Euler flows and singular geometric structures

EULER FLOWS AND SINGULAR GEOMETRIC STRUCTURES ROBERT CARDONA, EVA MIRANDA, AND DANIEL PERALTA-SALAS Abstract. Tichler proved in [24] that a manifold admitting a smooth non vanishing and closed one-form fibers over a circle. More generally a manifold admitting kindependent closed one-forms fibers over a torus Tk. In this article we explain a version of this construction for manifolds with boundary using the techniques of b-calculus [18, 13]. We explore new applications of this idea to Fluid Dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold’s structure theorem and describe b-symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities [19] and what we call b-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance [8]). These results provide a new technique to analyze the geometry of steady fluid flows on non-compact manifolds with cylindrical ends. 1. Introduction The existence of closed one-forms on a manifold simplifies the topology of the manifold in a similar way in which the existence of first integrals of a dynamical system simplifies the topology of its invariant sets. This idea dates back to the work of Tichler who proved in 1970 that a compact manifold admitting a nowhere vanishing closed one-form is a fibration over a circle, or more generally, the existence of kindependent closed one-forms implies that the manifold is a fibration over a k-dimensional torus. In a dual language, the existence of first integrals also adds constraints on the topology of the invariant manifolds, and the classical Arnold-Liouville theorem shows that an integrable system on a symplectic manifold has tori as compact invariant submanifolds (see [5] for an application of Tichler’s ideas to provide a new proof of Arnold-Liouville theorem). This same order of ideas can be applied to a more general picture in order to consider Fluid Dynamics and, more concretely, steady Euler flows on manifolds. In particular, we give a new proof of Arnold’s structure theorem when the Bernoulli function is not constant, which is based on Tischler’s theorem for manifolds with boundary. This starting point takes us to consider manifolds with boundary and b2k-forms, thus providing a proof of the b2k-Tichler theorem. Additionally, we analyze the singular level sets of the Bernoulli function, which are not considered in Arnold’s theorem, and prove that under some assumptions they can be described as b-symplectic manifolds. When the Bernoulli function is constant, we reconsider the correspondence between Beltrami fields and contact structures and extend it to contact manifolds with cylindrical ends (compactified as b-manifolds) thus obtaining a new correspondence between Beltrami fields in this case with the Robert Cardona is supported by FPI-BGSMath doctoral grant. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016 and partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR). Daniel Peralta-Salas is supported by the ERC Starting Grant 335079, the MTM grant 2016-76702-P, and partially supported by the ICMAT–Severo Ochoa grant SEV-2015-0554. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while Eva Miranda was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester. 1 2 ROBERT CARDONA, EVA MIRANDA, AND DANIEL PERALTA-SALAS b-contact manifolds recently introduced in [19]. Several questions concerning the Hamiltonian and Reeb dynamics of b-contact manifolds, such as the existence of periodic orbits, can be extremely useful to understand some properties of the stream lines of Beltrami flows on manifolds with cylindrical ends. Organization of this paper: In Section 2 we introduce bm-forms and study their desingularization. A special focus is given to the study of bm-symplectic and b-contact forms. In Section 3 we prove a Tischler theorem for manifolds with