mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere

We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group ${\rm U}(2)$. We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in ${\rm U}(2)$. Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.


Introduction
Broadly speaking, the study of integrable geometric evolution equations for curves has led to fruitful discoveries of the interactions between the structures associated to integrability (e.g., Lax pairs, conservation laws, and Bäcklund transformations) and the geometric and topological features of solution curves.While several authors (including some of us) have extensively studied integrable realizations of the sine-Gordon and nonlinear Schrödinger equations in Euclidean geometry [2,3,16], additional integrable equations arise through investigations of flows in less familiar geometries.For example, the KdV equation appears in many geometric contexts, including flows for curves in the centroaffine plane [5,27] and null curves in Minkowski 3-space [23], and investigations of curve evolutions in the projective plane and higher-dimensional centroaffine spaces have uncovered geometric flows that realize the Boussinesq and Kaup-Kuperschmidt hierarchies [6,22].
In this article, we study flows for curves in 3-dimensional pseudohermitian Cauchy-Riemann (CR) geometry, specializing to the homogeneous geometry of the 3-sphere.Recall that a CR structure of hypersurface type (sometimes referred to as a pseudoconformal structure) on a manifold of real dimension 2n + 1 consists of a contact structure together with a compatible almostcomplex structure on the 2n-dimensional contact planes.(Additional non-degeneracy conditions are usually assumed, but these are vacuous in the case n = 1 on which we will focus.) The 3-sphere inherits its standard CR structure via its embedding as the hypersurface in C 2 comprised of the set of unit length of vectors with respect to the Hermitian inner product.The automorphism group of this structure is strictly larger than U(2); in fact, the pseudoconformal 3-sphere is preserved by the 8-dimensional group SU(2, 1) acting on C 3 by preserving a Hermitian inner product of split signature.In that setting, the 3-sphere is identified with the projectivization of the cone of non-zero null vectors in C 3 and the standard contact structure in S 3 can be expressed in terms of this inner product.
In previous work [4,21,24,25], we investigated geometric invariants and geometric evolution equations for Legendrian curves as well as curves transverse to the contact distribution in the pseudoconformal 3-sphere.The setting of this article is pseudohermitian CR geometry, a sub-geometry of CR geometry in which a contact form is specified, resulting in a compatible hermitian metric on the contact planes.The canonical connection and curvature for such structures were introduced by Webster [30].In dimension three, the simply-connected homogeneous pseudo-Hermitian CR manifolds with constant Webster curvature consist of the 3-sphere S 3 = U(2)/U(1), the universal cover of anti-de Sitter space A 3 = U(1, 1)/U (1), and the Heisenberg group H 3 .In each case, the group preserves a fibration to a 2-dimensional space form, where the fibers are transverse to the contact planes.For the case of the 3-sphere, this is the Clifford map π C : S 3 → S 2 , a geometrical version of the Hopf fibration (see Section 2.1 for details).
In this article, we will focus on closed Legendrian curves in pseudo-Hermitian S 3 , in particular their discrete invariants, how their geometry is related to that of their images under π C , and integrable geometric evolution equations that arise naturally for such curves.Among other results, we will show that there exists an infinite sequence of geometric evolution equations which induce the nth flow of the mKdV hierarchy for the curvature of the Legendrian curve γ.These evolution equations are related to geometric realizations of the mKdV hierarchy that have already appeared in the literature.Indeed, applying the Clifford projection to a Legendrian curve in S 3 produces a curve in S 2 with the same curvature (up to a factor of 1/2), and curves evolving by (1.1) project to curves evolving by flows previously identified by Goldstein and Petrich [10,11] as inducing the mKdV hierarchy (see also the works by Doliwa and Santini [7] and by Langer and Perline [17]).However, the flows we define in this work are geometrically distinctive for two reasons.First, each flow Z n is Hamiltonian with respect to a symplectic structure on the space of periodic Legendrian curves, which is not induced by any corresponding structure on S 2 .Moreover, while curves in S 2 can be lifted to Legendrian curves in S 3 , the Clifford projection does not induce a surjective map from the space of closed Legendrian curves to closed curves in S 2 .As we shall see, a closed curve in S 2 must satisfy a rationality condition on its total curvature in order to have a lift into S 3 as a closed Legendrian curve, where the resulting rational number is related to the discrete invariants of the lift, as a closed Legendrian curve.
We now summarize the contents of the paper: -In Section 2, we review the geometry of Legendrian curves in S 3 -in particular, the moving frame and curvature, and their relation to the projection under π C .We define the Legendrian lift of a regular curve in S 2 , and compute the lifts of constant curvature circles as an example.We also review the discrete invariants of closed Legendrian curves, and compute such invariants for the circle lifts.
-In Section 3, we define a symplectic structure on the space of periodic Legendrian curves in S 3 (modulo symmetries), and show that Hamiltonian vector field for total length induces mKdV evolution for curvature.More generally, we show that the entire mKdV hierarchy can be induced by geometric flows for Legendrian curves.
-In Section 4, we compute the stationary curves for the mKdV vector field Z 1 , determining which of these are closed and, possibly, periodic in time.For a selection of representative examples, we use the Heisenberg projection, a contactomorphism of the sphere punctured at a point with R 3 , to obtain planar projections that enable us to compute discrete invariants, such as the Maslov index and Bennequin number (see [8] and the literature therein for an introduction to Legendrian knots and their contact invariants.) -In Section 5, we discuss some open questions and directions for future research.

