Vortex sheets in ideal 3D fluids, coadjoint orbits, and characters
François Gay-Balmaz Cornelia Vizman


theoremTheorem[section]
definition[theorem]Definition
lemma[theorem]Lemma
remark[theorem]Remark
proposition[theorem]Proposition
corollary[theorem]Corollary
example[theorem]Example










footnote1
LMD, École Normale Supérieure/CNRS, Paris, France.

footnote1

Department of Mathematics,
West University of Timişoara,
Romania.

footnote1 



Vortex sheets in ideal 3D fluids, coadjoint orbits,
 and characters
    François Gay-Balmaz^1 and Cornelia Vizman^2 
     
===================================================================






AMS Classification: 53D20; 37K65; 58D10

Keywords: coadjoint orbits, nonlinear Grassmannians, vortex sheets in ideal fluid, prequantization, characters.

 
We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of ^3 associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces  enclosing a given volume and endowed with a closed 1-form describing the vorticity density. If the vorticity density has discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle.
We determine the Hamilton equations for such vortex sheets relative to the Hamiltonian function suggested in <cit.> and prove that there are no stationary solutions having rotational symmetries.
These coadjoint orbits are shown to be prequantizable if the period group of the 1-form and the volume enclosed by the surface satisfy an Onsager-Feynman relation, as argued in <cit.> for the case of open vortex sheets (tubes/ribbons).
We find a character for the prequantizable coadjoint orbits, 
as well as a polarization group on which the character extends, 
which is a first step beyond prequantization. 





§ INTRODUCTION



Coadjoint orbits of volume preserving diffeomorphism groups are the natural phase spaces for regular and singular solutions of the Euler equations of an ideal fluid  <cit.>. 
In this paper we consider the coadjoint orbits associated to the motion of closed vortex sheet singular solutions of the 3D Euler equations, i.e., vorticities supported by closed surfaces in ℝ ^3. Such coadjoint orbits are described as a certain class of nonlinear Grassmannians of vortex sheets indexed by the topological type of support, the type of vorticity density, and the enclosed volume. This identification is made via the momentum map associated to the action of the volume preserving diffeomorphism group on manifolds of vortex sheets.
In this realisation of the coadjoint orbits, the Kirillov-Kostant-Souriau symplectic form takes a particularly simple expression reminiscent of the expression of a canonical symplectic form.

If the closed one form describing the vorticity density has a discrete period group and is nonvanishing, the vortex sheet is shown to be given by a surface of genus one fibered by its vortex lines over a circle. For the Hamiltonian suggested in <cit.>, given by the total length of the vortex lines, we determine the Hamilton equations of evolution of a vortex sheet ( Σ (t), β _ Σ(t)) to be

    Σ̇= kB ^⊥_Σ ,     β̇_ Σ = _(kB^⊤_Σ)β _ Σ.

Here the surface Σ denotes the support of the vortex sheet while the closed 1-form β _ Σ describes the vorticity density. Equations (<ref>) involve the normal and tangential components of the binormal curvature kB, with respect to the support Σ. The first equation is reminiscent of the vortex filament equation written as a binormal curvature flow.

These equations preserve the subset of circle-invariant vortex sheets (hence supported on surfaces of revolution), on which it is shown that there are no stationary solutions. 

The expression of the Hamilton equations, as partial differential equations for the parametrization of the vortex sheet, is given on the connected component of circle-invariant vortex sheets fibered by parallel circles (vortex lines).

If a genus one vortex sheet ( Σ , β _ Σ ) satisfies the Onsager-Feynman relation, written as

    a ℓ∈ 2 πℤ

with a the volume enclosed by Σ and ℓ the smallest period of β _ Σ, we show that the corresponding coadjoint orbit is prequantizable, admits a polarization group, as well as a character that extends to that group. In particular we recover and rigorously justify 
several steps towards the quantization process of vortex sheets presented in <cit.> for the case of open vortex sheets (tubes/ribbons).




Descriptions of coadjoint orbits of diffeomorphism groups in terms of nonlinear Grassmannians have been carrried out for several situations.
For the group of Hamiltonian diffeomorphisms, two classes of coadjoints orbits were described in <cit.>, <cit.> and <cit.>. 
These classes were obtained via symplectic reduction for the dual pair of momentum maps associated to the Euler equations in <cit.> and were generalized to coadjoint orbits of nonlinear flag manifolds in <cit.>.


For the group of volume preserving diffeomorphisms, coadjoint orbits associated to codimension two singular solutions of the Euler equations (e.g. vortex filaments in 3D) were described as nonlinear Grassmannians in <cit.> and <cit.>. The Hamiltonian evolution of a vortex filament in 3D space is given by the binormal curvature equation (e.g. <cit.>). It has been extended to higher dimensional codimension two submanifolds (vortex membranes) in <cit.> and <cit.>, given by a skew mean curvature equation, with respect to a regularized Hamiltonian taken as the volume of the membrane.
Coadjoint orbits associated to codimension one singular solutions were considered in <cit.> for open vortex sheets in 3D ideal fluids. A Hamiltonian was suggested in <cit.> for vortex sheets, given by the total length of the vortex lines. The motion of vortex sheet was given a geodesic description in <cit.> and has been recently formulated within a groupoid framework in <cit.>.



The paper is structured as follows. In section <ref>, coadjoint orbits of vortex sheets are described as certain classes of  nonlinear Grassmannians and the KKS symplectic form is derived in this description. In section <ref> we focus on the Hamilton equations for vortex sheets with discrete period group and nonvanishing vorticity density. We also consider the invariant submanifold of circle-invariant vortex sheets and show that there are no stationary vortex sheets of this type.
In section <ref> it is shown that a coadjoint orbit satisfying the Onsager-Feynman condition is prequantizable, with prequantum bundle explicitly constructed as the quotient of a certain decorated nonlinear Grassmanian by a discrete group. In section <ref>, a polarization group is given and a character is geometrically constructed that integrates the vortex sheet momenta on the polarization group.

We project to explore the higher dimensional version of these results in a future work.



§ COADJOINT ORBITS OF VORTEX SHEETS


In this section, we describe coadjoint orbits of the group  of compactly supported volume preserving diffeomorphisms, _(^3), associated to the motion of vortex surfaces in ideal 3D fluids,
also called vortex sheets. We also determine the  KKS symplectic form on these orbits.
Coadjoint orbits for codimension two vortex membranes (e.g. vortex filaments in 3D) were studied in <cit.>, <cit.>. 

Let S be a compact oriented surface
and let Diff_+(S) denote the group of  orientation preserving diffeomorphisms of S. 
Let ∈^1(S) be a closed 1-form with isolated zeros. By compactness, the zero set () is finite.
The surface S describes the support of the singular vorticity distribution, while the 1-form β describes its direction and strength.



 §.§ Principal bundles of embeddings


Consider the Fréchet manifold Emb(S,^3) of embeddings of S into ^3. It is the total space of the principal Diff_+(S) bundle

    π : Emb(S,^3) →Gr^S,    f ↦=f(S) ,

where Gr^S denotes the nonlinear Grassmannian of all oriented surfaces in ^3 of type S (including the orientations). 
By the transitivity of the action of the Lie algebra of compactly supported divergence free vector fields _c(^3) on Emb(S, ^3), see <cit.>,  the tangent space at f can be written
as T_f Emb(S,^3)={ u ∘ f: S → T^3 : u ∈𝔛 (^3)}. 
The tangent space at  to the nonlinear Grassmannian
can be identified with the space of sections of the normal bundle T^⊥ 
with respect to the Euclidean metric on ^3:

    T_^S=(T^⊥) ≃ C^().

With n denoting the unique unit normal vector field to  compatible with the orientations of  and of ^3, the second identification becomes u_↔ u_· n,
for any normal vector  field u_ on .
We define the decorated nonlinear Grassmannian  ^S,
as a bundle over ^S associated to the principal bundle (<ref>), namely:

    ^S,:=(S,^3)__+(S)Ø_,

with Ø_⊆^1(S) denoting the _+(S) orbit of .
The map 

    [f,]↦(f(S),f_*)=(,_),   f∈(S,^3), ∈Ø_,
 
makes the identification of (<ref>) with the space of oriented surfaces in ^3 of type S, each of them endowed (decorated) with a 1-form of type β

    Gr^S, β={(, β _) : ∈Gr^S, β _∈Ω ^1 ()  s.t. ∃ Ψ∈Diff_+(S,), Ψ ^∗β _ =  β}.

Note that β _  is closed with finite zero set, just like . 
The associated bundle projection becomes the forgetful map

    ^S,→^S,   (,_)↦.

Moreover, formally,

    π ^β : Emb(S,^3) →Gr^S, β,    f ↦ (f(S), f _∗β ) =(, β _ )

is a principal bundle with structure group Diff_+(S, β ), the group of -preserving diffeomorphisms that preserve the orientation. Note that it is not a Lie group in general, 
still it has an associated Lie algebra of  -preserving vector fields 

    (S,)
    ={v∈(S):(v)=constant}.