boundary using b2k-forms. In the smooth case, the result holds under suitable hypotheses and we use it to provide a new proof, in Section 4, of Arnold’s structure theorem for steady Euler flows. We analyze some singular level sets of the Bernoulli function in Section 5, in the context where Arnold’s theorem holds. Assuming the Bernoulli function is Morse-Bott, we find singular symplectic structures in some of these sets after resolving their topological singularities. Finally, in Section 6 we study steady Euler flows on manifolds with cylindrical ends and provide a correspondence between Beltrami fields on b-manifolds and b-contact structures. Acknowledgements: We are thankful to the referees of this paper for their careful and efficient work, and their interesting observations. 2. A crash course on bm-forms and their desingularization In this section we follow closely [13] and [19] to introduce singular symplectic and contact structures that will be of utter relevance in the study of Fluid Dynamics on manifolds with boundary. 2.1. b-symplectic manifolds. The language of b-forms was introduced by Melrose [18] in order to study manifolds with boundary. The subject gained attention in the realm of Poisson geometry as a special class of Poisson manifolds can be studied using b-calculus [13]. Most definitions can be used replacing the boundary by any given hypersurface of a manifold without boundary: Definition 2.1. Ab-manifold (M, Z) is an oriented manifold Mwith an oriented hypersurface Z. Remark 2.2. It is possible to extend this definition to consider non-orientable manifolds. See for instance [12] and [20]. In order to have the b-category we introduce the notion of b-map. Definition 2.3. Ab-map is a map f: (M1, Z1)−→ (M2, Z2) so that fis transverse to Z2and f−1(Z2) = Z1. Vector fields and differential forms have to be redefined also. Definition 2.4. Ab-vector field on a b-manifold (M, Z) is a vector field which is tangent to Zat every point p∈Z. Observe, in particular, that a b-vector field is tangent to the hypersurface Z, so from a dynamical point of view Zis an invariant manifold by the flow of these vector fields. These b-vector fields form a Lie subalgebra of vector fields on M. Let tbe a defining function of Zin a neighborhood Uand (t, x2, ..., xn) be a chart on it. Then the set of b-vector fields on Uis a free C∞(U)-module with basis t∂ ∂t,∂ ∂x2 ,..., ∂ ∂xn. EULER FLOWS AND SINGULAR GEOMETRIC STRUCTURES 3 We deduce that the sheaf of b-vector fields on Mis a locally free C∞-module and therefore it is given by the sections of a vector bundle on M. This vector bundle is called the b-tangent bundle and denoted by bTM. Its dual bundle is called the b-cotangent bundle and is denoted bT∗M. By considering sections of powers of this bundle we obtain b-forms. Definition 2.5. Let (M2n, Z) be a b-manifold and ω∈bΩ2(M) a closed b-form. We say that ωis b-symplectic if ωpis of maximal rank as an element of Λ2(bT∗ pM) for all p∈M. In the class of Poisson manifolds there is the distinguished subclass of b-Poisson manifolds which is indeed formed by b-symplectic manifolds together with a bi-vector field naturally associated to the b-symplectic forms. Definition 2.6. Let (M2n,Π) be an oriented Poisson manifold. Let the map p∈M7→ (Π(p))n∈Λ2n(TM) be transverse to the zero section. Then Π is called a b-Poisson structure on M. The hypersurface Zwhere the multivectorfield Πnvanishes, Z={p∈M|(Π(p))n= 0} is called the critical hypersurface of Π. The pair (M, Π) is called a b-Poisson manifold. The transversality condition is equivalent to saying that 0 is a regular value of the map p−→ (Π(p))n. The hypersurface Zhas a defining function obtained by dividing this map by a nonvanishing section of V2n(TM). The set of b-symplectic manifolds is in one-to-one correspondence with the set of b-Poisson