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Symplectically convex and symplectically star-shaped curves -- a variational problem

In this article we propose a generalization of the 2-dimensional notions of convexity resp. being star-shaped to symplectic vector spaces. We call such curves symplectically convex resp. symplectically star-shaped. After presenting some basic results we study a family of variational problems for symplectically convex and symplectically star-shaped curves which is motivated by the affine isoperimetric inequality. These variational problems can be reduced back to two dimensions. For a range of the family parameter extremal points of the variational problem are rigid: they are multiply traversed conics. For all family parameters we determine when non-trivial first and second order deformations of conics exist. In the last section we present some conjectures and questions and two galleries created with the help of a Mathematica applet by Gil Bor.


Introduction
In the paper we make a step toward expanding some notions and results of equiaffine differential geometry of the plane to symplectic spaces.
Let γ be a smooth, closed, strictly convex, positively oriented plane curve. One can give the curve a parameterization γ(t) such that [γ (t), γ (t)] = 1 for all t, where the bracket denoted the determinant made by two vectors. This is called an equiaffine parameterization and, accordingly, one defines the equiaffine length of the curve.
The affine isoperimetric inequality between the equiaffine length L and the enclosed area A assert with equality if and only if γ is an ellipse, see, e.g., [6,11]. Note that the inequality goes in the "wrong" direction, compared to the usual isoperimetric inequality! Assume that the origin is inside the curve, then γ is also star-shaped, in addition to being convex. For any parameterization γ(s), one has a well-defined (i.e., independent of the parameterization) differential 1-form and a cubic form Thus the affine isoperimetric inequality relates the integrals of these two 1-forms along a convex closed curve. Let γ(t) be a smooth closed curve in the standard symplectic vector space (R 2n , ω). Call γ symplectically star-shaped if ω(γ(t), γ (t)) > 0 for all t, and symplectically convex if ω(γ (t), γ (t)) > 0 for all t.
Remark 1. 1 We point out that an alternative definition of symplectically star-shaped resp. convex is to require = instead of > above. That this is actually more general is explained in Section 2 where we construct examples of curves with all possible sign combinations.
Both forms are well defined and have no zeroes. Inspired by the affine isoperimetric inequality, we are interested in the relative extrema of γ 3 ω(γ , γ ) dt constrained by γ ω(γ, γ ) dt. In fact, we consider a more general variational problem: describe the curves γ that are the relative extrema of γ ω(γ , γ ) a dt constrained by γ ω(γ, γ ) dt where a is a real exponent. We point out that the case a = 1 3 corresponds to the affine isoperimetric inequality. Before we come to our main results we point out that the corresponding metric problem, extremizing the L 2 -norm of the (metric) curvature on a class of plane curves, is a widely studied 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 are studied and [7] where the corresponding gradient flows are developed. The latter is a natural next step also for the affine context from this article. See [11] for an affine analog of the Euclidean curve shortening flow.

Main results
We prove that such extremal curves of this variational problem lie in symplectic affine 2-planes, and therefore the problem reduces to a 2-dimensional one (Proposition 3.4).
For a ∈ [ 1 2 , 1], one has rigidity: the extremal curves are multiply traversed conics (Propositions 3.7 and 3.8). The same rigidity result holds, although for a different reason, in the case of the affine isoperimetric inequality, a = 1 3 , (Theorem 3). In Theorem 1 we describe non-trivial infinitesimal deformations of multiply traversed conics in the class of extremal curves: if a = 1 3 , then the n-fold ellipse is infinitesimally rigid; otherwise, a non-trivial infinitesimal deformation of the n-fold ellipse exists if and only if for some positive integer k = n.
Theorem 2 concerns second order deformations of conics: for a < 0, the circle γ 0 is a local minimum of B a ; for a ∈ (0, 1 3 ), it is a local maximum; for a > 7 5 , 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 2 − 2 k 2 − 4 for some positive integer k.
In Section 7 we present examples of extremal curves and formulate some conjectures about them.