Moving frames and curvature
Let ⟨−, −⟩ be the standard Hermitian inner product on C 2 , and S 3 ⊂ C 2 denote the set of unit vectors.We will think of a regular parametrized curve γ : R → S 3 as a C 2 -valued function of parameter x.Differentiating ⟨γ, γ⟩ = 1 shows that ⟨γ x , γ⟩ is pure imaginary.The curve γ is Legendrian if it satisfies the condition We will say that γ is a unit-speed curve if ⟨γ x , γ x ⟩ = 1 identically, and we will use s in place of x whenever we are assuming a unit-speed parametrization.We let P denote the set of regular parametrized Legendrian curves in S 3 , and P 1 denote the subset of those with unit speed.The group U(2) of unitary matrices acts transitively on S 3 , preserving the Hermitian inner product, and thus inducing actions on P and P 1 .Differentiating ⟨γ, γ⟩ = 1 and using both the Legendrian condition (2.1) and the unit-speed condition ⟨γ s , γ s ⟩ = 1, shows that the matrix Γ = γ γ s takes value in U(2).We will refer to this as the U(2)-valued moving frame of γ.
Lemma 2.1.The moving frame Γ satisfies the Frenet-type equation where we designate k(s) as the curvature function of γ.
Proof .Expanding γ ss in terms of the moving frame gives where the first coefficient is determined by differentiating (2.1), and differentiating the unitspeed condition ⟨γ s , γ s ⟩ = 1 implies the second coefficient is purely imaginary.Its imaginary part is computed by k = Im⟨γ ss , γ s ⟩. ■ For a more general parametrization, we have the following.
Lemma 2.2.If γ is a regular Legendrian curve parametrized by an arbitrary variable x, its curvature is given by Moreover, if β := ⟨γ x , γ x ⟩ 1/2 denotes the speed of γ, then the analogue of (2.3) is Proof .Since any arclength parameter satisfies ds/dx = β, then It follows that Since β is real, we obtain (2.4) by taking the imaginary part of each side of the last equation.Using (2.3), we write which, when solved for γ xx , gives (2.5).■ Definition 2.3.If γ is periodic with minimal period L, we say that γ is a closed Legendrian curve of period L, and we let |[γ]| ⊂ S 3 denote the image or trace of γ.We let P 1 L denote the space of closed unit-speed Legendrian curves in S 3 with fixed length L. If the map γ : S 1 L → S 3 is injective, we say that γ is a (parametrized) Legendrian knot.
We remark that the elements of P 1 L can be viewed as isometric immersions of the circle S 1

L
of circumference L into S 3 .Also note that the curvature function of a closed curve of period L must satisfy k(x + L) = k(x), but L is not necessarily the minimal period of k.

Clifford projections and Legendrian lifts
The well-known diffeomorphism between S 3 and the group SU(2) can be defined by mapping a unit vector w = (w 1 , w 2 ) T ∈ S 3 to the matrix We also identify the Lie algebra su(2) with R 3 via the linear isomorphism Combining these allows us to define the 2-to-1 spin-covering homomorphism σ : S 3 → SO(3) as where Ad denotes the adjoint representation of SU(2), and we take the standard inner product on R 3 .We define the Clifford map π C : S 3 → S 2 in terms of σ as follows.Let {e 1 , e 2 , e 3 } be the standard basis on R 3 ; then Since σ e iθ w differs from σ(w) by rotation that fixes e 1 , this endows S 3 with the structure of a circle bundle whose fibers are the integral curves of the characteristic vector field w → iw.Thus, condition (2.1) can be interpreted as saying that a curve in S 3 is Legendrian if and only if it is orthogonal to the fibers of the Clifford map.Proof .Let w = (w 1 , w 2 ) T be the vector of components of γ.By applying (2.7) to the standard basis, we can write σ(w) in matrix form as and compute where cos α (The fact that this has unit modulus follows from the Legendrian condition w 1 w 1 ′ + w 2 w 2 ′ = 0 and the unit-speed condition ) The last equation shows that σ(w) gives a parallel orthonormal along the Clifford projection.
In order to construct a Frenet frame for the Clifford projection, we modify σ(w).From (2.8), one can check that (Note that the first column of σ(w), which gives the value of the Clifford map, is unchanged.)In particular, setting ϕ = −α/2 and differentiating this formula shows that F = σ e −iα/2 w satisfies On the other hand, the component-wise form of (2.3) gives from which we compute This gives α s = k and the result follows.■ Remark 2.6.Conversely, suppose η : R → S 2 is a curve parametrized by x with constant speed 2 and curvature function k/2.Identify SO(3) with the oriented orthonormal frame bundle of S 2 , with the basepoint map SO(3) → S 2 is given by the first column.Then the Frenet frame F = (η, T, N ) is a lift of η into SO(3) satisfying In turn, let γ : R → S 3 be a lift of F relative to the double cover σ i.e., such that F = σ • γ , which is unique up to a minus sign.Then, if α(x) is an antiderivative of k(x), γ(x) = e iα/2 γ(x) is a unit-speed Legendrian lift of η, unique up to multiplication by a unit modulus constant.
2.2 Closed Legendrian curves in S 3 and their discrete invariants The classical invariants of closed Legendrian curves and Legendrian knots are the Maslov index (or rotation number) and the Bennequin invariant (see, e.g., [8]).We first discuss how these are computed, before passing to less familiar discrete invariants.For a given unit vector w = (w 1 , w 2 ) T ∈ S 3 , let w * = (−w 2 , w 1 ) T (i.e., the second column in (2.6)).Then vector fields w → w * and w → iw * give an orthogonal parallelization of the contact distribution on S 3 .Hence, for any γ ∈ P 1L there is a unique smooth function θ : R/(LZ) → R/(2πZ) such that and the Maslov index µ γ is the degree of θ.
| is a Legendrian knot K, we define the Bennequin invariant tb γ by the following construction.For ε ∈ R, let K ε = e 2πiε K. Then there exists a connected open interval I ⊂ (0, 1) such that K ∩ K ε = ∅ for every ε ∈ I, and the Bennequin invariant tb γ equals the linking number Lk(K, K ε ).Definition 2.7.Let η be the Clifford projection of γ ∈ P 1 L and let L η be its length.Since the Clifford map doubles the speed of the curve, there is a positive integer cl γ called the Clifford index such that 2L = cl γ L η .
We say that γ has spin 1 if σ • γ has trivial homotopy class in π 1 (SO(3)) = Z/2, or has spin 1/2 if the homotopy class is non-trivial.Similarly, we say a closed curve in S2 has spin 1 if its Frenet frame F has trivial homotopy class, and has spin 1/2 if the Frenet frame has non-trivial homotopy class.Proposition 2.8.Let γ ∈ P 1 L .Then Hence, the total curvature 1 of a closed Legendrian curve is an integer.In addition, this implies that the total curvature of the Clifford projection is the rational number µ γ /cl γ .
Proof .Differentiating (2.9), we get Comparing with (2.3) shows that θ s = k.Hence, deg(θ) = 1 2π L 0 k(s) ds.On the other hand, if we use š to denote the arclength parameter along the Clifford projection, then š = 2s and κ(š) = 1  2 k(2s), so and the second assertion follows.■ Remark 2.9.Conversely, let η be a closed curve in S 2 of length 2T , with total curvature equal to a/b, where a ∈ Z and b ∈ Z + have no common divisors.Let γ be a unit-speed Legendrian lift of η constructed as described in Remark 2.6.Then Hence, in the case of spin 1, the Clifford index is cl γ = b if a is even and cl γ = 2b if a is odd.In the case of spin 1/2, cl γ = b if a + b is even and cl γ = 2b if a + b is odd.Definition 2.10.The CR-analogue of stereographic projection is the Heisenberg projection p H , a contactomorphism from the punctured sphere to R 3 with the contact form dz − ydx + xdy.
Taking S = (−1, 0) ∈ C 2 as the omitted point, then for z 1 , z 2 ∈ S 3 \ {S}, Remark 2.11.The Maslov index and the Bennequin invariant can be computed directly from the Heisenberg projection γ The Maslov index is the turning number of the Lagrangian projection of γ H , defined by α = π z • γ H , where π z denotes the orthogonal projection from R 3 onto the xy-coordinate plane.If γ is a knot, the Bennequin number is the writhe of α with respect to upward oriented z-axis (see, e.g., [8]).