manifolds. This correspondence, detailed in [13], can be formulated as Proposition 2.7. A two-form ωon a b-manifold (M, Z)is b-symplectic if and only if its dual bivector field Πis a b-Poisson structure. In this context we have a normal form theorem analogous to Darboux theorem for symplectic manifolds. This result is also proved in [13]. Theorem 2.8 (b-Darboux theorem). Let (M, Z, ω)be a b-symplectic manifold. Then, on a neighborhood of a point p∈Z, there exist coordinates (x1, y1, ..., xn, yn)centered at psuch that ω=1 x1 dx1∧dy1+ n X i=2 dxi∧dyi. Note that with this chart, the symplectic foliation of (M, Π) has a specific form. It has two open subsets where the Poisson structure has maximal rank given by {x1>0}and {x1<0}. The hyperplane {x1= 0}contains leaves of dimension 2n−2 given by the level sets of y1. One of the research directions has been to generalize b-structures and consider more degenerate singularities of the Poisson structure. This is the case of bm-Poisson structures, for which ωnhas a singularity of An-type in Arnold’s list of simple singularities [2] [3]. A dual approach is also possible and interesting, working with forms instead of bivector fields. Definition 2.9. A symplectic bm-manifold is a pair (M2n, Z) with a closed bm-two form ωwhich has maximal rank at every p∈M. Such as in the b-symplectic case, an analogous bm-Darboux theorem holds. A decomposition for these forms is given in [22]. 4 ROBERT CARDONA, EVA MIRANDA, AND DANIEL PERALTA-SALAS Definition 2.10. A Laurent Series of a closed bm-form ωis a decomposition of ωin a tubular neighborhood Uof Zof the form ω=dx xm∧( m−1 X i=0 π∗(ˆαi)xi) + β, where π:U→Zis the projection, where each ˆαiis a closed form on Z, and βis form on U. It is proved in [22] that every closed bm-form admits in a tubular neighborhood Uof Za Laurent form of this type, when fixing a semi-local defining function. 2.2. b-contact manifolds. Following these ideas and in analogy with contact structures, b-contact structures are developed in [19]. Definition 2.11. Let (M, Z) be a (2n+1)-dimensional b-manifold. A b-contact structure is the distribution given by the kernel of a one b-form ξ= ker α⊂bTM,α∈bΩ1(M), that satisfies α∧(dα)n6= 0 as a section of Λ2n+1(bT∗M). We say that αis a b-contact form and the pair (M, ξ) ab-contact manifold. As in contact geometry one can define the Reeb vector field that satisfies (iRαdα = 0 α(Rα)=1. A Darboux type theorem can be proved, providing a normal local form for these structures. Theorem 2.12. Let αbe a b-contact form inducing a b-contact structure ξon a b-manifold (M, Z) of dimension (2n+ 1) and p∈Z. We can find a local chart (U, z, x1, y1, . . . , xn, yn)centered at p such that on Uthe hypersurface Zis locally defined by z= 0 and (1) if Rp6= 0 (a) ξpis singular, then α|U=dx1+y1 dz z+ n X i=2 xidyi, (b) ξpis regular, then α|U=dx1+y1 dz z+dz z+ n X i=2 xidyi, (2) if Rp= 0, then ˜α=fα for f(p)6= 0, where ˜αp=dz z+ n X i=1 xidyi. Remark 2.13. There is also a dual correspondence between b-contact structures and other structures that play the role of Poisson in the contact context: Jacobi manifolds. The particular subclass is the one of b-Jacobi manifolds that satisfy also a transversality condition. For more details you may consult [19]. EULER FLOWS AND SINGULAR GEOMETRIC STRUCTURES 5 2.3. Desingularizing bm-forms. In [14] a desingularization procedure for bm-symplectic manifolds associates a family of folded symplectic or symplectic forms to a given bm-symplectic structure depending on the parity of m. Namely, Theorem 2.14 (Guillemin-Miranda-Weitsman, [14]).Let ωbe a bm-symplectic structure on a compact orientable manifold Mand let Zbe its critical hypersurface. •If m= 2k, then there exists a family of symplectic forms ωwhich coincide with the bmsymplectic form ωoutside an -neighborhood of Zand for which