48) ) = 0. Then J γ (cid:48) is a vector normal to γ inside the contact plane.Here J is the complex structure on C . The following lemma is well known, see, e.g., [3]. Lemma 2.1
Pushing a closed Legendrian curve γ slightly in the direction of J γ (cid:48) , resp. − J γ (cid:48) ,inside S yields a negative, resp. positive, transverse curve. Thus, a Legendrian curve in S ⊂ C can be deformed into a symplectically star-shaped curve. Proof.
Let us assume that γ is parametrized by arc-length and set Γ = γ + εJ γ (cid:48) . Then ω (Γ , Γ (cid:48) ) = ε (cid:2) ω ( γ, J γ (cid:48)(cid:48) ) + ω ( J γ (cid:48) , γ (cid:48) ) (cid:3) + O ( ε ) .
3f we denote by · the inner product then ω ( J γ (cid:48) , γ (cid:48) ) = − γ (cid:48) · γ (cid:48) = −
1. From γ · γ = 1 we conclude γ · γ (cid:48) = 0, i.e., ω ( γ, J γ (cid:48) ) = 0. Differentiating this equality gives ω ( γ, J γ (cid:48)(cid:48) ) + ω ( γ (cid:48) , J γ (cid:48) ) = 0and hence ω ( γ, J γ (cid:48)(cid:48) ) = ω ( J γ (cid:48) , γ (cid:48) ). It follows that for sufficiently small ε one has ω (Γ , Γ (cid:48) ) < γ − εJ γ (cid:48) is similar. (cid:50) Let us say that a Legendrian curve γ in S has an inflection point at p ∈ γ if it is second-order tangent to its tangent great circle at p . A generic curve in S does not have inflectionpoints but, as we shall see, a generic Legendrian curve has finitely many of them.The tangent Gauss map sends a Legendrian curve in S to the space of oriented Legendriangreat circles, that is, to the oriented Lagrangian Grassmannian Λ +2 . The image of the Gaussmap is a smooth curve which has vanishing differential precisely at the points correspondingto inflection points of the Legendrian curve. Lemma 2.2
A necessary and sufficient condition for a Legendrian curve γ having an inflec-tion point at γ ( t ) is ω ( γ (cid:48) ( t ) , γ (cid:48)(cid:48) ( t )) = 0 . Proof.
The condition ω ( γ (cid:48) , γ (cid:48)(cid:48) ) = 0 is invariant under reparameterization of γ , so we mayassume that γ is parameterized by arc-length.First, we claim that the orthogonal projection of γ (cid:48)(cid:48) to S is γ + γ (cid:48)(cid:48) . Indeed, γ · γ (cid:48) = 0implies that γ (cid:48) · γ (cid:48) + γ · γ (cid:48)(cid:48) = 0, hence γ · γ (cid:48)(cid:48) = −
1. Now γ · ( γ + γ (cid:48)(cid:48) ) = 1 − γ + γ (cid:48)(cid:48) .Next, we claim that the tangential acceleration vector γ + γ (cid:48)(cid:48) lies in the contact structure.Indeed, ω ( γ, γ (cid:48) ) = 0, hence, after differentiating, ω ( γ, γ (cid:48)(cid:48) ) = 0. Therefore, ω ( γ, γ + γ (cid:48)(cid:48) ) = 0,as needed.In addition, γ · γ = γ (cid:48) · γ (cid:48) = 1 implies that γ (cid:48) and γ + γ (cid:48)(cid:48) are orthogonal to each other.Finally, since γ is Legendrian, i.e., ω ( γ, γ (cid:48) ) = 0, the tangent vector γ (cid:48) lies also in the contactstructure. To summarize, we have two orthogonal vectors, γ (cid:48) and γ + γ (cid:48)(cid:48) , in the contactstructure. In particular, ω ( γ (cid:48) , γ + γ (cid:48)(cid:48) ) = ±(cid:107) γ (cid:48) (cid:107)(cid:107) γ + γ (cid:48) (cid:107) = ±(cid:107) γ + γ (cid:48) (cid:107) since (cid:107) γ (cid:48) (cid:107) = 1.The Lemma now follows from ω ( γ (cid:48) , γ (cid:48)(cid:48) ) = ω ( γ (cid:48) , γ + γ (cid:48)(cid:48) ) = 0 if and only if γ + γ (cid:48)(cid:48) = 0, i.e., ifand only if γ has an inflection point. (cid:50) It follows that if we find a closed Legendrian curve γ with ω ( γ (cid:48) , γ (cid:48)(cid:48) ) (cid:54) = 0, then, accordingto Lemma 2.1 its small push in the normal direction in the contact plane will yield a curvewith ω ( γ, γ (cid:48) ) (cid:54) = 0 and ω ( γ (cid:48) , γ (cid:48)(cid:48) ) (cid:54) = 0. Let us show how to construct such a Legendrian curve γ . See [1, 8] for related matters.Consider the Reeb field of the standard contact form on S and let pr : S → S be therespective Hopf fibration. The projection p takes the Legendrian great circles in S to thegreat circles in S . Closed Legendrian curves are projected to closed immersed curves in S that bound a region with signed area that is a multiple of 2 π and, conversely, such a sphericalcurve lifts (non-uniquely) to a closed Legendrian curve in S .Since pr is a Riemannian submersion it follows that the inflections of a Legendrian curve γ correspond to the inflections of the spherical curve pr ( γ ). Accordingly, every closed sphericalcurve with everywhere positive geodesic curvature and bounding area 2 πk for some k ∈ Z liftsto a Legendrian curve which is free from inflections. Starting with a closed spherical curvewith everywhere positive geodesic curvature, the area condition is easily arranged by addingappropriately sized kinks to the curve. 