Constant-curvature examples
To illustrate the Legendrian lift, we consider lifting circles obtained by intersecting the unit sphere S 2 ⊂ R 3 with the plane and constant curvature h/ℓ as a curve on S 2 , the total curvature of C(h) is equal to h. (Note that the curvature may be negative due to choice of orientation.)Thus, the only circles with closed Legendrian lifts are those for which h is rational.Proposition 2.12.For a pair of relatively prime positive integers m, n, let C m,n denote be the circle Proof .Let γ m,n : R → S 3 be defined by The Clifford projection of γ m,n is where we define ℓ := 2 √ mn/(m + n) for short.Thus, η m,n is a parametrization with constant speed 2 of C m,n .The least period of η m,n is πℓ, while the least period of γ m,n is 2π √ mn; it follows from Proposition 2.8 that m + n is the Clifford index of γ m,n .
As in the proof of Proposition 2.8, we use š = 2s to denote the arclength parameter along the Clifford projection, and let ηm,n (š) = η m,n (š/2) be its unit-speed reparametrization.This has least period 2πℓ and curvature h/ℓ.Thus, its Frenet frame is of the form F m,n (š) = A exp (šM m,n ), where A is some fixed matrix in SO(3) and Since the Lie algebra isomorphism induced by the spin-covering homomorphism σ is On the other hand, the eigenvalues of Mm,n are ±i/(2ℓ), so it follows that the least period of Fm,n is 4πℓ, which is twice the least period of F m,n .This proves that C m,n has spin 1/2.■ Remark 2.13.Proposition 2.12 is an adaptation to the context of pseudo-Hermitian geometry of the classification of Legendrian curves with constant CR-curvature given in [25].It follows from Propositions 2.8 and 2.12 that the Maslov index of γ m,n is m − n.Computing the writhe of the Lagrangian projection of p H • γ m,n , we find that the Bennequin invariant of γ m,n is −mn.Thus, according to the classification of the contact isotopy classes of Legendrian torus knots [9], closed Legendrian curves with constant curvature provide explicit models for negative torus knots with maximal Maslov index and maximal Bennequin number.
Remark 2.14.We also note a rather curious fact: the inversion of the Lagrangian projection of p H • γ m,n with respect to the origin is an epicycloid traced by the path of a point at distance √ m+ √ m + n / √ 2n from the origin, generated by rolling a circle of radius r = m/ 2n(m + n) along a fixed circle of radius R = n/ 2n(m + n) centered at the origin (see Figure 2).Indeed, from (2.10) and (2.11), it follows that the Lagrangian projection of p By comparison, the epicycloid traced by the path of the point (a + R + r, 0), generated by rolling of a circle with radius r on the fixed circle of radius R centered at the origin, can be parametrized by Hence, if we set a = dm,n , then |[β m,n ] is obtained by rotating the epicycloid by π/2 about the origin.