the family of bivector fields (ω)−1converges in the C2k−1-topology to the Poisson structure ω−1as →0. •If m= 2k+ 1, then there exists a family of folded symplectic forms ωwhich coincide with the bm-symplectic form ωoutside an -neighborhood of Z. This desingularization can be applied to any bm-form as detailed in [6]. Let us describe how the desingularization works in the even and odd case. Case I: even m. Assume m= 2kand let f∈ C∞(R) be an odd smooth function such that f0(x)>0 for all x∈[−1,1] as shown below, and satisfying f(x) = (−1 (2k−1)x2k−1−2 for x < −1 −1 (2k−1)x2k−1+ 2 for x > 1 outside the interval [−1,1]. Scaling the function consider the function f(x) := 1 2k−1fx . And outside the interval, f(x) = (−1 (2k−1)x2k−1−2 2k−1for x < − −1 (2k−1)x2k−1+2 2k−1for x> Replacing dx x2kby dfin the semi-local expression on Uwe obtain ω=df∧α+β. We call this form an f-desingularization of ω. 6 ROBERT CARDONA, EVA MIRANDA, AND DANIEL PERALTA-SALAS Case II: odd m. Consider m= 2k+ 1, and consider a function f∈C∞(R) satisfying •f(x) = f(−x) •f0(x)>0 if x > 0 •f(x) = x2−2 if x∈[−1,1] •f(x) = log(|x|) if k= 0, x∈R\[−2,2] •f(x) = −1 (2k+2)x2k+2 if k > 0, x∈R\[−2,2]. Taking the width of a tubular neighborhood of Zdefine f(x) := 1 2kfx  and consider the form ω=df∧α+β. The f-desingularization is again smooth and dfvanishes transversally at Z. When ωis closed, its Laurent decomposition can be used as done in [14] to conclude that ωis also closed. 3. A Tischler theorem for manifolds with boundary Let us recall Tischler theorem [24] as presented in [5]. Theorem 3.1. Let Mnbe a closed manifold endowed with rlinearly independent closed 1-forms βi, i = 1, . . . , r which are nowhere vanishing. Then Mnfibers over a torus Tr. As a remark in Tischler’s original paper, the theorem also holds for compact manifolds with boundary with an extra assumption. Theorem 3.2. Let Mnbe a compact connected manifold with boundary endowed with rlinearly independent closed 1-forms βi, i = 1, . . . , r which are nowhere vanishing and satisfy these conditions when restricted to the boundary. Then Mnfibers over a torus Tr. Using the language of b2k-forms and the deblogging procedure, one can state a Tischler theorem for manifolds with boundary. This theorem gives more information than the one we would get by simply applying the classical Tichler theorem restricted to the boundary. EULER FLOWS AND SINGULAR GEOMETRIC STRUCTURES 7 Definition 3.3. Let Mbe a manifold with boundary. Its double ¯ Mis obtained by taking two copies of Mand gluing along their boundary. ¯ M=M× {0,1}/∼, where (x, 0) ∼(x, 1) for all x∈∂M. Theorem 3.4. Let α1, ..., αrbe closed one b2k-forms in a b2k-manifold Msuch that α1∧· · ·∧αr6= 0 everywhere in M. If the pullback of the forms to the boundary are also independent then Mfibers over Tr. Otherwise the double ¯ Mfibers over Trand the glued boundary fibers over Tr−1. Proof. If the forms are also independent when pullbacked to the boundary, we can apply the desingularization that we will detail for the second case in the manifold with boundary and apply Theorem 3.2. Otherwise at least one of the forms has a singular part and one considers the extension of the forms αiinto ¯ Mby symmetry. In this way we obtain a b2k-manifold ¯ Mwith critical hypersurface Zwhere the boundaries have been glued. We can proceed to desingularize the 1-forms following [14]. Namely, the forms are closed and admit Laurent series in a neighborhood Uof Z, αi= ( 2k−1 X j=0 αj itj)dt t2k+βi, for ta positively oriented defining function. Here each αj iis a constant function and βiis smooth in Z. The term α0 iis constant and the only non vanishing term of the singular part at the hypersurface Z. The rest of terms αj ifor j6= 0 are paired with powers of tthat vanish at Z. The dividing term of dt t2kdoes not cancel the powers of tbecause