  
Splitting of the tangent space. The Euclidean metric on ^3 naturally endows the principal bundle (<ref>) with a principal connection:

    ∈^1((S,^3),(S)),  (uø f)=f^*(u|_Σ^⊤),
       u∈(^3).

with Σ = f(S). The horizontal lift of u_ Σ∈ T_ Σ^S,β at f ∈(S,^3) is ( (u_ Σ· n) n )  ∘ f ∈ T_f (S,^3).
This connection, on its own turn, induces a connection on the associated bundle 
Gr^S, β.
This splits in a natural way the tangent space to the decorated Grassmannian 
into its horizontal and vertical parts
(similarly to the weighted nonlinear Grassmannians addressed in <cit.>).
The tangent space to Ø_, the _+(S) orbit of ,
is  {_v:v∈(S)}= d (β( 𝔛  (S)))
=d C_^(S), with C^_(S) denoting the space of smooth functions vanishing on the finite subset ().
This yields the horizontal/vertical decomposition:

    T_(,β _)Gr^S,β
    =T_^S × d (β _( 𝔛  ()))
    (<ref>)=C^() × d C^__().


The bundle projection π^ in (<ref>) being (^3) equivariant
for the natural action on the decorated Grassmannian, ·(,_)=((),_*_), the infinitesimal generators _u^ and _u^ for u∈(^3) are π^ related.
Thus the tangent map to π^β can be written 
in the splitting (<ref>) as

    T_f π^(_u^)=T_f π ^β (u ∘ f)= ( u|_^⊥,  _ u|_^⊤β _)=(u|_· n, d  (  i  _u|^⊤_β _))=:^_u(,_)

where u|_=u|_^⊤+u|_^⊥ is the orthogonal decomposition along =f(S)⊆^3.






 §.§ Enclosing a constant volume


Let μ be the volume form on ^3 induced by the Euclidean metric.
Given a>0, we define the manifold

    Emb_a(S,^3) := { f ∈Emb(S,^3) : ∫_Sf ^∗ν =a },

where  ν∈^2(^3) is such that μ =  d  ν. One notices that Emb_a(S,^3) is independent of the choice of ν.
By restriction of the principal bundle (<ref>),  we get another 
principal Diff_+(S) bundle
π : Emb_a(S,^3) →Gr^S_a,
this time over  the manifold

    Gr^S_a:={∈^S: ∫_ν=a}

of oriented surfaces in ^3 that enclose a constant volume a.

The tangent spaces are: 

    T_f Emb_a(S,^3)={ u ∘ f: S → T^3 : u ∈𝔛  (^3), ∫_S f ^∗  i  _u μ=0}

and
T_Gr^S_a={ u_∈Γ (T^⊥) : ∫_ i  _u_μ =0}.
Using  the volume form induced on the surface  by the Euclidean metric, 
namely  μ_= i _nμ,
the identification in (<ref>) yields, by restriction,

    T_Gr_a^S≃ C_0^():={f∈ C^(): ∫_ fμ_=0}.

Indeed, ∫_ i _u_μ
=∫_ (u_· n) i _nμ=∫_ fμ_, for all u_∈(T^⊥).

We consider also the decorated Grassmannian enclosing a constant volume

    Gr _a^S, β:={(, β _) ∈^S, : ∈Gr^S_a},
 
together with the (formal) principal Diff_+(S, β ) bundle,
obtained from (<ref>),

    π ^β : Emb_a(S,^3) →Gr^S, β_a.

This time the projection is _ vol(^3) equivariant.
From now on, we will refer to the decorated Grassmannian (<ref>) formally as a manifold of vortex sheets of type (S, β ).




 §.§ Symplectic form on the manifold of vortex sheets



The 1-form  on S and the volume form μ=dν
on ^3 jointly define, in a natural way, a 2-form  on  (S,^3):

    (u_1ø f,u_2ø f)=∫_S f^* i _u_2 i _u_1μ∧,   u_1,u_2∈(^3).

In the hat calculus notation <cit.>, this form  is written as

    =μ·=d(ν·).

It is invariant under the commuting actions
of the reparametrization group _+(S,)  and of the ambient space
diffeomorphism group  _ vol(^3), with infinitesimal generators 
ζ_v(f):= Tf∘ v and ζ_u(f):= uø f respectively.


The restriction of  to  _a(S,^3)
descends via (<ref>) to a symplectic form 
 on Gr^S, β_a:

    (π^)^*=.




We know that  in (<ref>) is _+(S,)-invariant.
We prove below that the kernel of  is generated by the (S,) action.
It follows that the closed 2-form  on _a(S,^3) is  π^β-basic, 
thus it  descends to a closed non-degenerate 2-form  on the manifold of vortex sheets Gr _a^S, β. In general  is not an exact form, even though its pullback  is exact.

Notice that all the infinitesimal generators ζ_w(f):=Tfø w with w∈(S)
are tangent to _a(S,^3) at f.
We choose an arbitrary element u_f:=uø f∈ T_f_a(S,^3) in the kernel of .
Besides the condition ∫_Sf^* i _uμ=0 from (<ref>), we also get

    0=(u_ f,Tfø w)=-∫_S(w)f^* i _uμ,   for all w∈(S).

The 1-form  having a finite number of zeros, we deduce that
f^* i _uμ=0, which implies that u∈(^3) is tangent to the surface f(S).
This ensures the existence of v∈(S) such that u_ f=Tfø v.
It belongs to the kernel of , hence 0=(Tfø v,u'ø f)=∫_S (v)f^* i _u'μ for all u'∈(^3) such that ∫_Sf^* i _u'μ=0.
We get that the function (v) must be constant,
so, by (<ref>), the vector field v∈(S,) and u_ f=_v(f). 




Since the zero set of _ is finite, the nondegenerate pairing 

    C_0^() d  C^__() →,  
    (,d)=∫_μ_

permits to write the tangent space to _a^S,

 as

    T_(,β _)Gr^S,β_a=T__a^S ×  d  (β _( 𝔛  ()))=C_0^() ×
    C_0^()^*,

using also (<ref>) and (<ref>).
That these are canonical coordinates for the symplectic form  is shown in the next proposition.


The symplectic form    on 
Gr _a^S, β,  given in Proposition <ref>,
becomes

    Ω _(,_)( (_1,d_1),(_2,d_2)) =
    -∫_(_1_2 -_2_1 )μ_

in the description (<ref>) of the tangent space.




First we notice that the expression of  does not depend on the choice of the functions λ_1 and _2 in C^__(), since _1μ_ and _2μ_ are exact forms on .

Because of the infinitesimal transitivity result in Lemma <ref>,
it is enough to check the identity on infinitesimal generators
^_u_i(,_)=(_i, d_i),
for u_i∈_(^3), u_i|_^ ⊥ =_in and 
i  _u_i|_^ ⊤β _=_i, for  i=1,2.
Hence

    _(, β _)( (_1, d  λ_1 ), (_2,  d  _2))
    (<ref>)=∫_ ( i  _ u_2 i  _u_1 μ) 
    ∧_  
    =∫_ ( i  _ u_2|_^ ⊥ i  _ u_1|_^ ⊤μ) 
    ∧_  + ∫_  ( i  _ u_2|_^ ⊤ i  _u_1|_^ ⊥μ)∧_
    
    = ∫_ ( i  _u_1|_^ ⊤_)( i  _ u_2|_^ ⊥μ )- ∫_  ( i  _ u_2|_^ ⊤_)( i  _ u_1|_^ ⊥μ) =-∫_(_1_2-_2_1)μ_.

Only the mixed terms survive at step two:
i  _ u_2|_^ ⊥ i  _ u_1|_^ ⊥μ =0
since all the normal vectors to  are colinear
and  i_^*( i  _ u_2|_^ ⊤ i  _u_1|_^ ⊤μ)  =0 since i_^*μ=0.





 §.§ Hamiltonian action on manifolds of vortex sheets


The action of _ vol(^3)
on the manifold of embeddings that enclose a constant volume, _a(S,^3), descends to an action on the manifold of vortex sheets
_a^S,:

    ·(,_)=((),_*_).

Every divergence free vector field in _ vol(^3) admits a potential 1-form
∈^1(^3), i.e. i _uμ=d , which we denote u=X_ α. 


The action of _ vol(^3) on the symplectic manifold
(_a^S,,) is Hamiltonian with _ vol(^3) equivariant momentum map

    J: Gr _a^S, β→𝔛 _ vol(^3)^∗ ,   ⟨ J(,β _), X_α⟩ =∫_∧β _,  ∈^1(^3).




The map J  doesn't depend on the choice of the potential 1-form .
Any two potential forms differ by a closed  1-form on ^3, hence by an exact one, and ∫_ d h∧_=0 for all h∈ C^(^3).

To show that J is a momentum map, we check that for any u=X_, the infinitesimal generator _u^ 
is a Hamiltonian vector field on _a^S,  with Hamiltonian 
function H_u(,_):=∫_∧_.
We use the hat calculus in <cit.>:
on one hand we have 

    (π^)^* i __u^= i __u^ 
    = i _uμ·=d·=d(·),

on the other hand, the pullback of the function H_u by π^ is 

    ((π^)^*H_u)(f)=H_u(f(S),f_*)=∫_f(S)∧ f_*
    =∫_S f^*∧=(·)(f),
 
for all f∈_a(S,^3).
Knowing that for all ∈Diff_ vol(^3 ) we have Ad_φ X_ α = X_φ _∗α,
the equivariance  of the momentum map J follows easily.