Symplectic structure
Recall that P L denotes the space of regular periodic parametrized Legendrian curves γ : R → S 3 with period L in parameter x.The space P L has the structure of an infinite-dimensional manifold. 2We begin by characterizing its tangent spaces.
Lemma 3.1.Assume γ ∈ P L and let p γ(x, t) be a variation of γ, i.e., p γ(x, t) is smooth, belongs to P L for every fixed t (for |t| sufficiently small) and p . Then the variation vector field V along γ is of the form where p, q, r are real-valued L-periodic functions of x satisfying r x = 2q⟨γ x , γ x ⟩. 2 It can be shown that PL has a Fréchet manifold structure by adapting the argument used in Section 43.19 of Kriegl and Michor's monograph [15] for the group of contact diffeomorphisms.See also Lerario and Mondino [19], where a Hilbert manifold structure is discussed, and Haller and Vizman [12], where it is proven that the space of weighted Legendrian knots in a contact 3-manifold is a co-adjoint orbit of the group of contact diffeomorphisms, endowed with a Fréchet manifold structure.
Proof .We compute where several terms vanish because ⟨γ x , γ⟩ = 0. Differentiating the latter gives ⟨γ xx , γ⟩ + ⟨γ x , γ x ⟩ = 0. Using this, and ⟨γ, γ⟩ = 1, we get In particular, the deformations generated by vector fields H = iγ and R f = f (x)γ x , where f (x) is any L-periodic function, preserve the Legendrian condition.These deformations are, respectively, a constant-speed rotation along the fibers of the Clifford map, and reparametrization in x (while fixing the period).We wish to consider functionals on P L that are invariant under these transformations (which form a group); accordingly, we define Q L to be the quotient of P L by the action of this group, and let [γ] ∈ Q L denote the equivalence class of γ ∈ P L .
A tangent vector V ∈ T [γ] Q L corresponds to an equivalence class of vector fields V along γ, any two of which differ by adding the sum of a constant multiple of H and a vector field of the form R f .Thus, V has a unique representative of the form where the notation det R means that we apply a standard3 identification C 2 ∼ = R4 to each vector before taking the determinant of the resulting 4 × 4 matrix.(Note that the value of Ω [γ] (V, W) does not depend on the choice of representative.)If we write and similarly for W, then it is easy to compute that The fact that Ω is a closed 2-form follows as a special case of the calculus developed by Vizman in [28] for a 'hat pairing' of differential forms.Let S be a compact oriented k-dimensional manifold and M a finite-dimensional manifold, and let ω be a differential p-form on M and α a differential q-form on S. Then the pairing defines a differential (p + q − k)-form on F(S, M ), the space of smooth functions from S to M .Here ev : S × F(S, M ) → M is the evaluation map ev(x, f ) := f (x), pr : S × F(S, M ) → S the projection pr(x, f ) := x, and − S denotes the fiber integration.Then by [28,Theorem 1], We will show that the 2-form Ω in (3.1) can be realized as the hat pairing p ν = y ν • 1 = − S 1 ev * ν, where ν is the canonical volume form on S 3 and 1 is the constant function on S 1 .Note that ν is the pullback to the sphere of the interior product ı E µ, where µ is the standard volume form on R 4 and E is the Euler vector field r∂/∂r.
When γ is an embedding of the circle into the 3-sphere, p ν applied to a given V and W in T γ F S 1 , S 3 reduces to the so-called transgression map (see [28]) The closure of Ω follows immediately from the hat calculus formula (3.2), since dν = 0, as ν is a volume form on S 3 , and d1 = 0. ■ The symplectic form defines a correspondence between functionals on Q L (Hamiltonians) and vector fields in T Q L in the usual way.Namely, given a smooth functional H : Q L → R, the associated Hamiltonian vector field W H is defined by the correspondence In other words, whenever p γ(x, t) is a one-parameter family of Legendrian curves such that dp γ/dt| t=0 = V, then Proposition 3.3.Let A denote the total arclength functional on Q L , defined by Then the Hamiltonian vector field for 4A is equivalent to Proof .Let γ ∈ P L , let p γ(x, t) be a variation of γ, and let Using (2.4), we compute Because A is invariant under period-preserving reparametrizations, we may assume that γ has constant speed c = ⟨γ x , γ x ⟩ 1/2 .Then ⟨γ xx , γ x ⟩ = ic 3 k from (2.4), and integrating (3.4) gives Setting this equal to we get r W = 1 2 k.Hence, Corollary 3.4.Let γ ∈ P 1 be a unit-speed curve (not necessarily periodic), let p γ(x, t) be a variation of γ through unit-speed curves, and let V be as in (3.3).Then p s = kq and r s = 2q.
Proof .If we set p γ x , p γ x = 1 identically and x = s in (3.4), we obtain Then substituting the expansion (2.3) for γ ss gives p s = kq, while r s = 2q follows from Lemma 3.1.■ We now consider the evolution of the curvature function k induced by vector fields in T γ P. (Again, for what follows it is not necessary to assume periodicity.)Proposition 3.5.Let γ(x, t) be a family of regular Legendrian curves and let for real-valued functions p, q, r depending on x and t.Let β(x, t) = ⟨γ x , γ x ⟩ 1/2 be the speed function.Then the curvature k satisfies the PDE In particular, if we have a family of unit-speed curves, then x = s and Proof .From differentiating (2.4), we have Recall from (2.5) that γ xx = −β 2 γ + bγ x where b := β −1 β x + iβk.Next, we differentiate the expansion (3.5) twice, substitute into (3.8), and use the inner product formulas As well, from the proof of Proposition 3.3, we have The expression for k t then follows by substituting these expressions into (3.8) and simplifying.Then (3.7) follows by setting β = 1 and using the relation p s = kq from Corollary 3.4.■ Along a unit-speed curve γ ∈ P 1 L , the Hamiltonian vector field W 4A from Proposition 3.3 has a representative which satisfies the conditions from Corollary 3.4, and is thus tangent to P 1 L .By Proposition 3.5, this vector field induces the curvature evolution which differs by a Galilean transformation from the mKdV equation (with s as spatial variable).
In what follows, we will identify an infinite hierarchy of Hamiltonian vector fields that induce evolution equations for curvature which belong to the mKdV hierarchy.