This introduction would not be complete if we failed to mention another reason for our interest in centroaffine differential geometry, namely its close relation with the Korteweg-de Vries 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 Lamé's equation thoroughly studied in this context in a recent paper [4].
In this section we construct curves γ with all possible sign combinations of the quantities ω(γ, γ ) = 0 and ω(γ , γ ) = 0. In particular, we assume that all curves are immersed. We start with a remark concerning symplectically star-shaped curves. The sphere S 2n−1 ⊂ R 2n carries a contact structure defined by the symplectic orthogonal complement to the position vector. A symplectically star-shaped curve projects to a transverse curve in S 2n−1 . A similar remark applies to the contact RP 2n−1 , the projectivization of R 2n . If ω(γ, γ ) > 0 then γ is positively transverse and < corresponds to negatively transverse. A somewhat similar interpretation for the condition ω(γ , γ ) = 0 is derived in Lemma 2.2 in case of R 4 .
We consider the unit sphere S 3 ⊂ C 2 = R 4 with its standard contact structure. The standard contact form at a point q ∈ S 3 is ω(q, ·)| TqS 3 . Let γ(t) be a smooth closed Legendrian curve in S 3 , i.e., ω(γ, γ ) = 0. Then Jγ is a vector normal to γ inside the contact plane. Here J is the complex structure on C 2 . The following lemma is well known, see, e.g., [3]. Lemma 2.1 Pushing a closed Legendrian curve γ slightly in the direction of Jγ , resp. −Jγ , inside S 3 yields a negative, resp. positive, transverse curve. Thus, a Legendrian curve in S 3 ⊂ C 2 can be deformed into a symplectically star-shaped curve.
The tangent Gauss map sends a Legendrian curve in S 3 to the space of oriented Legendrian great circles, that is, to the oriented Lagrangian Grassmannian Λ + 2 . The image of the Gauss map is a smooth curve which has vanishing differential precisely at the points corresponding to inflection points of the Legendrian curve.
Proof. The condition ω(γ , γ ) = 0 is invariant under reparameterization of γ, so we may assume that γ is parameterized by arc-length.
Consider the Reeb field of the standard contact form on S 3 and let pr : S 3 → S 2 be the respective Hopf fibration. The projection p takes the Legendrian great circles in S 3 to the great circles in S 2 . Closed Legendrian curves are projected to closed immersed curves in S 2 that bound a region with signed area that is a multiple of 2π and, conversely, such a spherical curve lifts (non-uniquely) to a closed Legendrian curve in S 3 .
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 spherical curve with everywhere positive geodesic curvature and bounding area 2πk for some k ∈ Z lifts to a Legendrian curve which is free from inflections. Starting with a closed spherical curve with everywhere positive geodesic curvature, the area condition is easily arranged by adding appropriately sized kinks to the curve.
Finally, we note that the map (z 1 , z 2 ) → (z 1 ,z 2 ) of C 2 preserves the contact structure and the property of γ being Legendrian but changes the sign of ω(γ (t), γ (t)) to the opposite. Therefore we can have all four combinations of signs of the quantities ω(Γ, Γ ) and ω(Γ , Γ ) 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 different Hopf fibration pr : S 3 → S 2 , whose fibers are Legendrian. This Hopf fibration takes Legendrian great circles to circles (of some, possibly zero radius) on S 2 . The projection of a smooth Legendrian curve γ is a wave front, possibly with cusps, and the inflections of γ correspond to the vertices of the spherical curve pr(γ). If pr(γ) is smooth and convex, it must have at least four vertices (the 4-vertex theorem), without convexity at least two vertices, but if pr(γ) has cusps it may be vertex-free.

Towards the solution of the variational problem
In this section we describe the setting of our variational problem and prove that extremizers are contained in symplectic affine 2-planes. This allows us to reduce the problem to the 2-dimensional case.