 §.§ Manifolds of vortex sheets as coadjoint orbits



In this section we show that the connected  components of the manifold of vortex sheets _a^S,
can be seen as  coadjoint orbits of _(^3), the identity component of the group of compactly supported volume preserving diffeomorphisms of ^3.
For this we use the following well known fact (found for instance in
the appendix in <cit.>):


Suppose the action of the Lie group G on the symplectic manifold (M,) is transitive and infinitesimally transitive, with injective equivariant momentum map J:M→^*.
Then J is one-to-one onto a coadjoint orbit of G.
Moreover, it pulls back the Kostant–Kirillov–Souriau symplectic form ω_ on the coadjoint orbit to the symplectic form .


It is easy to see that the _(^3) equivariant momentum map (<ref>)
is injective. Indeed, if J(_1, β _1)= J(_2, β _2), then necessarily the surfaces coincide, _1=_2, because the 1-forms β _1 and β _2 have a finite number of zeros. We thus have 
_2∈^1(_1) and ∫__1 ∧ (β _1- β _2) =0, for all α∈^1(^3), hence β _1= β _2.

Next we  show that the action of the identity component of _(^3)
on connected  components of _a^S, is transitive and
infinitesimally transitive.

 Let N be an orientable codimension one submanifold of the m-dimensional manifold M with inclusion i_N:N→ M.
Given   Y ∈𝔛  (N), there exists ∈Ω ^m-2(M) such that i_N ^∗ =0 and X_  |_N= Y.
 

 We endow M with a Riemannian metric g
and we denote by μ the induced volume form. Since N is orientable, TN^⊥ is a trivial line bundle, the trivialization being given by
the unit normal vector field n along N, namely
 TN^⊥→ N ×ℝ, u_x↦ (x, u _x · n). Thus, we can identify a tubular neighborhood V of N in M with N ×  (-1,1).
With this identification, the volume form μ on V reads dt ∧ω, where  is a t-dependent (m-1)-form on N.

We define Y∈𝔛  (V) by Y(x,t):=Y(x)+0 ∂ _t on V.
We consider the form t  i  _Yω∈Ω ^m-2(V) and extend it to ∈Ω ^m-2(M). Let us check that  satisfies the two required conditions.
On one hand, we have i_N ^∗ = i_N ^∗( i  _tYω) =0, since N⊆ V is characterized by t=0. 
On the other hand, on V  we have
i  _X_  - Yμ=  d   -   i  _Yμ =  d  ( t  i  _Yω) -  i  _Y(dt ∧ω )=t  d   i  _Yω. We get (X_-Y)|_N=0,
so X_  |_N=Y.






The Lie algebra _(^3) of compactly supported 
divergence free vector fields acts transitively  on the manifold _a(S,^3).



We consider uø f∈ T_f Emb_a(S,^3) with u ∈𝔛_c  (^3), hence ∫_ i  _u μ =0, where =f(S). Therefore, there exists _∈^1() with i_ ^∗   i  _u μ =  d _. 
We choose α∈Ω_c ^1(^3) 
such that i_^∗α = α _. We have i^*_ (  i  _u- X_ αμ )=  d  α _- i_ ^∗  d  α =0, hence Y:=(u- X_ α)|_∈𝔛  ().
By the Lemma <ref> one can find 
∈Ω ^1(^3) with i_ ^∗=0 and such that X_  |_ =Y. Because  is compact,  can be chosen with compact support.
Hence the vector field X_α + ∈_(^3)
has the property u|_=X_α + |_, so uø f=X_α + ø f, which shows the infinitesimal transitivity of the 𝔛_(^3) action  on Emb_a(S,^3).




It is clear that the construction in the proof of Lemma <ref> 
can be done smoothly depending on a parameter.
Since _(^3) is locally connected by smooth arcs,
we conclude that the action is locally transitive.


The identity component of _(^3) acts transitively on 
connected  components of the manifold of vortex sheets _a^S,.


Now Proposition <ref> can be applied to the momentum map for vortex sheets.

 
The restriction of the momentum map 
J: Gr _a^S, β→𝔛 _(^3)^∗
in (<ref>) to any connected   component of the manifold of vortex sheets _a^S, is one-to-one onto a coadjoint orbit of the identity component of  _(^3), the Lie group of compactly supported volume preserving diffeomorphisms. 
The Kostant–Kirillov–Souriau symplectic form _ KKS on the coadjoint orbit satisfies J^*_ KKS=, for the symplectic form  in (<ref>).
 





  
Vector field approach to vorticity density.
In <cit.>, open sheets (ribbons/tubes) in 3D are treated and a vector field _ tangent to the vortex lines of  is used to describe the vorticity density (also called vortex sheet strength).

In our description of vortex sheets using closed 1-forms β _ Σ∈Ω ^1 ( Σ ) for the vorticity density,  the vector field γ _ Σ on Σ is defined by i __μ_=_ with μ_= i _nμ the volume form on the oriented surface Σ induced by the Euclidean metric. Note that γ _ Σ is a divergence free vector field with respect to μ _ Σ
and _=n_^♯.

In terms of the vorticity density vector field γ _ Σ, the momentum map (<ref>) becomes

    ⟨ J(,_),X_⟩=∫_∧ i __μ_
    =∫_(_)μ_,  ∈_c^1(^3).

Let χ:=^♯∈_c(^3) be the vector potential of u=X_∈_(^3), obtained from the potential ∈^1(^3)
by rising indices with the Euclidean metric, i.e, u = curlχ.
Using χ, the momentum map is given, as in <cit.>, by

    ⟨ J(,_),u⟩=∫_ (_·χ)μ_.

Note that the description of vorticity density with a closed 1-form β _ Σ is more natural since, unlike the divergence free vector field γ _ Σ,  the 1-form is naturally transported via push-forward by diffeomorphisms in the coadjoint action, and hence is associated to a given closed 1-form β on S, see (<ref>).





§ VORTEX SHEETS DYNAMICS IN 3D FLUIDS


In this section we characterize the vortex lines on vortex sheets (,_), having discrete period group ℓ and non-vanishing vorticity density, as the fibers of a certain fibration  Σ→_ℓ. 
We then determine the Hamilton equations governing the motion of such vortex sheets. Denoting by kB the binormal curvature to the vortex lines, these equations read

    {[                      Σ̇= k B^⊥_Σ; β̇_ Σ = d  ( β _ Σ ( kB ^⊤_ Σ)). ].
 
In particular, the deformation of the support of the vortex sheet is governed by the normal component of the binormal curvature of the vortex lines, while the deformation of the vorticity density is governed by the variation of the volume 
(γ _ Σ× kB) · n
generated by the vorticity density vector field, the binormal curvature, and the normal to the support. Finally, we focus  on the invariant
submanifold of circle-invariant vortex sheets, we show that there are no stationary solutions of this
type, and we determine the form of the Hamiltonian PDE (<ref>) on the connected component of
circle-invariant vortex sheets fibered by parallel vortex lines.





 §.§ Vortex lines and their curvatures on vortex sheets





  
Vortex lines as fibration over a circle. Recall that the period group of a closed 1-form β∈Ω ^1 (S) is the subgroup of ℝ defined by

    Per(β) := {∫_ γβ : [γ ] ∈ H_1(S,)}.



 
If the period group of  is discrete, Per( β)= ℓℤ for some ℓ>0, then there is a map

    b: S →ℝ/ℓℤ=:_ℓ ,
 
such that β = b ^∗ϑ_ℓ, where ϑ_ℓ∈Ω ^1 (_ℓ) denotes the angular form of 𝕋_ℓ, ., 

p _ℓ ^* ϑ_ℓ = dx, where p_ℓ: ℝ→_ℓ is the projection p _ℓ(x)= x +ℓℤ.





In other words  is the logarithmic derivative of b, also denoted by = b, and b is called a Cartan developing of .



From now on we will assume that the period group of the closed 1-form β is ℓ,
and that  has no zeros. 
This forces the Cartan developing b:S→_ℓ to be a submersion, hence a fibration (by compactness),
and the surface S to be of genus one.
Notice that the 1-form  cannot be exact.
Consequently, every element (,_) of the manifold of vortex sheets
_a^S, has similar properties: the oriented surface ⊆^3 is of genus one, it is endowed with a closed non-exact 1-form _  with period group ℓ, without zeros, and equal to the logarithmic derivative of a fibration b_:→_ℓ.
Its fibers 

    C_z:=b_^-1(z),   z∈_ℓ,
 
are closed curves on , called the vortex lines of the vortex sheet .
They integrate the kernel of β _ Σ,
thus they don't depend on the choice of the Cartan developing b_ of _. 





  
Frames, orientations, and curvatures for vortex lines. The Darboux frame of the vortex line C_z on the surface 
is the orthonormal frame {T,n_g,n} with T a unit tangent vector to C_z
(especially _(T)=0),
n the unit normal vector field to , and n_g=n T the  geodesic normal vector field (tangent to ). The orthonormal frame {T,n_g}  is positively oriented on Σ:
μ_(T,n_g)=1.