4inally, we note that the map ( z , z ) (cid:55)→ (¯ z , ¯ z ) of C preserves the contact structure andthe property of γ being Legendrian but changes the sign of ω ( γ (cid:48) ( t ) , γ (cid:48)(cid:48) ( t )) to the opposite.Therefore we can have all four combinations of signs of the quantities ω (Γ , Γ (cid:48) ) and ω (Γ (cid:48) , Γ (cid:48)(cid:48) )where Γ is a small push-off as in Lemma 2.1. Remark 2.3
Another interpretation of the above construction is by looking at a differentHopf fibration (cid:101) pr : S → S , whose fibers are Legendrian. This Hopf fibration takes Legen-drian great circles to circles (of some, possibly zero radius) on S . The projection of a smoothLegendrian curve γ is a wave front, possibly with cusps, and the inflections of γ correspond tothe vertices of the spherical curve (cid:101) pr ( γ ). If (cid:101) pr ( γ ) is smooth and convex, it must have at leastfour vertices (the 4-vertex theorem), without convexity at least two vertices, but if (cid:101) pr ( γ ) hascusps it may be vertex-free. In this section we describe the setting of our variational problem and prove that extremizersare contained in symplectic affine 2-planes. This allows us to reduce the problem to the2-dimensional case.The problem is to find the closed symplectically star-shaped and symplectically convexcurves that are extrema of B a ( γ ) := (cid:82) γ β = (cid:82) ω ( γ (cid:48) , γ (cid:48)(cid:48) ) a dt , a ∈ R , subject to the constraintgiven by A ( γ ) := (cid:82) γ α = (cid:82) ω ( γ, γ (cid:48) ) dt . More precisely, we consider the space P T of T -periodic symplectically star-shaped and symplectically convex curves in R n and consider A , B a : P T → R and ask for extrema of B a subject to A = c , that is, we want to describethe set Crit( B a | {A = c } ) ⊂ P T . Let us note that both functionals, A and B a , are translation invariant, that is, do not dependon the choice of the origin (as long as the curve remains star-shaped). We start with a fewsimple observations. Lemma 3.1
For c > and c > there is a natural bijection (by rescaling) from P T toitself inducing a bijections {A = c } ∼ = {A = c } and Crit( B a | {A = c } ) ∼ = Crit( B a | {A = c } ) . Proof.
Let γ ∈ P T and consider γ ( t ) := (cid:16) c c (cid:17) γ ( t ) ∈ P T . If γ ∈ {A = c } then A ( γ ) = c c A ( γ ) = c , i.e., γ ∈ {A = c } . From B a (cid:18)(cid:16) c a (cid:17) γ (cid:19) = (cid:16) c a (cid:17) a B a ( γ ) (1)5s follows that the bijection γ (cid:55)→ (cid:16) c c (cid:17) γ just rescales B a by a fixed factor, i.e., induces theclaimed bijection Crit( B a | {A = c } ) ∼ = Crit( B a | {A = c } ). (cid:50) Remark 3.2
Equation (1) implies that, if B a has critical points at all, they appear in R > -family and the critical value is necessarily 0. Lemma 3.3
The bijection P T → P given by Γ( t ) := γ ( t/T ) preserves A and rescales B a by T a − . Proof.
We consider the bijection P T → P given by Γ( t ) := γ ( t/T ). Then A (Γ) = (cid:90) ω (Γ( t ) , Γ (cid:48) ( t )) dt = (cid:90) ω (cid:0) γ ( tT ) , T γ (cid:48) ( tT ) (cid:1) dt = (cid:90) T ω (cid:0) γ ( s ) , γ (cid:48) ( s ) (cid:1) ds = A ( γ ) , as it has to be since A (Γ) = A ( γ ) is twice the enclosed area. Similarly, we compute B a (Γ) = (cid:90) ω (Γ (cid:48) ( t ) , Γ (cid:48)(cid:48) ( t )) a dt = (cid:90)
10 1 T a ω (cid:0) γ ( tT ) , γ (cid:48) ( tT ) (cid:1) a dt = T a − (cid:90) T ω (cid:0) γ ( s ) , γ (cid:48) ( s ) (cid:1) ds = T a − B a ( γ ) , as claimed. (cid:50) Now, we begin to study the relative extrema of B a constrained by A . We call the extrema critical curves . The previous two lemmata imply that we may consider curves of fixed periodand with fixed constraint. Proposition 3.4
A critical curve is contained inside a symplectic affine 2-plane of ( R n , ω ) . Proof.