of the structure of the b2k-cotangent bundle: one has to think of dt t2kas if it was a d˜ tfor a coordinate ˜ t. Since at least one of these α0 iis non vanishing, we can assume α0 16= 0. Redefining αi:= αi−α0 i α0 1 α1,for i= 2, ..., n we can assume that only the first form has a singular part at the hypersurface and independence of the forms still holds. Proceeding to the desingularization, one can take a suitable and the desingularized forms αi, =df∧( k X j=0 αj itj) + βi. Since we have α1∧... ∧αr6= 0,df6= 0 and at least one singular form (for instance the first one α0 16= 0) we deduce that α1, ∧... ∧αr, 6= 0 using elementary linear algebra as αi, and αidetermine the same matrix of coefficients. One has simply changed the form dt dt2kof the basis by df. Applying Theorem 3.1 we deduce that ¯ Mfibers over Tr. Observe that in Zthe form α1was the only one with a non vanishing singular term. Hence its the only one with a non vanishing term for df: we deduce that α2,, ..., αr, are independent when restricted to Zagain by linear algebra. In particular, Zfibers over Tr−1.  Remark 3.5. The parity (evenness) of mcomes from the desingularization procedure. The desingularized form obtained from a non-vanishing bm-form is non-vanishing only when mis even. For odd m, as explained in Section 2.3, the resulting form has a zero. This zero cannot be eliminated because the singular part of the form changes sign when crossing the hypersurface. This is why the conditions of the second statement of Theorem 3.4 cannot be met for odd m. However, the 8 ROBERT CARDONA, EVA MIRANDA, AND DANIEL PERALTA-SALAS first part can be obtained by adding a constant to the desingularization formula to prevent the desingularized form from vanishing at the boundary. Example 3.6. An easy example to consider is the compact cylinder Cvisualized as a subset of the torus T2(as quotient of the plane T2∼ =(R/Z)2). Consider the b2k-forms 1 sin(2πx)2kdx and dy on R2. The critical set is the boundary of a compact cylinder. The forms descend to the quotient, and in the compact cylinder they satisfy the hypotheses of the theorem. Figure 1. The double of a compact cylinder Remark 3.7. The second statement can also be applied for honest De Rham forms with the following changes. Instead of one of the forms having a singular part, we ask one of the forms to be transversal to the boundary everywhere. Secondly we need that the forms can be extended to the doubling of the manifold by symmetry which might not be true in general. As an easy corollary we obtain, Corollary 3.8. An n-dimensional manifold admitting nindependent and closed b2k-forms is a compact cylinder Tn−1×[0,1]. 4. Euler equations on 3-manifolds The Euler equations model the dynamics of an inviscid and incompressible fluid flow on a 3dimensional manifold, see e.g. [4, 21]. For a smooth domain in R3if we denote by Xthe velocity field of the fluid and Pthe pressure, which is a scalar function, then the equations can be written as follows, (∂X ∂t + (X· ∇)X=−∇P div X= 0 . Another vector field that has an important role in fluid dynamics is the vorticity, which is defined as ω:= curl X. This vector field is related to the local rotation of the fluid. Using the vorticity, one can rewrite the Euler equations as, (∂X ∂t −X×ω=−∇B div X= 0 , where B=P+1 2|X|2is the Bernoulli function. For any Riemannian 3-manifold (M, g) one can write the Euler equations (∂X ∂t +∇XX=−∇P div X= 0 . EULER FLOWS AND SINGULAR GEOMETRIC STRUCTURES 9 where ∇Xis the covariant derivative, and the operators ∇and div are computed with the metric g. Using the Riemannian volume form µ, the second condition can be expressed as follows, LXµ= 0. The vorticity is then the only vector field satisfying ιωµ=dα, where α(·) = g(X, ·) is the dual one-form of Xusing the Riemannian metric. In terms of the vorticity, the