The surface  is oriented, so the choice of n is fixed by the compatibility with the orientations of  and ^3. The vortex lines C_z are oriented by using β _ Σ and imposing the condition _(n_g)>0, which provides the orientation given by T=n_g n. 

The vortex lines are the integral curves of the (nowhere vanishing) vorticity density vector field  _ with i __μ_=_.
The vector field _ has the same orientation as the unit tangent vector field T,
since μ_(_,n_g)=_(n_g)>0.
It follows that β _ Σ (n_g) = γ _ Σ
and _=_(n_g)T.

Let {T,N,B} denote the (orthonormal) Frenet frame of the vortex line C_z⊆^3.
The mean curvature vector field of the curve C_z⊆^3 is H_C_z= k N,
where k≥ 0 denotes the curvature of C_z.
Its decomposition in the normal frame {n_g,n} to the vortex line C_z is

    H_C_z= k N=k_gn_g+k_nn,
 
with k_g and k_n denoting
the geodesic curvature and the normal curvature of the fiber C_z on the surface
. Note that N, k, and k_n depend only on the fibers C_z (not on their orientation given by β _ Σ) and on the orientation of Σ. However, the pointing direction of the vectors T, B, n_g, and the sign of κ _g depend on the orientation
of the fibers. Note also that the binormal curvature reads

    k B=k_gn-k_nn_g.





  
Gluing. The surface  is a disjoint union of the fibers C_z of the fibration b_,
thus the curvature functions k, k_g, and k_n on the fibers 
glue together to functions on the surface. We denote them by the same letters:
k,k_g,k_n∈ C^().
The same gluing can be done with the unit tangent vector field T to the fibers, 
as well as with n_g=n T, obtaining vector fields T,n_g∈().
It follows that the pullback of the 1-form 

    σ _:=- i _n_gμ_=- i _n_g i _nμ∈^1()

to each fiber,
namely i_C_z^* σ _, is the volume form μ_C_z on the fiber, z∈_ℓ. The following holds true:

    σ _∧_=_(n_g)μ_,
 
since both sides evaluated on the orthonormal frame {T,n_g} give _(n_g).


The smooth function k_g∈ C^() is the divergence of -n_g∈() (with respect to μ_), thus ∫_ k_gμ_=0.


Let ∇ denote the   Levi-Civita connection on .
We compute

    ÷ n_g=(_n_gμ_)(T,n_g)
    =-( d σ _)(T,n_g)
    = σ _([T,n_g])=T· [T,n_g]
    =-∇_TT· n_g
    =-k_g,

using the identities ∇_TT=k_gn_g and T^♭= σ _.




 §.§ Hamiltonian function and Hamilton's equations


On the symplectic manifold _a^S, we consider the Hamiltonian function suggested in <cit.> and given by the total length of the vortex lines:

    h(,_)
    =∫__ℓ (C_z) ϑ_ℓ ,

where (C_z)=∫_ C_zμ_ C_z, z∈_ℓ, denotes the length of the vortex line and ϑ_ℓ is the angular form defined in Lemma <ref>.
By the Fubini theorem for the fibration b_:

    ∫__ℓ( ∫_C_z) ϑ_ℓ=∫_∧_,  for all ∈^1(),

we can write the Hamiltonian as 

    h(,_)
    =∫__ℓ( ∫_C_zμ_C_z)  ϑ_ℓ =∫_ Σσ  _ Σ∧ b_ Σ ^* ϑ_ℓ = ∫_ Σσ  _ Σ∧β _ Σ(<ref>)=∫ _ Σβ _Σ(n_g) μ _ Σ,

since μ_C_z=i_C_z ^* σ  _ Σ. The last equality 
 shows how h differs from the area of Σ.


In the description (<ref>) of the tangent space to the coadjoint orbit, namely,

    T_(,β _)Gr^S,β_a=C_0^() ×  d  C^(),

the differential of the Hamiltonian function h is given by

    ( d_(,_)h)(, d)= - ∫_ Σ(   k_nβ _ Σ(n_g) + k_g λ) μ _ Σ,

with k_g and k_n the geodesic and normal curvatures of the vortex lines on Σ.



By the transitivity result in Lemma <ref>,
it is enough to check the identity on infinitesimal generators of u∈_(^3):

    ^_u(,_)=(u|_· n, d _(u|_^⊤))
    =:(, d ).

Let ϕ^u_t denote the flow of u. We notice that
b_ϕ^u_t()^-1(z)=ϕ_t^u(b_^-1(z)) for all z∈_ℓ.
Then

    ( d _(,_)h)(^_u)
       =d/dt|_0 h(ϕ^u_t(,_))=d/dt|_0h(ϕ^u_t(),(ϕ^u_t)_*_)
       =d/dt|_0∫__ℓ(b_ϕ^u_t()^-1(z))ϑ_ℓ
    =-∫__ℓ(∫_C_z(u· H_C_z)μ_C_z)ϑ_ℓ
       = - ∫_ Σ( (u· n)  k_nβ _ Σ(n_g) + k_g _(u|_^⊤) ) μ _ Σ= - ∫_ Σ(  k_nβ _ Σ(n_g) + k_g λ) μ _ Σ.

At step four we use the standard fact from the literature
that the mean curvature vector field is the direction where the volume of the submanifold decreases the fastest:

    d/dt|_0(b_ϕ^u_t()^-1(z))
    =d/dt|_0(ϕ^u_t(C_z))
    =d/dt|_0∫_C_z(ϕ^u_t)^* μ_ϕ^u_t(C_z)
    =-∫_C_z(u· H_C_z)μ_C_z.

For the step five we used (<ref>) and we applied the Fubini theorem 
(<ref>) for the fibration b_.
Here = (u· (k_gn_g+k_nn))σ _∈^1(), since 
μ_C_z=i_C_z ^* σ  _ Σ,
so that by (<ref>) we get

    ∧_=(k_gu· n_g+k_nu· n)σ _∧_
    =(k_g_(u|_^⊤)+k_n(u· n)_(n_g)) μ_,
 
using also the identity _(u|_^⊤)=(u· n_g)_(n_g).



The Hamiltonian vector field associated to the Hamiltonian function (<ref>), 
written in the decomposition (<ref>), is

    X_h(,_)=(k_g,- d (k_n_(n_g))).




Notice that k_g∈ C^_0() by Lemma <ref>,
so that the right hand side does belong to the tangent space (<ref>).
Let (f, d ) be an arbitrary tangent vector at (,_).
Then we compute with the help of Lemma <ref>:

    (X_h(,_),(, d ))   =( d _(,_)h)(, d )
    =-∫_(k_g+ k_n _ (n_g))μ_.

The result follows now from the expression (<ref>) of .




In terms of the vorticity density vector field  γ _ Σ, the Hamiltonian function and the Hamiltonian vector field read:

    h(,_) =∫ _ Σ_μ _ 
    ,    
    X_h( Σ , γ _ Σ ) = (k_g, -  d  ( k_n γ _ Σ)).







  
Hamilton's equations in terms of binormal curvature. 
From (<ref>) the Hamiltonian vector field can be written in terms of the binormal curvature kB as

    X_h( Σ , β _ Σ ) = ( (kB· n) n,  d ( ( kB· n_g) β _ Σ (n_g)))= ( kB^⊥_ Σ,  d  ( β _ Σ ( kB ^⊤_ Σ))),

with kB^⊥_ Σ and k B^⊤_ Σ the components of kB normal and tangent to the surface Σ. 
This expression of the Hamiltonian vector field X_h leads to the system (<ref>), equivalently written as

    Σ̇= kB ^⊥_Σ ,     β̇_ Σ = _(kB^⊤_Σ)β _ Σ.

Thus the deformation of the surface Σ is governed by the normal component of the binormal curvature of the vortex lines (but not by the magnitude of their strength β _ Σ (n_g)=γ _ Σ). The deformation of the vorticity density 1-form β _ Σ is governed by the variations of the volume β _ Σ ( kB ^⊤_ Σ)= (γ _ Σ× kB) · n which involves both the tangential component of the binormal curvature of the vortex lines and their vorticity strength. 






 §.§ Invariance and surfaces of revolution





  
(3)-invariance and momentum map. 
The Hamiltonian h  on the decorated Grassmannian
is invariant under the restriction of the
_ vol(^3) action (<ref>) to the special Euclidean group 
(3), whose action  on ℝ ^3 is simply denoted 
x ↦Φ_g(x)= Ax + a, g= (A,a) ∈(3). The associated SE(3)-momentum map is determined as follows.
A Lie algebra element, ξ =  ( ω , v) ∈𝔰𝔢(3), gives rise to the divergence free vector field u(x) = ω× x + v with vector potential χ(x)= α^♯ (x) = ( ω· x) x + 1/2 v × x. 
From (<ref>), the momentum map 𝕁: Gr_a^S, β→𝔰𝔢(3) ^* is

    ⟨𝕁(,_),(,v)⟩
    =1/2∫_(||x||^2^♭+(v x)^♭)∧_.