The problem is to find the closed symplectically star-shaped and symplectically convex curves that are extrema of B a (γ) := ω(γ , γ ) a dt , a ∈ R \ {0}, subject to the constraint given by A(γ) := ω(γ, γ ) dt. More precisely, we consider the space P T of T -periodic symplectically star-shaped and symplectically convex curves in R 2n and consider A, B a : P T → R and ask for extrema of B a subject to A = c 0 , that is, we want to describe the set Let us note that both functionals, A and B a , are translation invariant, that is, do not depend on the choice of the origin (as long as the curve remains star-shaped). We start with a few simple observations. Lemma 3.1 For c 0 > 0 and c 1 > 0 there is a natural bijection (by rescaling) from P T to itself inducing a bijections Proof. Let γ 0 ∈ P T and consider it follows that the bijection γ → c 1 c 0 1 2 γ just rescales B a by a fixed factor, i.e., induces the (1) implies that, if B a has critical points at all, they appear in R >0family and the critical value is necessarily 0. Proof. We consider the bijection P T → P 1 given by Γ(t) := γ(t/T ). Then as it has to be since 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 period and with fixed constraint. Proposition 3.4 A critical curve is contained inside a symplectic affine 2-plane of (R 2n , 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 v The derivative of the first integral in direction of v computes to where we used integration by parts and skew-symmetry of the symplectic form in the equality sign.
Similarly, setting F := ω(γ , γ ) a−1 , the derivative of the second integral can be expressed as Since v is arbitrary and ω is non-degenerate, the criticality condition implies that the vectors γ , γ , γ are linearly dependent, and since F = 0, one has γ = f γ + gγ for some periodic functions f (t), g(t).
It follows that the bivector γ ∧ γ satisfies the differential equation (γ ∧ γ ) = g(γ ∧ γ ), that is, remains proportional to itself. Hence the curve γ is planar. It follows by integration that the curve γ lies in an affine 2-plane, parallel to the plane spanned by γ and γ . 2 Proposition 3.4 allows us to reduce the problem to two dimensions as follows. A critical curve γ is contained in an affine 2-plane. Let V be the two dimensional linear space parallel to this affine 2-plane, and W be its symplectic orthogonal. Then we may write γ = γ 1 ⊕ γ 2 ∈ V ⊕ W . Since γ ∈ V we conclude that γ 2 is constant. We claim that γ 1 is also a critical curve. Indeed, since W is symplectically orthogonal to V we compute i.e., γ is symplectically star-shaped and A(γ) = A(γ 1 ). Trivially, γ 1 is also symplectically convex with B a (γ) = B a (γ 1 ). Now, that we have reduced the problem to two dimensions, we denote 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-shaped curve so that [γ, γ ] = 1, that is, γ is in its centroaffine parameterization.
Thus, we can reformulate our goal as describing all extremal curves of the functional B a (γ) := p a (t)dt on the space of periodic curves in centroaffine parameterization with p > 0. Here a ∈ R \ {0} is fixed. To be quite precise, we point out that we, of course, consider the space of all periodic curves in centroaffine parameterization of a fixed period, mostly, 2π or a multiple of 2π. As explained in the introduction, the case a = 1 3 is of special interest since this corresponds to the affine isoperimetric inequality. is invariant under translations. This is the only value of a with this additional symmetry.
Proof. Point-wise multiplication of a curve with a matrix from SL(2, R) does not change its centroaffine parameterization nor its centroaffine curvature.
A sufficiently small translation of a curve γ keeps the properties of being symplectically star-shaped and convex but in general not the centroaffine parameterization of γ. After reparametrization, the shifted curveγ = γ + c is in centroaffine parameterization, i.e.γ(τ ) = γ(t(τ )) + c and [γ, dγ dτ ] = 1. Its centroaffine curvaturep satisfies for all symplectically star-shaped and convex curves γ. 2 We now derive a critical point equation for B a in terms of the function F = p a−1 .
Proposition 3.7 For a = 1 the extremal curves γ of the functional p a (t)dt satisfy where F = p a−1 and b = 1 a−1 . If a = 1 or a = 1 2 , then p is constant and γ is therefore a conic.