We first derive the equation for a critical curve using a Lagrange multiplier λ ∈ R .For that let v ( t ) be a vector field along γ and consider an infinitesimal deformation γ ε = γ + εv of γ . Then γ is critical if for some λ for all vddε (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 (cid:20) λ (cid:90) ω ( γ ε , γ (cid:48) ε ) dt + (cid:90) ω ( γ (cid:48) ε , γ (cid:48)(cid:48) ε ) a dt (cid:21) = 0 . The derivative of the first integral in direction of v computes to (cid:90) ( ω ( v, γ (cid:48) ) + ω ( γ, v (cid:48) )) dt = 2 (cid:90) ω ( v, γ (cid:48) ) dt, where we used integration by parts and skew-symmetry of the symplectic form in the equalitysign.Similarly, setting F := ω ( γ (cid:48) , γ (cid:48)(cid:48) ) a − , the derivative of the second integral can be expressedas a (cid:90) F (cid:0) ω ( v (cid:48) , γ (cid:48)(cid:48) ) + ω ( γ (cid:48) , v (cid:48)(cid:48) ) (cid:1) dt = a (cid:90) (cid:0) F (cid:48) ω ( v (cid:48) , γ (cid:48) ) + 2 F ω ( v (cid:48) , γ (cid:48)(cid:48) ) (cid:1) dt = a (cid:90) (cid:0) F (cid:48)(cid:48) ω ( γ (cid:48) , v ) + 3 F (cid:48) ω ( γ (cid:48)(cid:48) , v ) + 2 F ω ( γ (cid:48)(cid:48)(cid:48) , v ) (cid:1) dt. v is arbitrary and ω is non-degenerate, the criticality condition implies that the vectors γ (cid:48) , γ (cid:48)(cid:48) , γ (cid:48)(cid:48)(cid:48) are linearly dependent, and since F (cid:54) = 0, one has γ (cid:48)(cid:48)(cid:48) = f γ (cid:48) + gγ (cid:48)(cid:48) for some periodicfunctions f ( t ) , g ( t ).It follows that the bivector γ (cid:48) ∧ γ (cid:48)(cid:48) satisfies the differential equation ( γ (cid:48) ∧ γ (cid:48)(cid:48) ) (cid:48) = g ( γ (cid:48) ∧ γ (cid:48)(cid:48) ),that is, remains proportional to itself. Hence the curve γ (cid:48) is planar. It follows by integrationthat the curve γ lies in an affine 2-plane, parallel to the plane spanned by γ (cid:48) and γ (cid:48)(cid:48) . (cid:50) Proposition 3.4 allows us to reduce the problem to two dimensions as follows. A criticalcurve γ is contained in an affine 2-plane. Let V be the two dimensional linear space parallelto this affine 2-plane, and W be its symplectic orthogonal. Then we may write γ = γ ⊕ γ ∈ V ⊕ W . Since γ (cid:48) ∈ V we conclude that γ is constant. We claim that γ is also a criticalcurve. Indeed, since W is symplectically orthogonal to V we compute0 (cid:54) = ω ( γ, γ (cid:48) ) = ω ( γ ⊕ γ , γ (cid:48) ) = ω ( γ , γ (cid:48) ) , i.e., γ is symplectically star-shaped and A ( γ ) = A ( γ ). Trivially, γ is also symplecticallyconvex with B a ( γ ) = B a ( γ ). Now, that we have reduced the problem to two dimensions, wedenote the symplectic form ω in the plane by brackets [ · , · ]. Convention 3.5
To fix the constraint A , from now on we always parametrize a star-shapedcurve so that [ γ, γ (cid:48) ] = 1 , that is, γ is in its centroaffine parameterization. In centroaffine parameterization we have γ (cid:48)(cid:48) = − pγ . The function p is called the centroaffinecurvature of the curve γ . For some computations below we record that p = [ γ (cid:48) , γ (cid:48)(cid:48) ] > γ (cid:48)(cid:48)(cid:48) = − p (cid:48) γ − pγ (cid:48) .Thus, we can reformulate our goal as describing all extremal curves of the functional B a ( γ ) := (cid:82) p a ( t ) dt on the space of periodic curves in centroaffine parameterization with p >
0. Here a ∈ R is fixed. To be quite precise, we point out that we, of course, consider thespace of all periodic curves in centroaffine parameterization of a fixed period , mostly, 2 π or amultiple of 2 π . As explained in the introduction, the case a = is of special interest sincethis corresponds to the affine isoperimetric inequality. Lemma 3.6
The properties of a curve being symplectically star-shaped and convex are invari-ant under point-wise multiplication by an element in SL (2 , R ) and sufficiently small trans-lation. For any value of a , the functional B a is invariant under this SL (2 , R ) action. Inaddition, if a = , the functional B is invariant under translations. This is the only valueof a with this additional symmetry. Proof.
Point-wise multiplication of a curve with a matrix from SL (2 , R ) doesn’t change itscentroaffine parameterization nor its centroaffine curvature.A sufficiently small translation of a curve γ keeps the properties of being symplecticallystar-shaped and convex but in general not the centroaffine parameterization of γ . Afterreparametrization, the shifted curve ¯ γ = γ + c is in centroaffine parameterization, i.e. ¯ γ ( τ ) = γ ( τ ( t )) + c and [¯ γ, d ¯ γdτ ] = 1. Its centroaffine curvature ¯ p satisfies¯ p ( τ ) = (cid:18) dtdτ (cid:19) p ( τ ) , p = (cid:20) d ¯ γdτ , d ¯ γdτ (cid:21) = (cid:34) dtdτ d ¯ γdt , (cid:18) dtdτ (cid:19) d ¯ γdt + d tdτ d ¯ γdt (cid:35) = (cid:18) dtdτ (cid:19) (cid:20) d ¯ γdt , d ¯ γdt (cid:21) = (cid:18) dtdτ (cid:19) p since d ¯ γdt = dγdt = γ (cid:48) . We conclude that precisely for a = the functional B a satisfies B a (¯ γ ) = (cid:90) ¯ p a ( τ ) dτ = (cid:90) p a ( t ) dt = B a ( γ )for all symplectically star-shaped and convex curves γ . (cid:50) We now derive a critical point equation for B a in terms of the function F = p a − . Proposition 3.7
For a (cid:54) = 1 the extremal curves γ of the functional (cid:82) p a ( t ) dt satisfy F (cid:48)(cid:48)(cid:48) = − b ) F b F (cid:48) (2) where F = p a − and b = a − . If a = 1 or a = , then p is constant and γ is therefore a conic. Proof.