Its reformulation in terms of the vortex density _,
as in (<ref>), gives:

    𝕁( Σ ,  _ Σ ) =  ( ∫_ Σ ( γ_ Σ· x)   x  μ _ Σ ,1/2∫_ Σ (x ×γ _ Σ)  μ _ Σ).
  
From Noether theorem it follows that the momentum map 𝕁 
is preserved along any solution ( Σ (t), β _ Σ (t)) of Hamilton's equations.




  
Flux homomorphism and isotropy groups. The flux homomorphism of a closed 1-form  with Per( β)= ℓℤ, is defined by

    c_β : Diff(S, β ) →_ℓ,   c_ β (φ )
    :=(b ∘φ ^-1) /b,
 
where b is a Cartan developing for β. Because =  b is preserved by φ, 
the map (b ∘φ^-1 ) /b:S →_ℓ is  constant and hence c_ is well-defined.
Also, it does not depend on the choice of the Cartan developing b:S→_ℓ, since c_ is defined up to multiplication of b by a constant in _ℓ.
The derivative of c_ β at the identity is the Lie algebra homomorphism 

    d  _e c_ β : 𝔛 (S, β )→ℝ,    d  _e c_ β (v)=-β (v).


Let us consider a diffeomorphism ∈_ vol(^3) in the isotropy group of (,_),
 |_∈() and ^*_=_.
Since  b_=_=|_^* b_=(b_ø|_) it follows that |_ satisfies b_ø|_=cb_, for the constant 
c=c__(|_^-1)∈_ℓ.
Thus  |_ is an automorphism of the fibration, that covers a rigid rotation of the base _ℓ,
namely z↦ cz.



  
Invariant submanifolds of circle-invariant vortex sheets. 
We consider  the subgroup _2π of rotations around a given axis in ^3.
The subset of _2π-invariant elements of the decorated Grassmannian _a^S,,

    ℛ:= {( Σ , β _ Σ )∈_a^S,|Φ_ g( Σ )= Σ ,  Φ ^* _g β _ Σ = β _ Σ,  for all g ∈_2 π},
 
consists of surfaces of revolution around the axis with circle-invariant fibration.
It means that for each (,_)∈, the isotropy subgroup
contains the subgroup of rotations _2π.
In particular the fibration b_ satisfies Φ_g(C_z)= C_c z, for all z ∈_ℓ and all g ∈_2π. As above, the  constant c depends on g through the flux homomorphism c__.


Since both the symplectic form  on _a^S, and the Hamiltonian function h are _2π-invariant, we have the following general result.

 The Hamiltonian vector field X_h is tangent to  and hence to each of its connected components _m,n, which are thus preserved by the Hamiltonian dynamics.
 




  
Description of circle-invariant vortex sheets. By rotating a closed plane curve , parametrized by (ξ(ρ),(ρ)),
with ξ ( ρ  ) >0, around the vertical axis, we obtain the parametrization of
a surface of revolution

    : (ξ(ρ)cosþ,ξ(ρ)sinþ,(ρ)).

Any _2π-invariant closed 1-form _ on the surface of revolution
is of the form 

    β _ Σ = ζ _ ρ  d ρ  + c d θ .

Since (_)=ℓ, the two terms must have period groups
(_ d)=mℓ and (cdþ)=nℓ, with m,n coprime natural numbers. In particular the constant must be c=nℓ/2π
and the real valued function  satisfies the condition (ρ+2π)=(ρ)+mℓ.
The fibration projection b_:→_ℓ (a Cartan developing of _) takes the form b_(ρ,þ)=p_ℓ(()+nℓ2πþ),
for the canonical projection p_ℓ:→_ℓ.

We denote by _m,n, m,n coprime, the subset of all circle-invariant vortex sheets 
with vorticity density of the form _=_ d+nℓ/2πdþ
with (_ d)=mℓ.
We show that _m,n is connected.
Any two surfaces of revolution can be continuously deformed one into the other,
so let us consider two vortex sheets in _m,n supported on the same
surface: (,_0) and (,_1). Then 
_t:=(1-t)_0+t_1=((1-t)_^0+t_^1)d+nℓ/2πdþ is a path from _0 to _1 
consisting of vorticity densities supported on  with (,_t)∈_m,n.

We conclude that  is a union of connected components:

    =
    ⋃_m,n coprime _m,n.

The connected component _1,0 contains surfaces of revolution fibered by parallel circles, where β _ Σ = ζ _ ρ  d ρ with (_ d)=ℓ.
The connected component _0,1 contains surfaces of revolution with meridian-like fibrations and β _ Σ of the form β _ Σ = ζ _ ρ  d ρ  + ℓ/2πd θ, with exact 1-form _ d. 


Note that on ℛ _m,n with n≠ 0 (hence c 0), the vortex lines have only one associated 1-form β _ Σ (up to sign), because its period group is fixed: ℓ. Thus the dynamics of the vortex lines C_z⊆ determines the dynamics of the vorticity density 1-form _. This is not the case for ℛ _1,0 whose vortex lines stay parallel, with vorticity density changing according to ζ _ ρ.





 §.§ Looking for stationary points


In this section we show there are no stationary points of the Hamilton equations for vortex sheet dynamics in the _2π-invariant subset .
We conjecture that there are no stationary points on the whole 
decorated Grassmannian _a^S,. Notice that also the vortex filament equation 
(on the knot space in ^3) doesn't possess stationary points.


The vortex sheet (,_) is a stationary point of the Hamiltonian vector field X_h if and only if all the vortex lines are geodesics of the surface  and  the product k_(B) (with k the curvature of the fiber and B its binormal vector field) is constant on .



A stationary point of the vector field X_h  in (<ref>) is characterized by k_g=0 and 
d (k_n_(n_g))=0 for the fibers of b_. 
The first condition ensures that the fibers are geodesics, so k_n=± k and n_g=∓ B the unit binormal vector field.
The second condition means that k_n_(n_g)=- k_(B) is a constant function on .




Let  be the torus in ^3 obtained by rotating a circle of radius r around the vertical axis,

    ((R+rcosρ)cosþ,
    (R+rcosρ)sinþ,rsinρ).
 
We assign an orientation so that the unit normal vector field of the surface points outwards:
n=(rcosρcosþ,rcosρsinþ,rsinρ).
We endow  with the 1-form  _=ℓ2π d þ: closed, without zeros, and with period group ℓ, thus (,_)∈_0,1.
Its Cartan developing b_ is the geodesic fibration 
by meridian circles.

We show that the second condition in Lemma <ref> is not satisfied.
The imposed condition _(n_g)>0 ensures that n_g=1/R+rcosρ_þ, so T=1/r_.
The unit normal vector field N of the meridian circle points inwards,
so N=-n. Thus the unit binormal vector field satisfies B=n_g.
The geodesic curvature k_g of the meridian circle vanishes and the normal curvature k_n=-k=-1r is constant.
Still, the function k_(B)=-ℓ2π r(R+rcosρ) is not constant, hence
(,_) is not a stationary point of the Hamiltonian vector field X_h.


More generally, let us see what happens for 𝕋_2 π-invariant vortex sheets (, _)∈,
parametrized as in (<ref>). One computes that it induces a geodesic fibration b_
only if the 1-form is

    β _ Σ = -c r( ρ  )/ξ ( ρ  )√(ξ ( ρ  )^2 -^2)d+cdþ,

where r ^2 = ξ _ ^2 + _^2,
and  is a constant.
A straightforward computation yields:

    k_n β _ Σ (n_g)=  c/ r ^2ξ√( r ^2 + ξ^2  _ ^2  /c^2 )( η _ξ _- η _ξ _ + ξη _ζ _ ^2/c^2 ),

which cannot be constant for our _=-c r/ξ√(ξ ^2 -^2) imposed by the geodesic fibration.
Once more the Lemma <ref> implies that (,_) is not a stationary point of X_h, hence the following result.



There are no stationary points for the Hamiltonian vector field X_h 
in (<ref>) restricted to .



Conjecture: X_h has no stationary points on the whole _a^S,.







 §.§ Surfaces of revolution fibered by parallel circles



In this section we determine the expression of the Hamilton equations as partial differential equations for the parametrization of the vortex sheet. This is important for further analytical and numerical studies of these equations. We focus on the invariant submanifold _1,0 of vortex sheets fibered by parallel circles, while the corresponding forms on the other connected components ℛ _m,n will be treated in a future work.
The component _1,0 is the only one in which the vorticity density
is not fully determined by the fibration, see Remark <ref>.

Let  denote the surface obtained by rotating the closed plane curve , parametrized by (ξ(ρ),(ρ)), as in (<ref>).
We endow it with the orientation inducing the unit normal vector field pointing inwards:

    n=1/√(ξ_^2+_^2)(-_cosþ,-_sinþ,ξ_).

The induced metric on  is 
(ξ_^2+_^2)d^2+ξ^2dþ^2 with orientation compatible
volume form 

    μ_=i_nμ=ξ√(ξ_^2+_^2)d∧ dþ.

Let us consider the fibration of the surface of revolution  by its parallel circles
(the coordinate lines =constant), which correspond to the connected component ℛ_1,0.
This means that the fibration projection b_ depends only on the parameter , hence _=_ d with _>0.