The mKdV hierarchy
In this subsection, we review the recursive construction of the mKdV hierarchy.What follows is essentially the same as in Olver [26], with some changes in notation; for example, we use u to denote a scalar function of spatial variable s and time t, and let u 0 = u and u 1 , u 2 , . . .denote its successive s-derivatives.In this notation, the mKdV takes the form On the space J[u] of differential polynomials in finitely many of the u j , we will need the total s-derivative operator δ and the Euler operator E, defined respectively by The mKdV hierarchy is an infinite sequence of evolution equations where M 1 = u 1 and M 2 is the right-hand side of (3.10).It can be constructed recursively as follows.There is an infinite sequence {ρ j } of differential polynomials in u with ρ 1 = 1 2 u 2 and satisfying where Then M j = DEρ j (e.g., M 1 = u 1 ).Since D, E are a Hamiltonian pair of skew-adjoint differential operators (in the sense of [26, Definition 7.19]) each member of the mKdV hierarchy is a bi-Hamiltonian system; moreover, each ρ j is a conserved density for each of these flows (see [26,Theorem 7.24]).It follows from (3.11) that the M j satisfy the recurrence M j+1 = RM j , where while the densities satisfy the recurrence Eρ j+1 = R * Eρ j for R * = D −1 • E. For example, It also follows from M j+1 = δ 2 + δuδ −1 u M j that there are differential polynomials N j such that uM j = δN j .Let L j = 2Eρ j .The connection between the ingredients of the mKdV hierarchy and deformations of Legendrian curves becomes apparent when we observe that if then by Corollary 3.4, p, q and r satisfy the same relationships as N j , M j and L j , respectivelyprovided we replace u by k.Moreover, from (3.7) the curvature of a unit-speed curve, under flow by V, evolves by where again we replace u by k in the expression for R. Matters being so, we define the following mKdV-type vector fields (From now on, we will take L j , M j , N j , as well as the operators for the mKdV hierarchy, as having u replaced by k.)As noted, these are tangent to the submanifold P 1 , and it is easy to see that on unit-speed curves they induce curvature evolutions by linear combinations of members of the mKdV hierarchy (see below).However, these vector fields are defined on the larger space P, and in the periodic case we will explore their properties in relation to the symplectic structure.For example, when we restrict our attention to the space P L of periodic curves, the vector field is equivalent to the Hamiltonian vector field from Proposition 3.3.Similarly, we will show below that the rest of the vector fields V j represent Hamiltonian vector fields on Q L .
Proof .Let γ ∈ P L .By definition of Ω [γ] , the statement is equivalent to L 0 L m (L j ) x dx = 0.Because this equation is invariant under arbitrary reparametrizations, it suffices to verify this when γ ∈ P 1  L and x = s, an arclength parameter.The rest of the argument is essentially the same as the last step in the proof of [26,Lemma 7.25].■ To show that the V j are Hamiltonian, we need to compute the variations of reparametrizationinvariant integrals involving the curvature and its derivatives.Lemma 3.7.Let γ ∈ P L , let p γ( • , t) ∈ P L be a smooth variation of γ, and let V be as in (3.3), but with the subscript V omitted from the components p, q, r.Let k j denote the jth derivative of curvature with respect to arclength and let β = ⟨γ x , γ x ⟩ 1/2 .Then for the functional where Proof .By straightforward differentiation, where the dot indicates ∂/∂t.We wish to express km in terms of k.For this purpose, note that Applying this to k j , we obtain We substitute this in (3.12), use dx = β −1 ds to apply integration by parts with respect to s, and substitute expressions for k from (3.6) and for β from (3.9), giving It is easy to check that and thus the terms in dH[V] involving p make up an exact x-derivative of a periodic function, whose integral is zero.This gives the desired expression.■ Proposition 3.8.Each mKdV vector field V j represents a Hamiltonian vector field on Q L .
Proof .Let γ ∈ P L , and let functional H and vector field V be as in Lemma 3.7.Because of reparametrization invariance of H, we can assume that γ is parametrized with constant speed β.Then, using r x = 2β 2 q, we have Applying integration by parts to the second derivative term gives Since (kEf − Ff ) s = k(Ef ) s from (3.13), we can express this in terms of an mKdV recursion operator applied to Ef : where δ denotes the total s-derivative.
Recall that the third 'r' component of V j is L j .Since L j = 2Eρ j and R * is the recursion operator for the L j , then Using this inductively, we have In particular, if we set f = j k=1 (−4) j−k ρ k , we see that Since it has already been shown that V 1 is Hamiltonian, it follows that V n is Hamiltonian for any n ≥ 1. ■ From (3.7), when restricted to unit-speed curves, the Hamiltonian vector field V j induces the curvature evolution for j ≥ 1.For the sake of convenience, we define V 0 = γ x , which induces the translation flow k t = k s .If we define then Z n induces the curvature evolution since M 1 = k s .Using (3.14) and telescoping, this simplifies to Thus, Z n induces the (n + 1)st mKdV equation for curvature.
For later use, we now derive the evolution of the Frenet frame under flow Z 1 .
Lemma 3.9.Let p Γ(s, t) be the U(2)-valued Frenet frame for a smooth family of curves p γ(s, t) parametrized by arclength s, such that where p k(s, t) is the curvature.Then p ΓP , where is a solution of the third-order ODE where prime denotes d/ds.Obviously the converse also holds: if k is a solution of (4.1), then p k(s, t) := k(s − at) is a traveling wave solution of the mKdV equation.
Remark 4.1.Equation (4.1) implies that the Clifford projection of an s-loop is a closed elastic curve in S 2 (see, e.g., [13] and the literature therein).However, due to the constraint on the rationality of the total curvature, the Legendrian lifts of closed elastica of S 2 are, in general, not closed.
We will refer to a unit-speed Legendrian curve whose curvature is a non-constant periodic solution of (4.1) as an s-curve; thus, while s-loops are closed, s-curves are not necessarily closed despite having periodic curvature.
The study of s-loops is organized in two steps.The first is the analysis of the s-curves (obtained from periodic solutions of (4.1)) and the second is the derivation of the closure conditions, via integration of the Frenet equations (2.2).We describe this scheme for the general case, suppressing some details, before we apply it to a specific class of stationary curves.