First, we describe the vector fields v along γ that preserve the centroaffine param-eterization of γ . Write such a field as v = gγ + f γ (cid:48) . Since the deformation by v is assumedto preserve the centroaffine parameterization, we conclude that [ γ, v (cid:48) ] + [ v, γ (cid:48) ] = 0 and hence2 g + f (cid:48) = 0. Thus, the vector field v has the form − f (cid:48) γ + f γ (cid:48) ; the deformations keeping thecentroaffine parameterization depend on one periodic function.As in the proof of Proposition 3.4, linearizing (cid:82) p a ( t ) dt in v leads to (cid:90) F (cid:0) [ γ (cid:48) , v (cid:48)(cid:48) ] + [ v (cid:48) , γ (cid:48)(cid:48) ] (cid:1) dt = 0for every vector field v as above, where we recall that F = p a − . Using integration by partstwice and recalling that γ (cid:48)(cid:48)(cid:48) = − p (cid:48) γ − pγ (cid:48) , we rewrite the integral as (cid:90) (cid:16)(cid:0) F (cid:48)(cid:48) − pF (cid:1) [ γ (cid:48) , v ] − (cid:0) pF (cid:48) + 2 p (cid:48) F (cid:1) [ γ, v ] (cid:17) dt. Using [ γ, v ] = f , [ γ (cid:48) , v ] = f (cid:48) and another integration by parts, this integral becomes (cid:90) (cid:16) − F (cid:48)(cid:48)(cid:48) − pF (cid:48) − p (cid:48) F (cid:17) f dt. The critically condition is that this integral vanishes for all f , and we conclude that theintegrand is zero.If a = 1, then F ≡
1, and hence the criticality condition simply becomes p (cid:48) = 0. Therefore, γ solves γ (cid:48)(cid:48) = const · γ , i.e., γ is a conic.Otherwise, recall that p = F b , hence p (cid:48) = bF b − F (cid:48) . Substitute this to the integrand andcollect terms to obtain the claimed formula (2).If a = then b = −
2, and equation (2) reduces to F (cid:48)(cid:48)(cid:48) = 0. Since F is periodic andpositive, F is necessarily a constant, and so is p = F − . Thus, again γ is a conic in this case. (cid:50) Proposition 3.8
For a ∈ ( , , equation (2) has only constant solutions. roof. We can integrate equation (2) to F (cid:48)(cid:48) = − b + 2) b + 1 F b +1 + c (3)with some constant c . Note that b (cid:54) = −
1. Since F is a positive and periodic function, we getat the minimum m := min F and the maximum M := max F of F the usual inequalities for F (cid:48)(cid:48) , leading to 0 ≥ − b + 2) b + 1 M b +1 + c and 0 ≤ − b + 2) b + 1 m b +1 + c. Thus, we arrive at the two inequalities b + 2 b + 1 m b +1 ≤ b + 2 b + 1 M b +1 m ≤ M. (4)If b + 1 > b + 2 > > b + 1, then the first inequality in (4) is consistent with the second.However, the first inequality together with 0 > b + 2 implies m ≥ M and we conclude m = M .That is, 0 > b + 2 implies that F is constant. Now, recall that b = a − . Therefore, F isconstant if < a <
1, as claimed. (cid:50)
Lemma 3.9
For a (cid:54)∈ ( , , equation (2) has non-constant positive, periodic solutions. Remark 3.10
The solutions we construct in the proof of the lemma below are, by construc-tion, often times multiply covered.
Proof.