Proof. First, we describe the vector fields v along γ that preserve the centroaffine parameterization of γ. Write such a field as v = gγ + f γ . Since the deformation by v is assumed to preserve the centroaffine parameterization, we conclude that [γ, v ] + [v, γ ] = 0 and hence 2g + f = 0. Thus, the vector field v has the form − f 2 γ + f γ ; the deformations keeping the centroaffine parameterization depend on one periodic function.
As in the proof of Proposition 3.4, linearizing p a (t)dt in v leads to for every vector field v as above, where we recall that F = p a−1 . Using integration by parts twice and recalling that γ = −p γ − pγ , we rewrite the integral as Using [γ, v] = f , [γ , v] = 1 2 f and another integration by parts, this integral becomes The critically condition is that this integral vanishes for all f , and we conclude that the integrand is zero. If a = 1, then F ≡ 1, and hence the criticality condition simply becomes p = 0. Therefore, γ solves γ = const · γ, i.e., γ is a conic.
Otherwise, recall that p = F b , hence p = bF b−1 F . Substitute this to the integrand and collect terms to obtain the claimed formula (2).
If a = 1 2 then b = −2, and equation (2) reduces to F = 0. Since F is periodic and positive, F is necessarily a constant, and so is p = F −2 . Thus, again γ is a conic in this case. Proof. We can integrate equation (2) to with some constant c. Note that b = −1. Since F is a positive and periodic function, we get at the minimum m := min F and the maximum M := max F of F the usual inequalities for F , leading to Thus, we arrive at the two inequalities If b + 1 > 0 or b + 2 > 0 > 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 . Remark 3.10 The solutions we construct in the proof of the lemma below are, by construction, often times multiply covered.
Proof. As a preparation, we consider the Hamiltonian formulation of the ODE (3). Hamilton's equations for the Hamiltonian function (c ∈ R is a constant which we will choose appropriately below) and are, with Q = F and P = F , equivalent to the ODE (3). We compute dH(Q, P ) = P dQ and HessH(Q, P ) = 2(b + 2)Q b 0 0 1 .

Therefore, (Q, P ) is a critical point of H if and only if
and then is a local minimum if Our We are searching for non-constant positive, periodic solutions of equation (2). I.e. we are looking for non-constant periodic orbits of H with Q = F > 0.
In case b ∈ (0, ∞) we choose c ∈ R very large and positive. Then the equation 2(b+2) b+1 Q b+1 = c determines a local minimum of F at, say, (Q 0 , P 0 = 0) with Q 0 large and positive. Therefore, the linearized dynamics given by is a very fast rotation of R 2 . That is, the linearized dynamics is periodic with very small periodic. Since (Q 0 , 0) is a local minimum, any level set {H = E} with E slightly larger than the energy H(Q 0 , 0) = 2 b+1 Q b+2 0 −cQ 0 of (Q 0 , 0) consists of a small circle (and potentially other connected components). This small circle is then a periodic orbit with period approximately that of the linearized dynamics. Since b = 0, the Hamiltonian H is not quadratic and therefore varying the energy value E also changes the period of the periodic orbit in the level set {H = E} near (Q 0 , 0). I.e. by iterating and varying the level set we can arrange any period we wish.
In case b ∈ (−1, 0) we choose c ∈ R positive and small. Then the equation 2(b+2) b+1 Q b+1 = c determines a local minimum of F at (Q 0 , P 0 = 0) with Q 0 small and positive. The linearized dynamics is again given by This is still a very fast rotation of R 2 since b < 0, b + 2 > 0 and Q 0 > 0 is small. The argument proceeds as in the previous case. In case b ∈ (−2, −1) we choose c ∈ R 2 negative and large in absolute value. The equation

2(b+2)
b+1 Q b+1 = c determines a local minimum of F at (Q 0 , P 0 = 0) with Q 0 small and positive. The linearized dynamics is yet again a very fast rotation of R 2 since b < 0, b + 2 > 0 and Q 0 > 0 is small. The argument proceeds as in the first case. 2 Remark 3.11 Equation (2) can be further integrated, which is in terms of the Hamiltonian formulation just expressing the preservation of H along solutions.