The general scheme
By integrating (4.1) twice, we obtain the two conservation laws Thus, k is determined by inverting an Abelian integral along the phase curve y 2 + P a,b,c (x) = 0, where P a,b,c is the fourth-degree polynomial As a consequence of the Poincaré-Bendixon theorem, periodic solutions of (4.2b) exist if and only if P a,b,c possesses either four distinct real roots e 1 > e 2 > e 3 > e 4 = −(e 1 + e 2 + e 3 ), which we will call the dnoidal case; or two distinct real roots e 1 > e 2 and two complex conjugate roots − 1 2 (e 1 + e 2 ) ± ie 3 where e 3 > 0, which we will call the cnoidal case; or three distinct real roots, one them with multiplicity two, which is a degenerate limit of the cnoidal case.
In each case, the coefficients a, b and c of P a,b,c can be written as functions of real parameters (e 1 , e 2 , e 3 ).We will refer to e = (e 1 , e 2 , e 3 ) as the modulus of k.
Assuming that γ is an s-curve, we can arrange (by translating in s, and reversing the orientation of γ if necessary) that k(0) = e 2 , from which it follows that k ′ (0) = 0.With this choice of initial condition, the modulus e uniquely determines k(s).Moreover, any other s-curve with the same modulus is of the form Aγ 0 (s + c) for some matrix A ∈ U(2) and constant c, and is thus congruent to γ.Consequently, the congruence classes of s-curves of dnoidal type are in one-to-one correspondence with the elements of D = (e 1 , e 2 , e 3 ) ∈ R 3 | e 1 > e 2 > e 3 > −(e 1 + e 2 + e 3 ) , while the congruence classes of s-curves of cnoidal type are in one-to-one correspondence with the elements of Sets C and D are the moduli spaces of the s-curves of dnoidal and cnoidal type, respectively.
The curvature of an s-curve with modulus e = (e 1 , e 2 , e 3 ) is given by k(s) = q 11 + q 12 dn 2 (qs, m) q 21 + q 22 dn 2 (qs, m) , dnoidal case, k(s) = q 11 + q 12 cn(qs, m) q 21 + q 22 cn(qs, m) , cnoidal case, ( where cn and dn are the Jacobi elliptic functions, m is the square of the Jacobi modulus [18], and coefficients q ij and parameters q, m are certain functions of e (see [1, formulas (256.00) and (259.00)]).In the degenerate cnoidal case, m = 0 and cn(qs, m) is replaced by cos(qs); in the sequel, we will not consider this case.
The wavelength (i.e., the least period of the curvature) is ω = 2K(m)/q, dnoidal case, 4K(m)/q, cnoidal case, ( where K(m) denotes the complete elliptic integral of the first kind.By integrating the curvature over its wavelength, we obtain the quantum of total curvature which can be expressed in terms of K(m) and the complete elliptic integral of the third kind Π(n 1 , m), where A, B and n 1 are functions of e (see in [1, formulas (340.03) and (341.03)]).By Proposition 2.8, rationality of the quantum of curvature is a necessary condition for an s-curve to be closed; however, this is not sufficient.In order to deduce sufficient conditions, we next define the conserved momentum and show how explicit s-curves can be obtained through integration by quadratures.The momentum.Let h 0 (2) be the vector space of traceless 2 × 2 hermitian matrices.For every modulus e, we consider the map H : R → h 0 (2) defined by where k is as in (4.3).From (4.2), it follows that H has constant eigenvalues ±λ, where Given the expression for λ 2 in terms of the components of e, it is straightforward to use calculus to prove that it never vanishes in either the cnoidal or dnoidal cases.Moreover, equation (4.2) implies that It follows that, if Γ is the moving frame along an s-curve γ with curvature k, then where m is a constant element of h 0 (2), called the momentum of γ.By multiplying γ by a matrix in U(2) if necessary, we can assume that

Integrability by quadratures
Let V j : R → C 2 for j = 1, 2 be the periodic maps Proposition 4.2.For every s ∈ R, V 1 (s) and V 2 (s) belong to the ∓λ-eigenspaces of H(s), respectively.They are everywhere linearly independent if γ is of dnoidal type, or if γ is of cnoidal type with generic modulus (i.e., 8e 2 + b + 4λ ̸ = 0).In the exceptional (i.e., non-generic cnoidal) case, V 1 and V 2 vanish at s = hω for each h ∈ Z (where ω is the wavelength in (4.4)), and are linearly independent everywhere else.
Proof .The first part of the statement is a straightforward consequence of (4.9).Moreover, if V 1 and V 2 are nonzero then they are linearly independent as they are eigenvectors for different eigenvalues.Since V 1 , V 2 are periodic, we can assume that s ∈ [0, ω).Note that the derivative k ′ vanishes at s = 0 or s = 1  2 ω and is nonzero elsewhere, so V 1 (s), V 2 (s) are nonzero if s ̸ = 0 and s ̸ = 1 2 ω.Note also that k(0) = e 2 and k 1 2 ω = e 1 .Thus, we have Hence, it suffices to prove that V 1 is nowhere zero in the dnoidal or generic cnoidal cases, and that, in the exceptional case, V 1 vanishes only at s = 0. Consider the cnoidal case.In this case, expressing a in terms of e 1 , e 2 , e 3 gives for one half of the imaginary part of the upper entry of V 1 1 2 ω .Suppose this vanishes; then Using this equality and writing λ and b as a functions of e 1 and e 2 , minus one times the bottom On the other hand, if 4λ = −(8e 2 + b), then squaring both sides and using (4.10) gives e 2 2 + 2a − 8 = 0, so V 1 (0) = 0. Consider now the dnoidal case.In this case, the analogue of (4.10) is Suppose that left-hand side which is one half of the imaginary part of the upper entry of V 1 1 2 ω vanishes.Recall that the roots e 1 , e 2 , e 3 , e 4 have been ordered so that e 1 > e 2 > e 3 > e 4 .Since their sum is zero, root e 1 is positive.Then, setting the right-hand side of (4.11) to zero and solving gives Since e 2 > e 3 the left hand side is strictly negative, and thus V 1 (0) ̸ = 0. ■ Let J = R in the generic case and J = R \ {hω} h∈Z in the exceptional case.From (4.7), it follows that Γ(s)V 1 (s) and Γ(s)V 2 (s) are ∓λ-eigenvectors of m, for every s ∈ J. Since these vectors are fixed up to scalar multiple, it follows that there exist smooth functions ℓ j : J → C, j = 1, 2 such that d ds ΓV j = ℓ j ΓV j , j = 1, 2. (4.12) Using (2.2), we rewrite (4.12) in the form d ds From (4.6) and (4.9), we can express V j and its derivative in terms of k and k ′ .Solving (4.13) for ℓ 1 and ℓ 2 , we obtain Re On the other hand, (4.12) implies that Γ| J e − ℓ j ds V j = Ξ j , j = 1, 2, where Ξ 1 and Ξ 2 are locally constant maps with values in the ∓λ-eigenspaces of m.Since m is in diagonal form, Ξ 1 = ξ 1 1 , 0 T and Ξ 2 = 0, ξ 2 2 T , where ξ 1 1 , ξ 2 2 are nonzero constants.Choosing appropriately the primitives of the ℓ j -functions, we may assume ξ 1 1 = ξ 2 2 = 1.Hence, the Frenet frame along an s-curve with modulus e is given by where