The Darboux frame {T,n_g,n} that satisfies _(n_g)>0 has 

    n_g=1/√(ξ_^2+_^2)(ξ_cosþ,ξ_sinþ,_)=1/√(ξ_^2+_^2)_

and unit tangent vector field
T=(sinþ,-cosþ,0)=-1/ξ_þ.
Consequently, the Frenet frame {T,N,B} of a parallel circle has N=-(cosþ,sinþ,0), hence B=T N=(0,0,-1).

Now the normal curvature k_n and the geodesic curvature k_g can be computed from the identity kN=k_nn+k_gn_g, where k=1/ξ
is the curvature of a parallel circle. We obtain:

    k_n=_/ξ√(ξ_^2+_^2)    
    k_g=-ξ_/ξ√(ξ_^2+_^2) .






The Hamiltonian vector field on the manifold of vortex sheets
_a^S,, restricted to the component ℛ _1,0, reads

    X_h(,_)=(k_g,- d (k_n_(n_g)))=(-ξ_/ξ√(ξ_^2+_^2),
    
    - d (__/ξ(ξ_^2+_^2))).

This formula, not depending on the variable þ, 
ensures that ℛ _1,0 is preserved by the Hamiltonian dynamics, consistently with Lemma <ref>.

The Hamiltonian dynamics above induces a dynamics 
(t) on the space of oriented closed curves in the plane: _t=k_g n_,
where n_ denotes the unit normal vector field to the plane curve .
Let ^S^1(^2) denote the nonlinear Grassmannian of oriented
closed plane curves. The euclidean metric on ^2 defines in a natural way 
a principal connection on the principal bundle (S^1,^2)→^S^1(^2)
with structure group _+(S^1), as in (<ref>).
The horizontal lift of (t) is the curve f(t)=(ξ(t),(t))∈(S^1,^2)
of parametrizations that satisfies

    f_t=k_g n_ø f,  =f(S^1).
 
The next theorem spells out the dynamics of the parametrization functions ξ and ,
joined with that of the function  that describes the vorticity density. 



Let ((t),(t)) be a solution of the Hamilton equation starting at (,_) ∈_1,0, and let (t) denote the closed plane curve that describes the surface of revolution (t).
The parametrization f(t, )=(ξ(t,),(t,)) of this curve given in (<ref>)  is governed by the system of partial differential equations deduced from
 the first component of (<ref>):

    ξ_t=ξ__/ξ(ξ_^2+_^2)    _t=-ξ_^2/ξ (ξ_^2+_^2).

The variation of the vorticity density along the parallel circles is governed by the partial differential equation deduced from
 the second component of (<ref>): 

    _t=-__/ξ(ξ_^2+_^2).

It depends on a solution (ξ,) of (<ref>).


Note that we can't get rid of the factor ξ_^2+_^2 by starting with an arc-length parametrized curve , because the arc-length parametrization condition is not compatible with the Hamilton equations.




  
Constants of motion.
The dynamics (<ref>)–(<ref>) on _1,0 
preserves  the volume enclosed by the surface  a=π∫__2πξ^2_ d and the Hamiltonian function:

    h(,_)
    (<ref>)=∫ _ Σβ _Σ(n_g) μ _ Σ
    =∫__/√(ξ_^2+_^2)μ_=2π∫__2πξ_ d.

In addition, by Noether theorem, it also preserves
the momentum map (<ref>). The relevant subgroup here consists of translations in the vertical direction, which commutes with the subgroup of rotations around the vertical axis.
Setting  u(x)=e_3, and using the potential 1-form =1/2(e _3 x)^♭, whose pullback to the surface of revolution  is 1/2ξ^2dþ, the corresponding constant of motion is

    k(,_)=⟨𝕁(,_),(0,e_3)⟩=∫_∧_
    =∫_1/2ξ^2dþ∧_ d=π∫__2πξ^2_ d.
 
While the Hamiltonian represents the total length of the fibers, 
 h(,_)=∫__2π2πξ_ d=∫__ℓ Length(C_z)ϑ_ℓ, this new constant of motion represents the total area of the disks D_z determined by the fibers, k(,_)=∫__2ππξ^2_ d=∫__ℓ Area(D_z)ϑ_ℓ.







§ PREQUANTIZATION OF COADJOINT ORBITS


In this section we show that if the coadjoint orbit of vortex sheets _a^S, with Per()=ℓ satisfies the Onsager-Feynman condition

    aℓ∈ 2πℤ,
 
then it is prequantizable, with prequantum bundle ( 𝒫 , Θ ) explicitly constructed as a certain decorated Grassmannian quotiented
by a discrete subgroup of the circle.

The same Onsager-Feynman condition was found in <cit.> applied to coadjoint orbits of infinite vortex sheets (ribbons/tubes) in ^3 “enclosing" a finite volume 𝒱=a. Given such an unbounded vortex sheet (,_), this condition was written there as 𝒱_tot∈ 2π with Ω _ tot the total vorticity of the vortex sheet, defined as the integral

    _tot:=∫_ dl·(n_)

along a curve  crossing the ribbon/tube transversally and oriented such that the frame {γ _ Σ , dl, n} is positively oriented in ℝ ^3. This integral does not depend on the choice of Γ.
In our setting, using _= i __μ_= i __ i _nμ, the total vorticity can be simply written as _tot=∫__ and is equal to the smallest period ℓ of the closed 1-form _.


In this section S is a compact connected surface of genus one, endowed with a closed 1-form 
without zeros and with period group ℓ.
We fix a Cartan developing b: S →_ℓ of ,
 b ^∗ϑ_ℓ = β as in Lemma <ref>, uniquely defined up to a constant factor.




 §.§ Circle bundles over manifolds of vortex sheets


In this paragraph, we define circle bundles over ^S, and _a^S, as preliminary steps in the construction of the prequantum bundle. The total space of the circle bundle over ^S, is the new type of decorated nonlinear Grassmannian defined as
    _^S,:=(S,^3)__+(S)Ø_b,

(compare with the associated bundle (<ref>)). Here Ø_b denotes the _+(S) orbit of 
the Cartan developing b∈ C^(S,_ℓ) of .
The orbit Ø_b only depends on = b,
hence the new decorated Grassmannian doesn't depend on the choice of the Cartan developing b for .
Any another Cartan developing is a constant multiple b'=zb of b,
and by the surjectivity of c_ it can be written as b'=c_()b=bø^-1∈Ø_b for some ∈_+(S,).


The map 

    [f,b']↦(f(S),f_*b')=(,b_),   f∈(S,^3), b'∈Ø_b,
 
makes the identification of (<ref>) with 

    Gr^S, β_ ex   = {(, b_) ∈Gr^S(^3) × C^∞(,_ℓ):  ∃ Ψ∈Diff(S,), Ψ ^∗ b_ =  b }.


The decorated Grassmannian Gr^S, β_ ex can be seen as the base of a principal bundle 
with structure group the normal subgroup Diff _(S,)  ⊆Diff_+(S,β), defined as the kernel of 
 the flux homomorphism considered in (<ref>) 
restricted to _+(S,), namely

    Diff _(S,)
    := c_= {φ∈Diff_+ (S, β ) : b ∘=b}.

This group formally integrates the Lie subalgebra 
of (S,):

    𝔛  _ ex(S, β )
    :={ v ∈𝔛  (S,β ) :β (v)=0}.

Since the homomorphism c_ β is surjective, the quotient group _+(S,)/_(S,) is isomorphic to the circle _ℓ.



To summarize, we formally get three principal bundles with structure groups _+(S,), 
_(S,)
and _ℓ≃_+(S,)/_(S,) as illustrated in the following diagram

    (S,^3)[rrr]^_(S,)_π _ ex ^ β@/_1pc/[rrrd]^π ^ β__+(S,)         _^S,[d]^_ℓ _Π̅
             ^S,
 
with the principal bundle projections 

    π^_(f)=(f(S),f_*b)  and  Π̅(, b_) = (,  b_).

The commutativity of the diagram, Π̅øπ_^=π^,
follows from f_*b=f_* b=f_*. The principal _ℓ action  on Gr^S, β_ ex is reminiscent from the principal _+(S,)
action  (by reparametrization) on the manifold of embeddings:

    c_()·π_^(f)=π_^(fø),   ∈_+(S,), 
    f∈(S,^3).
 
It can be written simply as 

    z·(,b_)=(,zb_),   z∈_ℓ

because,  by the  identity c_()b=_*b, we have:

    z·(,b_)   =c_()·π_^(f)
    (<ref>)=π_^(fø)=(f((S)),f_*_*b)
    =(f(S),c_()f_*b)=(,zb_).



In the constant enclosed volume setting, a similar construction yields a circle bundle over _a, ^S,, obtained as the associated bundle _a, ^S,:=_a(S,^3)__+(S)Ø_b, in a similar way with (<ref>). Note the identification

    Gr^S, β_a,  ex= 
    {(,b_)∈_^S,:∈_a^S}.

See the left hand side of the diagram (<ref>) for the analogue to the commuting diagram  (<ref>) in the constant enclosed volume setting.