For a = 2, resp. a = 3 2 , we have b + 2 = 3, resp. b + 2 = 4, and hence F is the Weierstrass elliptic function. Since p = F b , in the first case p is also the Weierstrass function, and in the second case p is its square. In the former case, the respective equation γ = −pγ is called the Lamé equation, and it was thoroughly studied, see, e.g., [14].

Infinitesimal rigidity of multiple conics as critical curves
As a multiple conic we take the unit circle traversed n times. This is a critical curve of the functional B a (γ) = 2πn 0 p a dt, and we ask whether it admits a non-trivial infinitesimal deformation in the class of critical curves.
The functional B a is invariant under the action of SL(2, R); the respective deformations comprise a 3-dimensional space, and we consider them as trivial. In addition, if a = 1 3 , the functional is invariant under parallel translations, see Lemma 3.6. In this case, we add this 2-dimensional space to the deformations that we consider as trivial. The n-fold circle is infinitesimally rigid if it does not admit non-trivial infinitesimal deformations in the class of critical curves.
Let γ 0 (t) = (cos t, sin t) be the unit circle traversed n times, and let γ 1 = − f 2 γ 0 + f γ 0 be a vector field along it that defines its infinitesimal deformation. We assume that the period is 2πn, and accordingly, f (t) is a 2πn-periodic function.
If f (t) = 2 sin t then, again mod ε 2 , the curve γ 0 + εγ 1 is the unit circle (x + ε) 2 + y 2 = 1, and likewise for f = 2 cos t. 2 We are ready to describe the infinitesimal rigidity of the circle. Proof. The calculations below are modulo ε 2 . Let Γ = γ 0 + εγ 1 . We have Γ = −(p 0 + εp 1 )Γ, hence Therefore We calculate Compare with [9], where the Korteweg-de Vries equation is interpreted as a flow on centroaffine curves. The case of a = 1 was considered earlier (the only solution of the variational problem is a conic); therefore we assume that a = 1.
Since the perturbed curve is critical, equation (3) holds. Write the constant in this equation as c 0 + εc 1 . Then (3) implies Since q and q have zero average, we conclude that c 1 = 0. Therefore q = −2(b + 2)q. This equation has periodic solutions only when b + 2 > 0, and then q(t) is a linear combination of cos( 2(b + 2)t) and sin( 2(b + 2)t). For q to be 2πn-periodic, one has to have 2(b + 2) = k n , that is, for positive integers k, n. Note that since b = 0, one has k = 2n. Let us show that if condition (6) holds, the desired infinitesimal deformations of a multiple circle exists.
We find f (t) from the equation It follows that, modulo the kernel of the differential operator 1 2 d 3 + 2d, the function f is a linear combination of cos kt n and sin kt n . The kernel of the differential operator contributes trivial deformations, and we end up with a 2-dimensional space of deformations.
It remains to see when these deformations are trivial. Since k = 0 and k = 2n, the only "suspicious" case is k = n. In this case, a = 1 3 , and indeed, the first harmonics give trivial deformations, corresponding to parallel translations. 2 Remark 4.2 We note that, in agreement with Lemma 3.8, does not take values in [ 1 2 , 1]: if k > 2n, then a > 1, and if k < 2n, then a < 1 2 .

Second order deformations of conics
Here we investigate how the functional B a (γ) = p(t) a dt changes under a second order deformation of the unit circle. We recall that the functional is SL(2, R)-invariant, see Lemma 4.1.
We assume that the period is 2π and that the curves have the centroaffine parameterization [γ, γ ] = 1. Let Γ = γ 0 + εγ 1 + ε 2 γ 2 , ignoring the higher order terms in ε. As before, γ 0 (t) = (cos t, sin t), hence γ 0 = −γ 0 , and The condition [Γ, Γ ] = 1 implies We have; Γ = −P Γ, hence Then Proof. Let We have 4 − 5k 2 + k 4 = (k 2 − 1)(k 2 − 4) ≥ 0, and the sum is positive unless the only non-zero term is for k = 1, that is, f is a first harmonic. 2 Now consider the general case: In terms of the Fourier coefficients of f , this is The expression in the parentheses equals We also note that the quadratic term of P a contains the factor a.