As a preparation, we consider the Hamiltonian formulation of the ODE (3). Hamil-ton’s equations for the Hamiltonian function ( c ∈ R is fixed) H ( q, p ) := 12 p + 2 b + 1 q b +2 − cq are given by ˙ p = − ∂H∂q = − b + 2) b + 1 q b +1 + c ˙ q = ∂H∂p = p, and are, with q = F and p = F (cid:48) , equivalent to the ODE (3). We compute dH ( q, p ) = p dp + (cid:18) b + 2) b + 1 q b +1 − c (cid:19) dq and Hess H ( q, p ) = (cid:18) b + 2) q b
00 1 (cid:19) . Therefore, ( q, p ) is a critical point of H if and only if p = 02( b + 2) b + 1 q b +1 = c, b + 2) q b > . Our assumption a (cid:54)∈ ( ,
1) is equivalent to b + 2 > b = a − ).Therefore, for an appropriate choice of the constant c , the Hamiltonian function H admitsa local minimum ( q, p = 0) with q >
0. Since we are in dimension 2, a level set of H slightlyabove the critical value of ( q,
0) contains a small circle (and potentially other connectedcomponents). These circles are then periodic orbits which correspond to positive (since q > (cid:50)
Remark 3.11
Equation (2) can be further integrated, which is in terms of the Hamiltonianformulation just expressing the preservation of H along solutions.( F (cid:48) ) = − b + 1 F b +2 + 2 cF + d. (5)For a = 2, resp. a = , we have b + 2 = 3, resp. b + 2 = 4, and hence F is the Weierstrasselliptic function. Since p = F b , in the first case p is also the Weierstrass function, and in thesecond case p is its square. In the former case, the respective equation γ (cid:48)(cid:48) = − pγ is called theLam´e equation, and it was thoroughly studied, see, e.g., [14]. As a multiple conic we take the unit circle traversed n times. This is a critical curve ofthe functional B a ( γ ) = (cid:82) πn p a dt , and we ask whether it admits a non-trivial infinitesimaldeformation in the class of critical curves.The functional B a is invariant under the action of SL (2 , R ); the respective deformationscomprise a 3-dimensional space, and we consider them as trivial. In addition, if a = ,the functional is invariant under parallel translations, see Lemma 3.6. In this case, we addthis 2-dimensional space to the deformations that we consider as trivial. The n -fold circle isinfinitesimally rigid if it does not admit non-trivial infinitesimal deformations in the class ofcritical curves.Let γ ( t ) = (cos t, sin t ) be the unit circle traversed n times, and let γ = − f (cid:48) γ + f γ (cid:48) bea vector field along it that defines its infinitesimal deformation. We assume that the periodis 2 πn , and accordingly, f ( t ) is a 2 πn -periodic function.The trivial deformations are described in the following lemma. Lemma 4.1
The infinitesimal action of SL (2 , R ) corresponds to the functions f that arelinear combinations of , cos(2 t ) , sin(2 t ) , and parallel translations to the functions f that arelinear combinations of cos t, sin t . Proof.
The deformed curve is γ + εγ . The case f = 1 corresponds to the rotation of theunit circle.Let f ( t ) = sin(2 t ). In this case, computing up to ε , the curve γ + εγ is the ellipse(1 + 2 ε ) x + (1 − ε ) y = 1, and likewise for f = cos(2 t ).10f f ( t ) = 2 sin t then, again mod ε , the curve γ + εγ is the unit circle ( x + ε ) + y = 1,and likewise for f = 2 cos t . (cid:50) We are ready to describe the infinitesimal rigidity of the circle.
Theorem 1 If a = , then the n -fold unit circle is infinitesimally rigid. Otherwise, a non-trivial infinitesimal deformation of the n -fold unit circle exists if and only if a = k − n k − n for some positive integer k (cid:54) = n . Proof.
The calculations below are modulo ε .Let Γ = γ + εγ . We have; Γ (cid:48)(cid:48) = − ( p + εp )Γ, hence p + εp = [Γ (cid:48) , Γ (cid:48)(cid:48) ] = [ γ (cid:48) + εγ (cid:48) , γ (cid:48)(cid:48) + εγ (cid:48)(cid:48) ] = 1 + ε ([ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ]) . Therefore F = ( p + εp ) a − = 1 + ε ( a − γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ]) . We calculate γ (cid:48) = − (cid:18) f + 12 f (cid:48)(cid:48) (cid:19) γ + 12 f (cid:48) γ (cid:48) , γ (cid:48)(cid:48) = − (cid:18) f (cid:48) + 12 f (cid:48)(cid:48)(cid:48) (cid:19) γ − f γ (cid:48) , hence q := [ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ] = 2 f (cid:48) + 12 f (cid:48)(cid:48)(cid:48) . Compare with [9], where the Korteweg-de Vries equation is interpreted as a flow on centroaffinecurves.The case of a = 1 was considered earlier (the only solution of the variational problem is aconic); therefore we assume that a (cid:54) = 1.Since the perturbed curve is critical, equation (3) holds. Write the constant in this equa-tion as c + εc . Then (3) implies q (cid:48)(cid:48) = − b + 2) q + c a − . Since q (cid:48)(cid:48) and q have zero average, we conclude that c = 0. Therefore q (cid:48)(cid:48) = − b + 2) q. This equation has periodic solutions only when b + 2 >