  
Splitting of the tangent space. There is again a  natural splitting of the tangent space to the associated bundle
(S,^3)__+(S)Ø_b
into its horizontal and vertical parts, using the Euclidean metric on ^3.
The tangent space to the _+(S) orbit Ø_b of b:S→_ℓ,
is  { i _v:v∈(S)}=β( 𝔛  (S))
=C^(S), because the 1-form  has no zeros.
The analogue of the  decomposition (<ref>) is

    T_(,b _)_^S,
    =T_^S×β _( 𝔛  ())
    =C^() × C^∞().

The bundle projection π_^  being (^3)-equivariant, the tangent map  can be written, similarly to (<ref>), as

    T_fπ^ β _ ex (u ∘ f)= ( u|_· n, i  _u|_^⊤β _),
      _=f_*,

where u|_=u|_^⊤+u|_^⊥ is the Euclidean orthogonal decomposition of u∈(^3) along =f(S).
The right hand side is the  infinitesimal generator of the (^3) action 
on Gr_^S, β.





 §.§ The prequantum bundle over the manifold of vortex sheets


We show that the Onsager-Feynman prequantization condition aℓ∈ 2π permits the construction of a prequantum bundle ( 𝒫 , Θ ) over _a^S,.


Consider on _a(S,^3) the 1-form given by θ:= ν·, namely,

    þ(uø f)= ∫_S f^* i _uν∧,

for all u∈_ vol(^3) (see the transitivity Lemma <ref>).
We have seen that the identity = d þ with ω=μ· given in (<ref>)
follows from μ= d ν, using the hat calculus.
We have also seen that the 2-form  is π^ basic (see Proposition <ref>).
Now we check that the 1-form þ is π_^ basic.
Indeed, the infinitesimal generator _v of v∈_(S,), namely _v(f)=Tfø v, annihilates þ:

    þ(Tfø v)=∫_S i _vf^*ν∧=-(v)∫_Sf^*ν=-a(v)=0.

Thus  the _+(S,) invariant 1-form  þ descends to a _ℓ invariant 1-form  on _a,^S,, (π_^)^*=þ.
Now the calculation
(π_^)^*Π̅^*=(π^)^*=
= d þ= d (π_^)^*=(π_^)^* d
implies that Π̅^*= d.
We summarise these facts in the next lemma.


 
The principal _ℓ bundle  

    Π̅:Gr^S, β_a,  ex⟶Gr^S, β_a,
      Π̅(,b_)=(, b_)
 
is endowed with a _ℓ-invariant 1-form 
that satisfies Π̅^*= d.
 

In the splitting 
T_(,b _)_a,^S,
=C_0^() × C^∞() deduced from (<ref>),
the 1-form  is expressed as

    Θ̅_(, b_)( , λ )
    =∫_  i  _nν∧_-∫_λν .

By the transitivity Lemma <ref>, it is enough to check the formula on an infinitesimal generator 
(<ref>) for u∈_ vol(^3),
namely on  (,)=(u· n, i _u|_^⊤_).
Then by (<ref>) we have _(,b_)(,)=þ(uø f)=∫_  i _uν∧_
=∫_  i  _nν∧_-∫_λν.



In the next lemma, we show that the 1-form   reproduces the  infinitesimal generators of the principal _ℓ action only up to the factor a>0, hence a further step is needed to construct the prequantum bundle.


 For all (,b_)∈Gr^S, β_a,  ex,
the infinitesimal generator _s of the _ℓ action is

    (_s(,b_))=as,    s∈.




Given s∈, we choose v∈(S,) with (v)=-s and we
let ϕ_t^v∈(S,) denote its flow. 
By (<ref>),  the curve z(t):=c_(ϕ_t^v)∈_ℓ satisfies z'(0)=-(v)=s, hence the infinitesimal generator of s can be written as
_s(,b_)=
ddt|_0 (z(t)·(,b_)).
We consider f∈_a(S,^3) such that  (,b_)=π_^(f).
Then 

    z(t)·(,b_)=c_(ϕ_t^v)·π_^(f)(<ref>)=π_^(føϕ_t^v).

Now we use (π_^)^*=þ to compute

    (_s(,b_))
       =(ddt|_0π_^(føϕ_t^v))
    =(π_^)^*(ddt|_0(føϕ_t^v))=þ(Tfø v)
    (<ref>)=-a(v)=as,

which shows the identity (<ref>).




Now the prequantization condition k = aℓ2π∈ℤ comes into play: the multiplication by a is a well defined surjective group homomorphism

    m_a:_ℓ=ℝ/ℓℤ⟶_2π=ℝ/2πℤ,
       s+ ℓ↦ as+2π.

The kernel of m_a, denoted by K, is isomorphic to _k, and the quotient group is _ℓ/K≃_2π.

We define 𝒫 as the quotient of _a,^S, with respect to the principal _ℓ action restricted to the subgroup K:

    q:_a,^S,⟶:=_a,^S,/K.
 
Notice that the bundle projection  Π̅ in Lemma <ref> descends to 

    Π:⟶Gr^S, β_a,
      Πø q=Π̅,

a principal _2π bundle for the action on  that descends from the principal _ℓ action on _a,^S,:

    m_a(z)· q(,b_):=q(z·(,b_)),   z∈_ℓ.

Here m_a(z) runs through the whole  group _2π.
The diagram below illustrates these constructions.

    ( _a(S,^3), θ) [rrr]^_(S,)_π _ ex ^ β@/_1pc/[rrrd]^π ^ β__+(S,)         ( _a,^S,[d]^_ℓ _Π̅, Θ̅) [rrr]^K_q          ( 𝒫 = _a,^S,/K,Θ) @/^1pc/[llld]_Π^_2 π
             _a^S,
 



 
If the Onsager-Feynman condition aℓ∈ 2πℤ holds, the manifold of vortex sheets 
Gr_a^S, β endowed with the symplectic form  is prequantizable.
A prequantum bundle is given by  the principal _2π bundle Π:→Gr^S, β_a
endowed with the connection 1-form  that descends from , q^*=. In particular, the coadjoint orbits of vortex sheets satisfying the Onsager-Feynman condition are prequantizable.

 
The 1-form  is _2π≃_ℓ/K invariant because  is _ℓ-invariant.
Note that, by (<ref>), the infinitesimal generators 
_s on _a,^S, and _as on 𝒫 are q-related.
Now, since q^*=, the identity (<ref>) implies that
  reproduces the generators of the infinitesimal principal _2π-action.
Thus the 1-form  is a principal connection.
That its curvature  is the symplectic form , d =Π^*,
follows from Lemma <ref>














§ CHARACTERS AND POLARIZATIONS


In this section we present a polarization for the momentum J(,_)
∈_(^3)^*, with (,_)∈_a^S,,
polarization that extends the one defined in <cit.> to closed vortex sheets.
Assuming that the Onsager-Feynman condition k = aℓ∈ 2πℤ holds, with ℓ the smallest period of _, we build geometrically a character for this momentum. This character can be extended from the isotropy subgroup of (, β  _ Σ ) to the polarization subgroup.



 §.§ Polarization subgroup


A character associated to a momentum m∈^* is  a group homomorphism
χ:G_m→_2π defined on the isotropy subgroup associated to m
(for the coadjoint action),
that integrates the Lie algebra homomorphism obtained by the restriction of 
the momentum to its isotropy Lie algebra
_m,  d _eχ= m|_𝔤_m.  It defines a line bundle  G_χℂ→ G/G_m over the coadjoint orbit Ø_m. 

A polarization group associated to a momentum m∈^*
is a Lie group H  such that G_m⊂ H⊂ G, whose Lie algebra  satisfies m|_[,]=0 (m|_:→ Lie algebra homomorphism).
Moreover, if the character χ can be extended to a group homomorphism on H,
then we also get a polarization line bundle G_χℂ→ G/H 
over the configuration space.
In finite dimensions it is also asked that (G/H)=1/2 (G/G_m), so the configuration space  is
half the dimension of the coadjoint orbit.



Let (,_)∈_a^S, be a vortex sheet Σ fibered by
vortex lines with vorticity density described by the closed 1-form β _ Σ.
Let us consider the subgroup Diff_(^3)_⊆Diff_(^3) consisting of all compactly supported volume preserving diffeomorphisms of ^3 that leave  invariant as a set. It contains the isotropy subgroup Diff_(^3)_( Σ , _).


The identity component H of Diff_(^3)_
is a polarization group for the momentum J(,_).



The corresponding Lie algebra
=𝔛  _(^3)_, which consists of divergence free
vector fields that are tangent to , satisfies the 
polarization condition J(,_)|_[,]=0. Indeed, for all X_α _1 , X_α _2 ∈𝔥, we have

    ⟨ J(, _), [ X_α _1 , X_α _2 ] ⟩ =∫_  (  i  _ X_α _1  i  _ X_α _2 μ )∧_ =0,

using that  fact that i  _ X_α _1  i  _ X_α _2 μ is a potential form for
the divergence free vector field [ X_α _1 , X_α _2 ]
on ^3.

