Phase-symmetrical s-curve of cnoidal type
We now specialize the general scheme to s-curves and s-loops of cnoidal type whose phase curves are symmetrical about the imaginary axis; we call these ϕ-curves and ϕ-loops for short.The moduli of a ϕ-curve satisfy e 2 = −e 1 .We drop the dependence upon e 2 and we identify the moduli space of ϕ-curves with the quadrant C 0 = (e 1 , e 3 ) ∈ R The function Φ 2 is real-analytic on C 0 \ C * 0 and has a jump discontinuity on C * 0 .However, its regularization where 0 , is real analytic on C 0 .Since Φ 2 differs from Φ 2 by a rational number, we have the following.
Corollary 4.9.A ϕ-curve with modulus e is closed if and only if Φ 2 (e) ∈ Q. Definition 4.10.Let γ be a ϕ-loop with modulus e such that Φ 2 (e) = q ∈ Q.We call q the characteristic number of γ.
An experimental analysis of the symplectic gradient of Φ 2 leads to the following observations (see also Figure 3): The map Φ 2 is a submersion from C 0 onto the interval J 1/2 = (1/2, +∞).For r ∈ J 1/2 , let Σ r denote the fiber (or level set) Φ −1 2 (r).Then Σ r intersects C * 0 transversely in a single point.In particular, for every q ∈ J 1/2 ∩Q, the points along Σ q correspond to a 1-parameter family of distinct congruence classes of ϕ-loops with characteristic number q.
If r ∈ (1/2, 1), the fiber Σ r is bounded, lies above Σ 1 , and has two boundary points along the e 3 -axis: the upper boundary point e + r = 0, e + 3,r where e + 3,r > 4, and the lower boundary point e − r = (0, e − 3,r ) where 2 < e − 3,r < 4. When e ∈ Σ r tends to e ± r , the tangent to Σ r at e tends to a horizontal line.
If r > 1, the fiber Σ r lies below Σ 1 and has a unique boundary point e − r = (0, e − 3,r ), with e − 3,r ∈ (0, 2).As in the previous case, when e ∈ Σ r tends to e − r , the tangent to Σ r at e tends to the horizontal line e 3 = e − 3,r .In addition, the e 1 -axis is an asymptote of Σ r .For q ∈ J 1/2 ∩ Q, we call Σ q the modular curve of q.The exceptional point Σ q ∩ C * 0 is denoted by e * q .Then Σ q \ {e * q } has two connected components: Σ − q ⊂ C − 0 and Σ + q ⊂ C + 0 .

Congruence class representative
For a given modulus e, we pick a representative ϕ-loop γ e , uniquely specified by initial conditions where We call this is a standard ϕ-loop; clearly, any ϕ-loop is congruent to a standard one.From now on we assume that ϕ-loops are in their standard forms.Definition 4.11.We call the 1-parameter family G q = {γ e | e ∈ Σ q } of ϕ-loops the isomonodromic family of q.The terminology is motivated by the fact that every loop in G q has monodromy M q = e i2π q 0 0 e −i2π q .
(The diagonal form of the monodromy is the reason for our choice of initial conditions for standard loops.)Finally, we discuss the geometry of ϕ-loops, along with several examples.

Geometric features
Let e ∈ Σ q for q = m/n ∈ J 1/2 ∩ Q.From formula (4.15) for the Frenet frame, setting Φ 1 = 0, Φ 2 = q along Σ q , and using the properties of the map Φ 2 , we derive the following results: If n is odd, the spin s and the Clifford index cl of γ e are both 1.If n is even, s = 1/2 and cl = 2.
The image of γ e is invariant by the group of order n generated by the monodromy, while the image of the Clifford projection η e is invariant by the group generated by the rotation of 2π/(sn) around the x-axis.The image of η e * q passes through each of the poles N ± = (±1, 0, 0) ns times.If e ̸ = e * q , the image of η e is bounded by the planes x = η 1 e (0) ∈ (0, 1) and x = η 1 e (ω e /2) = −η 1 e (0) ∈ (−1, 0), where η 1 e denotes the first component of η e .
Let S 1 x ⊂ S 2 be the equator S 2 ∩ Oyz oriented counterclockwise with respect to i = (1, 0, 0).Taking the homotopy class of S 1 x as generator, we identify the fundamental group π 1 S 2 \ {N ± } with Z. Similarly, if we let S 1 z ⊂ R 3 \Oz be the unit circle centered at the origin and contained in the plane z = 0, equipped with the counterclockwise orientation with respect to upward oriented z-axis, then that generator allows us to identify the fundamental group π 1 R 3 \ Oz with Z.Using these identifications, for a given e ∈ Σ + q the homotopy class q , the homotopy classes [η e ] and γ e are both 2sm sign(n − m).
As e limits to e ± q , η e tends to the (standard) Legendrian lifts of the corresponding multiplycovered circles.
If e ∈ Σ q , the Clifford projection η e is a simple curve if and only if m = n − 1 and ∥e − e + q ∥ < ε q , where ε q > 0 depends on q = (n − 1)/n.There exist countably many e ∈ Σ q such that the evolution γe (−, t) of γ e is periodic in t.
More precisely, let P q : Σ q → R be the real-analytic function and suppose its range is the interval I q ⊂ R. From Proposition 4.5 and Remark 4.6, it follows that the evolution γe (−, t) of a ϕ-loop γ e ∈ G q is periodic in t if and only if P(e) = m/ n ∈ Q.The time period is 2π nn/hλ e , where h = gcd(n, m).