0, and then q ( t ) is a linearcombination of cos( (cid:112) b + 2) t ) and sin( (cid:112) b + 2) t ). For q to be 2 πn -periodic, one has tohave (cid:112) b + 2) = kn , that is, b = k − n n or a = k − n k − n (6)for positive integers k, n . Note that since b (cid:54) = 0, one has k (cid:54) = 2 n .Let us show that if condition (6) holds, the desired infinitesimal deformations of a multiplecircle exists. 11e find f ( t ) from the equation2 f (cid:48) + 12 f (cid:48)(cid:48)(cid:48) = a cos (cid:18) ktn (cid:19) + b sin (cid:18) ktn (cid:19) . It follows that, modulo the kernel of the differential operator d + 2 d , the function f is alinear combination of cos (cid:0) ktn (cid:1) and sin (cid:0) ktn (cid:1) . The kernel of the differential operator contributestrivial deformations, and we end up with a 2-dimensional space of deformations.It remains to see when these deformation are trivial. Since k (cid:54) = 0 and k (cid:54) = 2 n , the only“suspicious” case is k = n . In this case, a = , and indeed, the first harmonics give trivialdeformations, corresponding to parallel translations. (cid:50) Remark 4.2
We note that, in agreement with Lemma 3.8, a = k − n k − n does not take values in [ , k > n , then a >
1, and if k < n , then a < . Here we investigate how the functional B a ( γ ) = (cid:82) p ( t ) a dt changes under a second orderdeformation of the unit circle. We recall that the functional is SL (2 , R ), see Lemma 4.1.We assume that the period is 2 π and that the curves have the centroaffine parameterization[ γ, γ (cid:48) ] = 1. Let Γ = γ + εγ + ε γ , ignoring the higher order terms in ε . As before, γ ( t ) = (cos t, sin t ), hence γ (cid:48)(cid:48) = − γ , and γ = − f (cid:48) γ + f γ (cid:48) , where f ( t ) is a 2 π -periodicfunction.The condition [Γ , Γ (cid:48) ] = 1 implies[ γ , γ (cid:48) ] + [ γ , γ (cid:48) ] + [ γ , γ (cid:48) ] = 0 . (7)We have; Γ (cid:48)(cid:48) = − P Γ, hence P = [Γ (cid:48) , Γ (cid:48)(cid:48) ] = [ γ (cid:48) + εγ (cid:48) + ε γ (cid:48) , γ (cid:48)(cid:48) + εγ (cid:48)(cid:48) + ε γ (cid:48)(cid:48) ] = 1 + εq ( t ) + ε r ( t ) , where q = [ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ] , r = [ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ] . Then P a = 1 + εaq + ε a (cid:18) r + a − q (cid:19) . As we already know, q = [ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ] = 2 f (cid:48) + 12 f (cid:48)(cid:48)(cid:48) . We need to calculate (cid:90) π (cid:18) r + a − q (cid:19) dt. γ (cid:48)(cid:48)(cid:48) = − γ (cid:48) , we get (cid:90) rdt = (cid:90) ([ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ]) dt = (cid:90) (2[ γ , γ (cid:48) ] + [ γ (cid:48) , γ (cid:48)(cid:48) ]) dt. Integrate equation (7):0 = (cid:90) ([ γ , γ (cid:48) ] + [ γ , γ (cid:48) ] + [ γ , γ (cid:48) ]) dt = (cid:90) (2[ g , γ (cid:48) ] + [ γ , γ (cid:48) ]) dt, therefore (cid:90) rdt = (cid:90) ([ γ (cid:48) , γ (cid:48)(cid:48) ] − [ γ , γ (cid:48) ]) dt. We calculate[ γ , γ (cid:48) ] = f + 12 f f (cid:48)(cid:48) − f (cid:48) , [ γ (cid:48) , γ (cid:48)(cid:48) ] = f + 12 f f (cid:48)(cid:48) + 34 f (cid:48) + 14 f (cid:48) f (cid:48)(cid:48)(cid:48) , hence (cid:90) rdt = (cid:90) (cid:18) f (cid:48) + 14 f (cid:48) f (cid:48)(cid:48)(cid:48) (cid:19) dt = (cid:90) (cid:18) f (cid:48) − f (cid:48)(cid:48) (cid:19) dt. Next, (cid:90) q dt = (cid:90) (cid:18) f (cid:48) + 12 f (cid:48)(cid:48)(cid:48) (cid:19) dt = (cid:90) (cid:18) f (cid:48) − f (cid:48)(cid:48) + 14 f (cid:48)(cid:48)(cid:48) (cid:19) dt. In the case of most interest, a = , and we obtain (cid:90) (cid:18) r + a − q (cid:19) dt = (cid:90) (cid:18) r − q (cid:19) dt = − (cid:90) (4 f (cid:48) − f (cid:48)(cid:48) + f (cid:48)(cid:48)(cid:48) ) dt. (8) Lemma 5.1
The integral in (8) is non-negative, and it equals zero if and only if f is a firstharmonic. Proof.
Let f (cid:48) ( t ) = (cid:88) k c k e ikt , c − k = ¯ c k be the Fourier expansion. Then (cid:90) π (4 f (cid:48) − f (cid:48)(cid:48) + f (cid:48)(cid:48)(cid:48) ) dt = 2 (cid:88) k> (4 − k + k ) | c k | . We have 4 − k + k = ( k − k − ≥
0, and the sum is positive unless the only non-zeroterm is for k = 1, that is, f is a first harmonic. (cid:50) Now consider the general case: (cid:90) (cid:18) r + a − q (cid:19) dt = (cid:90) (cid:18) (2 a − f (cid:48) − a − f (cid:48)(cid:48) + a − f (cid:48)(cid:48)(cid:48) (cid:19) dt. In terms of the Fourier coefficients of f (cid:48) , this is2 (cid:88) k> (cid:18) a − − a − k + a − k (cid:19) | c k | . k − (cid:2) ( a − k − a − (cid:3) . (9)We also note that the quadratic term of P a contains the factor a . Theorem 2
For a < , the circle γ is a local minimum of B a ; for a ∈ (0 , ) , it is a localmaximum; for a > , it is a local minimum, and in other cases the Hessian is not sign-definite. The Hessian is degenerate (with 1-dimensional kernel) if and only if a = k − k − forsome positive integer k . Proof.