equations read as in the Euclidean case: (∂X ∂t −X×ω=−∇B div X= 0 , and the Bernoulli function is defined using the Riemannian form, as well as the vector product. Stationary solutions. We will be interested in equilibrium configurations, i.e. in stationary solutions of these equations. A well known fact is that the Bernoulli function is a first integral for both Xand ω. In particular the stream lines are confined into the level sets of B. The stationary Euler equations with the Bernoulli formulation are (X×ω=∇B div X= 0 . In the analytic setting, Arnold noticed the following fact about solutions to these equations. Let (M, g) be an analytic Riemannian manifold and let Xbe an analytic solution of the equations. If Bis constant and Xis non-vanishing, the vorticity is proportional to Xeverywhere, that is curl X=fX for some analytic function f. In this case, Xis called a Beltrami flow. If Bis not constant, its critical set Cr(B) := {p∈M| ∇B(p) = 0}has a stratified structure and its codimension is at least 1. In this case, Arnold showed that the structure of the stream lines of Xis very similar to the one of integrable systems described by the Arnold-Liouville theorem (see previous sections). We now provide a new proof using the existence of certain closed one-forms as in [5]. Theorem 4.1 (Arnold’s structure theorem).Let Xbe an analytic stationary solution of the Euler equations on an analytic compact manifold with non constant Bernoulli function. The flow is assumed to be tangent to the boundary if there is one. Then there is an analytic set Cof codimension at least 1such that M\Cconsists of finitely many domains Misuch that either (1) Miis trivially fibered by invariant tori of Xand on each torus the flow is conjugated to the linear flow, (2) or Miis trivially fibered by invariant cylinders of Xwhose boundaries lie on the boundary of M, and all stream lines are periodic. Proof. We define first the analytic set C. Consider C1={B−1(c) : cis a critical value of B}and C2the level sets such that they are tangent at some point to the boundary. Take C=C1∪C2. By compactness and analyticity [4], it is a finite union of level sets of the function Band hence it is an analytic set of codimension greater or equal to one. Consider the following one-forms. On the one hand, β=ιXµ2, 16 ROBERT CARDONA, EVA MIRANDA, AND DANIEL PERALTA-SALAS Let gbe the b-metric g(u, v) = 1 h(α(u)⊗α(v)) + dα(u, Jv). The vector field Y satisfies ιYg=α. Take µ=1 hα∧dα as b-volume form in M. It obviously satisfies ιYµ=dα. Hence Yis a Beltrami field (with constant proportionality factor) for this choice of g and µ.  Example 6.4 (ABC flows).A very well-known family of Beltrami flows in T3are the ABC flows: X(x, y, z) = [Asin z+Ccos y]∂ ∂x + [Bsin x+Acos z]∂ ∂y + [Csin y+Bcos x]∂ ∂z . Everything is computed in R3and then quotiented depending on which hypersurface we consider. Taking as hypersurface Z={z= 0}one can check for which values of the parameters the b-vector field X(x, y, z)=[Asin z+Ccos y]∂ ∂x + [Bsin x+Acos z]∂ ∂y + [Csin y+Bcos x]z∂ ∂z is a Beltrami field in the corresponding b-manifold. The metric and volume forms are g=dx2+dy2+ (dz z)2, µ =dx ∧dy ∧dz z. We compute the one form α=g(X, ·) = [Asin z+Ccos y]dx + [Bsin x+Acos z]dy + [Csin y+Bcos x]dz z, and the contraction by the volume ιXµ= [−Bsin x−Acos z]dx ∧dz z+ [Asin z+Ccos y]dy ∧dz z+ [Csin y+Bcos x]dx ∧dy. It is clear that dιXµ= 0, it remains to check the equation dα =fιXµ. Computing the derivative of alpha dα = [−Bsin x−zA cos z]dx ∧dz z+ [zA sin z+Ccos y]dy ∧dz z+ [Csin y+Bcos x]dx ∧dy. When differentiating with respect to zin the b-cotangent bundle, a zfactor appears. For dα =fιXµ to be satisfied, we need A= 0 and f= 1. The two-parameter family of vector fields X(x, y, z) = Ccos y∂ ∂x +Bsin x∂ ∂y + [Csin y+Bcos x]z∂ ∂z is b-Beltrami on the b-manifold