 §.§ A geometric construction of the character


Here S is again a compact connected oriented surface of genus one, endowed with a closed 1-form 
without zeros and with discrete period group ℓ.
Let us choose (,_)∈_a^S, and  b_:→_ℓ
 a Cartan developing for _.
The Fubini theorem on fiber bundles (<ref>) permits to write the momentum 
associated to it as:

    ⟨ J(,_),X_⟩=∫_∧_ =∫__ℓ(∫_b_^-1(z))ϑ_ℓ .


As a first step towards the construction of the character, we integrate the
Lie algebra homomorphism J(,_)|_ to a group homomorphism 
χ̃:H̃→, with H̃ the universal covering group of 
the polarization group H, the identity component of _(^3)_.
A point in H̃ is a homotopy class of smooth paths _t∈ H
with fixed endpoints: the identity _0 and _1=.
The fact that the group structure on this universal cover 
can be viewed either as composition of diffeomorphisms or as path juxtaposition, 
as done in <cit.> for the Hamiltonian group,
will be used in the proof of Lemma <ref> below.
 


For each z∈_ℓ, let C_z=b_^-1(z) be the fiber over z.
The isotopy _t preserves  ,
hence the 2-chain C_z^_t swept out by C_z under
the path of diffeomorphisms _t
from time 0 to time 1 stays on .
A smooth singular 2-chain D_z in ^3 with boundary C_z
sweeps out a smooth singular 3-chain D_z^_t under the path of diffeomorphisms _t from time 0 to time 1.
In the special case when D_z can be chosen inside the genus one surface ,
then D_z^_t stays inside , as illustrated in Figure 1.
With these notations, we state the following result.



 
The map assigning to the homotopy class {_t} the total volume of D_z^_t over the base  _ℓ,

    χ:H→,  χ({_t}):=∫__ℓ (∫_D_z^_tμ) ϑ_ℓ,

is a well defined group homomorphism.
 

 The volume ∫_D_z^_tμ swept out by D_z only depends on C_z and _t and not on the choice of D_z. Indeed, for D_z=D̅_z=C_z and μ=dν, we get

    ∫_D_z^_t-D̅_z^_tμ
    =∫__1(D_z-D̅_z)ν-∫_D_z-D̅_zν
    =∫_D_z-D̅_z(_1^*ν-ν)=0,
 
because D_z-D̅_z is a cycle and _1^*ν-ν∈^2(^3) is  exact.
The independence of b_ follows by the translation invariance of the volume form ϑ_ℓ on _ℓ, see the proof of Lemma <ref>.

Next we check the independence of (<ref>) on the path in the homotopy class
{_t}. Let _t be a path from the identity to 
homotopic to _t through the homotopy _s,t∈ H with fixed endpoints.
The boundary of the 3-chain swept out by C_z under this homotopy,
C_z^_s,t is C_z^_t-C_z^_t and coincides with the boundary of the 3-chain D_z^_t-D_z^_t. We get

    ∫_D_z^_tμ-∫_D_z^_tμ
    =∫_C_z^_t-C_z^_tν
    =∫_ C_z^_s,tν
    =∫_C_z^_s,tμ=0
 
by dimensional reasons: the 3-chain C_z^_s,t lies on the surface ,
since each diffeomorphism _s,t preserves .
Let _t and _t be two paths in H starting at the identity. 
We use the homotopy equivalence of the paths

    _tø_t   and   _t:=
    _2t,     t∈[0,12]
    _1ø_2t-1,     t∈[12,1]

in the next computation, taking into account that _1^*μ=μ,

    χ̃({_t}{_t})=χ̃({_t})
    =∫__ℓ (∫_D_z^_tμ) ϑ_ℓ
    +∫__ℓ (∫_D_z^_t_1^*μ) ϑ_ℓ
    =χ̃({_t})+χ̃({_t}).

This ensures the homomorphism property of χ̃.





Each path is the flow of a time dependent vector field u_t∈ =_(^3)_, namely ddt_t=u_tø_t.
The volume swept out by D_z can also be expressed 
with the help of potential 1-forms _t∈^1(^3), i _u_tμ= d _t, as

    ∫_D_z^_tμ=∫_0^1(∫_ D_z i __t^*u_tμ)dt=∫_0^1(∫_ D_z_t^*( i _u_tμ))dt
    =∫_0^1(∫_ C_z_t^*_t)dt.

A formula for the character in the style of (<ref>) follows:

    χ({_t})
    =∫__ℓ(∫_0^1(∫_ C_z_t^*_t)dt)ϑ_ℓ.




The group homomorphism χ:H→ integrates the restriction to the polarization Lie algebra =_(^3)_ 
of the momentum
J(,_) associated to (,_)∈_a^S,.



Let ϕ_t^u denote the flow at time t of the vector field u∈
and consider the path ↦{ϕ_t^u}_0≤ t≤ of homotopy classes in H. 
Its derivative at =0 is equal to  u, so  the associated Lie algebra homomorphism is

    u=X_↦d/d|_=0χ( {ϕ_t^u}_0≤ t≤)
       (<ref>)=d/d|_=0∫__ℓ(∫_0^(∫_ C_z(ϕ_t^u)^*)dt)ϑ_ℓ
       =
    ∫__ℓ(∫_ C_z)ϑ_ℓ(<ref>)=⟨ J(,_),X_⟩,

which is the required momentum.



Let (,_)∈_a^S, and
assume that the Onsager-Feynman condition aℓ∈ 2πℤ holds.
Then a  character on the polarization group H is given by

    χ:H→_2π,  χ():=
    p_2π(∫__ℓ(∫_D_z^_tμ)ϑ_ℓ),
 
where _t∈ H is any path from the identity to .



The fundamental group π_1(H) consists of homotopy classes of loops _t∈ H based at the identity. We only need to check that every such class {_t}∈π_1(H),
is mapped by the homomorphism  χ̃, given in (<ref>), to 2π.
The boundary of the 3-chain D_z^_t swept out by D_z
 under the loop of diffeomorphisms _t
is the  2-chain C_z^_t⊆^3
swept out by the fiber C_z= D_z⊆.
Notice that  each diffeomorphism _t∈ H
preserves the surface  and we get a smooth singular 2-cycle C_z^_t⊆, which is also integral. Thus its homology class must be 
an integer multiple  of the fundamental class of the surface .
The integer will not depend on z∈_ℓ (by continuity reasons), hence 
[C_z^_t]=k[] with k∈.
(In the special case when the smooth singular 2-chains D_z can be chosen inside , 
the whole 3-chain D_z^_t remains inside  and it covers it k times.)

The total volume enclosed by  is  a=∫_ν.
Then

    χ({_t})
    =∫__ℓ (∫_D_z^_tμ) ϑ_ℓ
    =∫__ℓ (∫_C_z^_tν) ϑ_ℓ
    =∫__ℓ (∫_kν) ϑ_ℓ
    =∫__ℓ(ka)ϑ_ℓ=kaℓ∈ 2π,

by the prequantization condition aℓ∈ 2πℤ.



The isotropy subgroups _(^3)_(,_) and _(^3)_J(,_) coincide.
We have seen in <ref> that an element  of the isotropy subgroup
maps the vortex line C_z into the vortex line C_cz with
c=c__().
We show below 
a  character formula that doesn't involve isotopies.



The restriction of the character χ from the Proposition <ref> to the isotropy subgroup of (,_)∈_a^S, is also given by the formula

    χ()=p_2π(∫_ d ^-1(^*ν-ν)∧_)m_a(c__()^-1),

where p_2π:→_2π denotes the canonical projection and  m_a:_ℓ→_2π the multiplication (<ref>).


Let u_t∈_(^3)_(,_) be the time dependent vector field
associated to the path _t∈_(^3)_(,_), and let _t be the potential 1-forms. Then d_t=_u_tν-di_u_tν.
We apply the Fubini theorem to (<ref>) and we compute

    χ({_t})   =∫__ℓ(∫_0^1(∫_ C_z_t^*_t)dt)ϑ_ℓ
    =∫_0^1∫__t^*_t∧_ dt
       =∫_0^1∫_ d^-1(_t^*_u_tν)∧_ dt
    -∫_0^1∫__t^*i_u_tν∧_ dt
       =∫_0^1∫_d/dt d^-1(_t^*ν-ν)∧_ dt
    -∫_0^1∫_ i_u_tν∧_ dt
       =∫_ d^-1(_1^*ν-ν)∧_
    +a∫_0^1_(u_t) dt,

using at step four  the fact that _t preserves  and _,
at step five the fact that u_t is tangent to , and at the last step
the fact that _(u_t) is a constant function on .
We know that p_2π(χ({_t}))=χ(_1), so the result follows from:

    p_2π(a∫_0^1_(u_t) dt)=m_a(p_ℓ(∫_0^1_(u_t) dt))
    =m_a(c__(_1)^-1),

knowing from (<ref>) that the derivative of the flux homomorphism c__ is -_.






  
Acknowledgement. Both authors were partially supported by the LEA Franco-Roumain “MathMode". FGB was also partially supported by the ANR project GEOMFLUID, ANR-14-CE23-0002-01. CV was also partially supported by CNCS UEFISCDI, project number  PN-III-P4-ID-PCE-2016-0778.
We would like to thank Stefan Haller and Boris Khesin  for very helpful comments and references.








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