Let a = 1. Then the sign of (9) is that of − ( k − k = 1 andnegative for k ≥ a >
1. Then the sign of (9) is positive for sufficiently large k . The Hessian is positive-definite if this sign is positive for all k . When k = 1, the first factor in (9) is negative, andso is the second one: 1 − a . When k ≥
3, the first factor is positive, and the second one ispositive if and only if 9( a − − a − >
0, that is, a > .Let 0 < a <
1. Then the sign of (9) is negative for sufficiently large k . The Hessianis negative-definite if this sign is negative for all k . When k = 1, the first factor in (9) isnegative, and the second factor equals 1 − a . Thus (9) is negative if and only if a < . Ifthis inequality is satisfied then, for k ≥
3, the second factor in (9) is 2 − a − (1 − a ) k < a <
0, then the analysis of the preceding paragraph still holds, but the factor a of thequadratic term in P a changes the sign to the opposite.Finally, the Hessian is degenerate when (9) is zero for some k . Solving this for a yieldsthe last claim of the theorem. (cid:50) The last statement of the theorem agrees with Theorem 1. The numbers a = k − k − formthe sequence 13 , , , , . . . that converges to 1. Each time that a crosses an element of this “spectrum”, the signatureof the Hessian changes by 1. a = We recall that we attempt to describe extremal curves of the functional B a = (cid:82) p a ( t ) dt onthe space of periodic curves in centroaffine parameterization with p >
0. The case a = corresponds to the affine isoperimetric inequality. In particular, the functional then istranslation invariant, see Lemma 3.6. In this case, b = − , and equation (3) reads F (cid:48)(cid:48) = 2 F − + c. (10)Since F (cid:48)(cid:48) ≤ F >
0, we conclude that c < F (cid:48) ) = 8 F + 2 cF + d (11)14this is the equation of a level curve of the Hamiltonian, see Section 3).Let F ( t ) = G ( t ) where G ( t ) is also a positive periodic function. Then (11) becomes( GG (cid:48) ) = c G + 2 G + d (12)(again renaming the constants).The right hand side of (12) is a quadratic polynomial in G , and it has at least two rootsbecause G is a periodic function that attains maximum and minimum. Since a quadraticpolynomial has at most two roots, G oscillates between its maximum and minimum, and hasno other critical values. Example 6.1
Let us examine the case of a parallel translated n times traversed circle, whicha critical curve: γ = ( a + r cos α, b + r sin α ) , where α ( t ) is a function of the centroaffine parameter t . We assume that the range of t is[0 , π ], that of α is [0 , πn ], and the radius of the circle is r .The range of the centroaffine parameter is twice the (algebraic) area bounded by the curve,hence r = n − .We calculate: [ γ, γ α ] = r + ra cos α + rb sin α, and since [ γ, γ t ] = 1, we have dtdα = r + ra cos α + rb sin α. This can be integrated: t = r α + ra sin α − rb cos α + C, or α + a sin α + b cos α = nt + C (cid:48) , with the constants renamed. Since a sin α + b cos α = − (cid:112) a + b cos( α + θ )with sin θ = a √ a + b , cos θ = − b √ a + b , we can change the parameter α to obtain a simplified equation α ( t ) − a cos α ( t ) = nt + C (13)(once again renaming the constants) Then dα/dt = n (1 + a sin α ) − . Next, p = [ γ t , γ tt ] = [ γ α , γ αα ] (cid:18) dαdt (cid:19) = n (1 + a sin α ) − . Since G = p − , we conclude that G ( t ) = n − (1 + a sin α ( t )) (14)(renaming the constant again). 15et us continue with the general case. Let 0 < m < M be the minimum and the maximumof G , and let c = − k . Write the right hand side of (12) as k ( G − m )( M − G ), then thedifferential equation becomes GG (cid:48) = k (cid:112) ( G − m )( M − G ) , (15)where we allow the square root to take values at its positive and negative branches.Set µ = M + m , ε = M − m . Note that since
G >
0, we have ε < µ .Since G oscillates between m and M , let us make another substitution: G ( t ) = µ + ε sin ϕ ( t ) , (16)where ϕ ( t ) is not necessarily a periodic function anymore. Since G is 2 π -periodic, ϕ ( t + 2 π ) = ϕ ( t ) + 2 πn where n is an integer.Substitute (16) into (15) to obtain ϕ (cid:48) ( µ + ε sin ϕ ) = k . This differential equation integratesto µϕ ( t ) − ε cos ϕ ( t ) = kt + C. (17)If | ε | < µ then the left hand side is a monotonic function of ϕ , therefore this functionalequation uniquely determines the function ϕ ( t ).Substituting ϕ ( t + 2 π ) in (17), we find that µ ( ϕ ( t + 2 π ) − ϕ ( t )) = 2 πk . Since ϕ ( t + 2 π ) = ϕ ( t ) + 2 πn with n ∈ Z , we have k = µn . Thus we have ϕ ( t ) − ε cos ϕ ( t ) = nt + C, (18)where, as before, we renamed the constants. Theorem 3 If a = then the relative extrema of the functional B a constrained by A aremultiply traversed conics. Proof.
Let γ ( t ) be a 2 π -periodic critical curve. Then equation (18) holds for some n ,defining function ϕ ( t ). Observe that equation (18) is identical to equation (13) from Example6.1. Therefore ϕ ( t ) coincides with the function α ( t ). Similarly, equations (16) and (14)coincide, and so function G ( t ) coincides with that from Example 6.1.It follows that the centroaffine curvature p ( t ) of the curve γ ( t ) is the same as that of theparallel translated n -fold circle γ ( t ). Consider the contact element ( γ (0) , γ (cid:48) (0)) Acting on γ ( t ) by an element of SL (2 , R ), we can arrange for ( γ (0) , γ (cid:48) (0)) to coincide with ( γ (0) , γ (cid:48) (0)).The action of SL (2 , R ) does not change the centroaffine curvature, hence the two curves, γ and γ , satisfy the same second order differential equation and have the same initial data.Therefore they coincide. (cid:50) Remark 6.2
The Lambert W function is the inverse function of the complex function z = we w , see [5]. The function ϕ defined by equation (18) is related to the Lambert function inthe following way. 16et us assume that C = 0 , n = 1 in (18). Consider the complex function given by theequation ξ = η − εe iη . If η is real then Re ξ = η − ε cos η , the expression that defines thefunction ϕ ( t ).Let z = we w , and set z = − iεe iξ , w = − iεe iη . Then ln z = w + ln w , that is, ξ = η − εe iη . Therefore the inverse function η ( ξ ) is conjugated to the Lambert function w ( z ) bythe exponential function. First, in Figures 1 – 4 we present of extremal curves obtained in numerical experiments usinga Mathematica applet created by Gil Bor.Figure 1: Curves having a = − , a = − , a = 0 . , ,
4, respectively.Figure 2: These curves have a = 1 . , a = 1 . , a = 1 .