T2×Rwith a T2as critical hypersurface. To obtain a vector field in a compact manifold, one can chose sin zinstead of zas defining function of the critical set (which is now defined in the quotient to T3) and hence work with sin z∂ ∂z and dz sin z. We obtain a Beltrami field on T3with two T2as critical hypersurfaces. It is an easy computation to check that the b-vector field Xis non-vanishing as a section of bT M if and only if |B| 6=|C|. Remark 6.5. For this b-manifold the corresponding original manifold can be thought as M= R×T2. When compactifying each of the cylindrical ends we obtain a manifold diffeomorphic to T2×[0,1]. When considering its double the resulting b-manifold is T3with two T2as critical hypersurfaces. EULER FLOWS AND SINGULAR GEOMETRIC STRUCTURES 17 As an example of b-contact structure, let us compute it in the simple case C= 0 and B > 0 for ABC fields in T3, i.e. the defining function is sin z. The one b-form αin this case is α=g(X, ·) = Bsin xdy +Bcos xdz sin z. It is clearly a b-contact structure since α∧dα =B2dx ∧dy ∧dz sin z, and its Reeb vector field is R=1 Bsin x∂ ∂y +1 Bcos xsin z∂ ∂z , a rescaling of the original Beltrami field. Remark 6.6. This correspondence holds true if we consider Beltrami fields on bm-manifolds. The associated structure is then a bm-contact structure. The Weinstein conjecture [25] on periodic orbits of Reeb flows claims that any Reeb vector field admits a periodic orbit on a compact manifold. In [19] a plug-like construction is used to give a counterexample to the Weinstein conjecture on a b-contact manifold: the associated Reeb vector field does not have a smooth periodic orbit and the notion of singular periodic orbits is introduced. Corollary 6.7. A Beltrami field on a b-manifold does not necessarily have a smooth periodic orbit. In [19] the authors conjecture a singular version of the Weinstein conjecture claiming that the Reeb vector field of any compact b-contact manifold possesses at least one periodic orbit which may be singular in the following sense: Definition 6.8. Let Mbe a manifold with hypersurface Z. A singular periodic orbit is either a periodic orbit in M\Zor an orbit γsuch that limt→±∞ γ(t)∈Z. One could obtain information about the stream lines of a b-Beltrami flow depending on the possible casuistics that this conjecture opens. In particular, this would allow to establish the existence of either a periodic orbit or an unbounded orbit that escapes (in both directions) through a cylindrical end. M γ2 γ1 Figure 6. Two possible singular periodic orbits References [1] V. I. Arnold, Sur la topologie des ´ecoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris 261 (1965) 17-20. [2] V. I. Arnold, Remarks on Poisson structures on a plane and on other powers of volume elements. (Russian) Trudy Sem. Petrovsk. No. 12 (1987), 37-46, 242; translation in J. Soviet Math. 47 (1989), no. 3, 2509–2516. [3] V.I. Arnold, Critical point of smooth functions. Vancouver Intern. Congr. of Math., 1974, vol.1, 19-39. [4] V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics. Springer-Verlag, New York 1998. [5] R. Cardona and E. 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Robert Cardona, Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Universitat Polit` ecnica de Catalunya, Barcelona e-mail: [email protected] Eva Miranda, Laboratory of Geometry and Dynamical Systems, Departament of Mathematics, EPSEB, Universitat Polit` ecnica de Catalunya BGSMath Barcelona Graduate School of Mathematics in Barcelona and, IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne Universit´ e, 77 Avenue Denfert-Rochereau, 75014 Paris, France Daniel Peralta-Salas, Instituto de Ciencias Matem´ aticas-ICMAT, C/ Nicol´ as Cabrera, no13-15 Campus de Cantoblanco, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain e-mail: dper[email protected]