Discussion
To summarize, we have shown that, in the context of pseudo-Hermitian geometry on S 3 , there are flows for Legendrian curves that induce curvature evolution by any integrable PDE in the mKdV hierarchy.Moreover, we constructed a natural symplectic structure on the space of periodic Legendrian curves relative to which each of these flows is Hamiltonian.For the flow Z 1 which induces evolution by the mKdV equation itself, we have carried out a detailed analysis of curves that are stationary (i.e., whose flows are congruent to the initial curve), identifying closure conditions and obtaining a complete description of periodic stationary curves in a significant special case.These results naturally suggest further questions and directions for research.First, it is worth highlighting that the closure conditions for stationary curves could only be obtained because these curves are integrable by quadratures (see Section 4.2).In fact, Z 1 -stationary curves arise as projections to S 3 of trajectories of a completely integrable contact-Hamiltonian system on U(2) × R 3 .It is natural to ask if this holds for higher flows in the hierarchy.That is, for n > 1 are Z n -stationary curves also the projections of the trajectories of some completely integrable finite-dimensional contact-Hamiltonian system?
Next, as a completely integrable PDE the mKdV equation has a rich structure, including infinitely many conservation laws (as mentioned above), but also a Bäcklund transformation which generates new solutions from old ones [14,29].Since this transformation can be realized as a gauge transformation or 'dressing' at the level of the Lax pair, and these Lax equations are equivalent to the AKNS-type system satisfied by our U(2)-valued Frenet frame, it is natural to ask if there is a geometric transformation for Legendrian curves that corresponds to the Bäcklund transformation for curvature.If available, this transformation could be used to generate new and interesting solutions to our flows, starting with some of the stationary curves we have obtained above.
Last, the pseudo-Hermitian 3-sphere has, as mentioned in the introduction, a non-compact dual A 3 , the 3-dimensional anti-de Sitter space equipped with its pseudo-Hermitian structure of constant Webster curvature −1.In this case, we expect that the defocusing mKdV equation and its associated hierarchy can be realized by flows of Legendrian curves in A 3 .It is also possible that there are integrable flows for null curves in this space, which would likely be related to the KdV hierarchy.If this is the case, it is natural to ask whether the Miura transformation [20] between KdV and defocusing mKdV equations has some geometric realization that mediates between flows of Legendrian and null curves in A 3 .

Definition 2 . 4 .Proposition 2 . 5 .
The Clifford projection of a Legendrian curve γ is the immersed curve η : R → S 2 defined by η = π C • γ. (Note that γ is regular if and only if η is regular.)Let γ be a Legendrian curve parametrized by arclength s, with curvature function k(s).Then its Clifford projection has speed 2 and Frenet curvature k/2.

has spin 1 / 2 ,
and its Legendrian lifts are left-handed torus knots of type (−m, n).Moreover, these lifts have constant curvature k m,n = (m − n)/ √ mn, total curvature equal to m − n, and Clifford index m + n.

. 15 )
Proof .Since p γ is the first column of p Γ, then entries of the first column of P are dictated by the flow of p γ. Since p Γ is U(2)-valued then P takes value in u(2), and this determines the upper-right entry.Finally, since p Γ s = p ΓU where U = 0 −1 1 i p k , then equating mixed partials leads to the necessary compatibility condition U t − P s − [U, P ] = 0, which determines the lower right entry in P .■ 4 Stationary curves Let γ be a periodic unit-speed Legendrian curve with curvature function k and let p γ(s, t) be its evolution by the flow of Z n .We say that γ is stationary if p γ(s, t) = A(t)γ(s−at) for some constant a ∈ R and A : R → U(2).If γ is stationary, then the evolving curvature p k(s, t) := k(s − at) is a periodic traveling wave solution of the (n + 1)st mKdV equation.This section focuses on closed curves with non-constant curvature which are stationary for Z 1 , which we will call s-loops.By substituting p k into the mKdV equation, we see that k(s) = p k(s, 0)

2 | e 1 >e 2 3 .e 2 3 − 2 (
0, e 3 > 0 .(The zero subscript indicates our symmetric assumption.)The curvature and the wavelength of a ϕ-curve with modulus e = (e 1 , e 3 ) are given by k = −e 1 cn(ℓs, From the curvature formula it follows that the quantum Φ 1 of total curvature of a ϕ-curve vanishes.As a consequence we have: Corollary 4.7.A ϕ-curve of modulus e is closed if and only if Φ 2 (e) ∈ Q.The Maslov index of a ϕ-loop is zero.Remark 4.8.The modulus e is exceptional (in the sense of Proposition 4.2) if and only if it is an element of C * 0 = {e ∈ C 0 | |e| = 4}.In order to compute Φ 2 (e) := 1 2π ωe 0 Λ ds, we start with the expressions for a and λ in terms of the modulus, take into account that b = 0, and use(4.19).Then expression (4.14) for the Λλe 1 cn(ℓs, m) 4λ − 8e 1 cn(ℓs, m) , 20), using the standard elliptic integral (341.03)of[1], we obtain Φ

Figure 3 .
Figure 3. Selected level sets Σ r for r-values specified in the legend.
CK(m) + DΠ(n 2 , m),where C, D and n 2 are functions of the modulus e.As a consequence of (4.5) and (4.15), we have the following.