Theorem 2 For a < 0, the circle γ 0 is a local minimum of B a ; for a ∈ (0, 1 3 ), it is a local maximum; for a > 7 5 , it is a local minimum, and in other cases the Hessian is not signdefinite. The Hessian is degenerate (with 1-dimensional kernel) if and only if a = k 2 −2 k 2 −4 for some positive integer k.
Proof. Let a = 1. Then the sign of (9) is that of −(k 2 − 4), which is positive for k = 1 and negative for k ≥ 3.
Let a > 1. Then the sign of (9) is positive for sufficiently large k. The Hessian is positivedefinite if this sign is positive for all k. When k = 1, the first factor in (9) is negative, and so is the second one: 1 − 3a. When k ≥ 3, the first factor is positive, and the second one is positive if and only if 9(a − 1) − 2(2a − 1) > 0, that is, a > 7 5 . Let 0 < a < 1. Then the sign of (9) is negative for sufficiently large k. The Hessian is negative-definite if this sign is negative for all k. When k = 1, the first factor in (9) is negative, and the second factor equals 1 − 3a. Thus (9) is negative if and only if a < 1 3 . If this inequality is satisfied then, for k ≥ 3, the second factor in (9) is 2 − 4a − (1 − a)k 2 < 0.
If a < 0, then the analysis of the preceding paragraph still holds, but the factor a of the quadratic 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 yields the last claim of the theorem. 2 The last statement of the theorem agrees with Theorem 1. The numbers a = k 2 −2 k 2 −4 form the sequence 1 3 , that converges to 1. Each time that a crosses an element of this "spectrum", the signature of the Hessian changes by 1.
6 The case of a = 1 3 We recall that we attempt to describe extremal curves of the functional B a = p a (t)dt on the space of periodic curves in centroaffine parameterization with p > 0. The case a = 1 3 corresponds to the affine isoperimetric inequality. In particular, the functional then is translation invariant, see Lemma 3.6. In this case, b = − 3 2 , and equation (3) reads Since F ≤ 0 at the maximum and F > 0, we conclude that c < 0. Next, integrate equation (10) to (this is the equation of a level curve of the Hamiltonian, see Section 3). Let F (t) = G 2 (t) where G(t) is also a positive periodic function. Then (11) becomes (again renaming the constants). The right hand side of (12) is a quadratic polynomial in G, and it has at least two roots because G is a periodic function that attains maximum and minimum. Since a quadratic polynomial has at most two roots, G oscillates between its maximum and minimum, and has no other critical values.
Example 6.1 Let us examine the case of a parallel translated n times traversed circle, which is a critical curve: where α(t) is a function of the centroaffine parameter t. We assume that the range of t is [0, 2π], that of α is [0, 2πn], and the radius of the circle is r.
This can be integrated: with the constants renamed. Since we can change the parameter α to obtain a simplified equation (once again renaming the constants) Then dα/dt = n(1 + A sin α) −1 . Next, Since G = p − 1 3 , we conclude that Let us continue with the general case. Let 0 < m < M be the minimum and the maximum of G, and let c = −2k 2 . Write the right hand side of (12) as k 2 (G − m)(M − G), then the differential equation becomes where we allow the square root to take values at its positive and negative branches.
Since 0 < ε < µ, the left hand side is a monotonic function of ϕ, therefore this functional equation uniquely determines the function ϕ(t).
Theorem 3 If a = 1 3 then the relative extrema of the functional B a constrained by A are multiply traversed conics.
It follows that the centroaffine curvature p(t) of the curve γ(t) is the same as that of the parallel translated n-fold circle γ 0 (t). Consider the contact element (γ(0), γ (0)). Acting on γ 0 (t) by an element of SL(2, R), we can arrange for (γ 0 (0), γ 0 (0)) to coincide with (γ(0), γ (0)). The action of SL(2, R) does not change the centroaffine curvature, hence the two curves, γ and γ 0 , satisfy the same second order differential equation and have the same initial data. Therefore they coincide. 2 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 in the following way.