4, respectively.17igure 3: These curves have a = 1 . , a = 1 . , a = 1 .
75, respectively.Figure 4: These curves have a = 2 , a = 2 . , a = 3 and the rotation numbers 3 , ,
5, respec-tively. 18et us comment on a common geometrical feature of these curves. Recall the notion ofthe osculating circle of a smooth curve in Euclidean geometry: this is a circle that is 2-ordertangent to a curve at a given point, that is, it shares the curvature with the curve. Informallyspeaking, the osculating circle passes through three infinitesimally close points of the curve.In centroaffine geometry, the role of osculating circles is played by osculating centralconics. The space of central conics is 3-dimensional, and for every point of a star-shapedcurve there exists a central conic that is 2-order tangent to it at this point. Central conicshave constant centroaffine curvature, and the osculating central conic at point γ ( t ) has theconstant centroaffine curvature p ( t ).Similarly to the case of osculating circles the osculating central conic goes from one sideof the curve to the other side if p (cid:48) ( t ) (cid:54) = 0. If p has a non-degenerate maximum or minimumat point t , that is, γ ( t ) is a centroaffine vertex, then the osculating central conic lies on oneside of the curve near this point.In the following lemma we prove that the centroaffine curvature of extremal curves takesonly two values at their centroaffine vertices, its maximum and minimum. Lemma 7.1
The function F , introduced in Section 3, and hence the centroaffine curvature p , has only two critical values, its maximum and minimum. Proof.
Since the curve is closed, the function F attains its maximum and minimum. It hasno other critical values because the right hand side of equation (5), as a function of F , hasno more roots than the number of sign changes among its three coefficients, see [10] (part 5,chapter 1, §
6, No 77). (cid:50)
Looking again at the above figures it is fairly obvious that the case a = 1 . a of the form a = k − k − , k ∈ Z > , a circle (which is a critical curve forany a ) admits a non-trivial infinitesimal deformation. Indeed, 1 . − − .A more systematic computer experiment (again using the Mathematica applet by Gil Bor)lead to Figure 5 where curves corresponding to k = 4 , , , , , a , the functional B a is invariant under SL (2 , R ). These correspond, of course,to trivial deformations.Figure 5 leads us to the conjecture that for a = k − k − there exists an extremal curve whichis a ”rounded ( k − k − B a . We point out that these ( k − k very large. This is, at least, consistent with a → k → ∞ and the circleis indeed rigid for B . As a rather special case, the egg (case a = 1 . • What the minimal rotation number of a periodic solution of B a in dependence of a ? Itseems that this minimal rotation number goes to infinity with a . • In general, what happens if a → ∞ respectively, a → −∞ ? • Is there some kind of duality for positive and negative values of a ? • Is there an appropriate gradient flow of B a similar to the metric case, see [7] and [13]?19 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Figure 5: a = , a = , a = , a = , a = , a = Appendix
In this appendix we describe a curious symmetry of equations (5). In the discovery of thissymmetry, we were motivated by Bohlin’s theorem as described in Appendix 1 of [2].Consider the family of equations (cid:18) dFdt (cid:19) = uF q + vF + w, (19)where F ( t ) is an unknown function and u, v, w, q are parameters. We are looking for changesof independent and dependent variables F ( t ) = G ( τ ) µ , dτdt = G λ that preserve the form of equation (19), but possibly change the parameters u, v, w , and q . Theorem 4
These changes of variables form a group, the group of permutations S . Theorbit of the exponent q is (cid:26) q, q , − q, − q , qq − , q − q (cid:27) . We note that this is precisely how the permutations of four points in the projective lineaffect their cross-ratio.
Proof.
Denote the new parameters by ¯ u, ¯ v, ¯ w , and ¯ q .Using the chain rule, one obtains the differential equation on G : (cid:18) dGdτ (cid:19) = uµ G µq − µ − λ +2 + vµ G − λ − µ + wµ G − λ − µ . For this equation to have the same form as (19), one needs the following relation between theexponents to hold: { , } ⊂ { µq − µ − λ + 2 , − λ − µ, − λ − µ } . Thus one needs to consider six cases. We present one of them: (cid:40) µq − µ − λ + 2 = 02 − λ − µ = 1 . Hence µ = 11 − q , ¯ q = 2 − λ − µ = qq − , ¯ u = w ( q − , ¯ v = v ( q − , ¯ w = u ( q − . The other five cases are analyzed in a similar way. (cid:50)
Returning to equation (5), one has q = b + 2 = a − a − . The S -orbit of the exponent a isas follows: (cid:26) a, − a, a − a − , a a − , a − a − , a − a − (cid:27) .
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