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) by the exponential function.     Let us comment on a common geometrical feature of these curves. Recall the notion of the osculating circle of a smooth curve in Euclidean geometry: this is a circle that is 2-order tangent to a curve at a given point, that is, it shares the curvature with the curve. Informally speaking, the osculating circle passes through three infinitesimally close points of the curve.

Pictures and open problems
In centroaffine geometry, the role of osculating circles is played by osculating central conics. The space of central conics is 3-dimensional, and for every point of a star-shaped curve there exists a central conic that is 2-order tangent to it at this point. Central conics have constant centroaffine curvature, and the osculating central conic at point γ(t 0 ) has the constant centroaffine curvature p(t 0 ).
Similarly to the case of osculating circles the osculating central conic goes from one side of the curve to the other side if p (t 0 ) = 0. If p has a non-degenerate maximum or minimum at point t 0 , that is, γ(t 0 ) is a centroaffine vertex, then the osculating central conic lies on one side of the curve near this point.
In the following lemma we prove that the centroaffine curvature of extremal curves takes only 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 has no other critical values because the right hand side of equation (5), as a function of F , has no more roots than the number of sign changes among its three coefficients, see [10] (part 5, chapter 1, §6, No 77). 2 Looking again at the above figures it is fairly obvious that the case a = 1.4 in Figure 2, the "egg" lacks the same symmetry all other curves have. This seems related to Theorem 1 asserting that for all a of the form a = k 2 −2 k 2 −4 , k ∈ Z >2 , a circle (which is a critical curve for any a) admits a non-trivial infinitesimal deformation. Indeed, 1.4 = 3 2 −2 3 2 −4 . A more systematic computer experiment (again using the Mathematica applet by Gil Bor) leads to Figure 5 where curves corresponding to k = 4, 5, 6, 7, 8, 9 are displayed. We recall that for any value 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 2 −2 k 2 −4 there exists an extremal curve which is a "rounded (k − 1)-gon". Unfortunately, our software is currently not powerful enough to verify this. We hope to return to this point in the future. It is worth pointing out that these (k − 1)-gons seem to approach a circle for k very large. This is, at least, consistent with a → 1 when k → ∞ and the circle is indeed rigid for B 1 . As a rather special case, the egg (case a = 1.4 in Figure 2) should be considered a rounded 2-gon.
We collect some further questions.
• What is the minimal rotation number of a periodic solution of B a in dependence of a? It seems that this minimal rotation number goes to infinity with 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]? In this appendix we describe a curious symmetry of equations (5). In the discovery of this symmetry, we were motivated by Bohlin's theorem as described in Appendix 1 of [2]. Consider the family of equations dF dt 2 = uF q + vF + w, where F (t) is an unknown function and u, v, w, q are parameters. We are looking for changes of independent and dependent variables 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 3 . The orbit of the exponent q is We note that this is precisely how the permutations of four points in the projective line affect their cross-ratio.
Using the chain rule, one obtains the differential equation on G: For this equation to have the same form as (19), one needs the following relation between the exponents to hold: Thus one needs to consider six cases. We present one of them: The other five cases are analyzed in a similar way. 2 Returning to equation (5), one has q = b + 2 = 2a−1 a−1 . The S 3 -orbit of the exponent a is as follows: In particular, the orbit of 1 is {1, 1/2, 0} and the orbit of 1/3 is {1/3, 2/3, ∞}, these are special values in our study. All other orbits consist of six elements. We end with the following remarks. First, the function G (or equivalently F ) needs to be positive in order for dτ dt = G λ being an actual change of coordinates. This is always satisfied in our situation. Moreover, F is periodic if and only if G is. However, in the transition from a solution F of (5), a special case of (19), to a curve γ it seems hard to see if γ is again periodic. Therefore, the groups of coordinate changes above might or might not map closed curves to closed curves.