A characterization of 3D steady Euler flows using commuting zero-flux homologies

Ergod. Th. & Dynam. Sys. (2021), 41, 2166–2181 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. doi:10.1017/etds.2020.25 c The Author(s) 2020. Published by Cambridge University Press. A characterization of 3D steady Euler flows using commuting zero-flux homologies DANIEL PERALTA-SALAS †, ANA RECHTMAN‡ and FRANCISCO TORRES DE LIZAUR§ †Instituto de Ciencias Matem´ aticas, Consejo Superior de Investigaciones Cient´ ıficas, 28049 Madrid, Spain (e-mail: [email protected]) ‡IRMA, Universit´ e de Strasbourg, 7 rue Ren´ e Descartes, 67084 Strasbourg, France (e-mail: r[email protected]a.fr) §Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany (e-mail: [email protected]) (Received 10 April 2019 and accepted in revised form 10 February 2020) Abstract. We characterize, using commuting zero-flux homologies, those volumepreserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved. Key words: smooth dynamics, symplectic dynamics, Euler equations, volume-preserving flows 2010 Mathematics Subject Classification: 57R30, 35Q31, 76B99 (Primary) 1. Introduction and main theorem The dynamics of an inviscid and incompressible fluid flow on a Riemannian 3-manifold Mis described by the Euler equations ∂tX+ ∇XX= −∇ P,div X=0, where Xis the velocity field of the fluid (which is a non-autonomous vector field on M) and Pis the pressure function, which is uniquely defined by the equations up to a constant. The operator ∇Xdenotes the covariant derivative of a vector field along Xand div is the divergence operator computed with respect to the Riemannian volume form. When the vector field Xdoes not depend on time, we say that it is a stationary solution of the Euler equations, which models a fluid flow in equilibrium. It is well known [1,11] https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 3D steady Euler flows 2167 that the stationary Euler equations can be equivalently written as X×curl X= ∇ B,div X=0, where B:= P+1 2|X|2is the Bernoulli function of the fluid. We recall that, given a metric gand a volume form µ, the curl operator and the vector product ×are defined as icurl Xµ=d(iXg), iX×Yg=iYiXµ, where iZgdenotes the 1-form dual to the vector field Zusing the metric. An important milestone in the study of the stationary Euler flows, which marked the birth of modern topological hydrodynamics, is Arnold’s structure theorem [1]. Roughly speaking, it shows that, when Xand curl Xare not collinear, they behave as integrable Hamiltonian systems with two degrees of freedom. For extensions of this theorem to higher dimensions, see [7]. The ‘degenerate’ case corresponds to the so-called Beltrami flows, which are defined as those divergence-free vector fields such that curl Xis proportional to X(via a not necessarily constant proportionality factor). The geometric wealth of steady Euler flows has been revealed in recent years and its study has attracted the attention of many people. Etnyre and Ghrist, developing an idea suggested by Sullivan, showed the equivalence between Reeb flows of a contact form and non-vanishing Beltrami fields with constant proportionality factor [5,6] (see also [2] for an extension of this result to b-contact structures); the case of general Beltrami fields corresponds to volume-preserving geodesible flows (or Reeb flows of stable Hamiltonian structures), as noticed by Rechtman [12,13]. More recently, Cieliebak and Volkov [4] constructed steady Euler flows that are not geodesible, and Izosimov and Khesin [8] characterized the vorticity functions of two-dimensional steady Euler flows using Reeb graphs. Nevertheless, we are still far from having a deep understanding of the space of stationary solutions to the Euler equations. Our goal in this paper is to give a complete characterization of steady Euler flows ` a la Sullivan using zero-flux 2-chains for a vector field that commutes with X. This contains, as a particular case, Sullivan’s characterization of geodesible flows in terms of tangent homologies [16] (for volume-preserving fields). To this end, we introduce the definition of an Eulerizable flow. Definition 1.1. (Eulerizable flow) Let Mbe a three-dimensional manifold endowed with a volume form µ. We say that a volume-preserving vector field Xis Eulerizable if there is a Riemannian metric gon M(not necessarily inducing the volume form µ) for which Xis a stationary solution to the Euler equations X×curl X= ∇ B,LXµ=0, for some (Bernoulli) function B:M→R. Here, LXdenotes the Lie derivative. Equivalently, a volume-preserving vector field Xis Eulerizable if there is a Riemannian metric gsuch that iXdα= −d B, where α:= iXgis the 1-form dual to X. Remark 1.2. When the Riemannian volume form µgdoes not coincide with the volume form µ, the Euler equations presented above describe the behavior of an ideal https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 2168 D. Peralta-Salas, A. Rechtman and F. Torres de Lizaur barotropic fluid, that is, a fluid whose density is a function of the pressure. Indeed, writing µ=ρ µg,ρ > 0 is a function which plays the role of the density. Then the vector field Xsolves the equation ρ∇XX= −ρ∇p=: −∇ ˆpprovided that ρis a function of p:= B−1 2α(X), and LρXµg=0. In fact, if Xis a non-vanishing Eulerizable flow, the proof of the implication (ii)⇒(i)of Theorem 1.3 below shows that the metric gcan be taken to be compatible with µ, that is, the Riemannian volume form µgcoincides with µ. To state our main theorem, let us first introduce some notation. For a vector field X, we denote by FXthe set of the boundaries of zero-flux 2-chains: FX=∂c:cis a 2-chain with Zc iXµ=0. Let ZXand CXbe the cone of foliation currents and of foliation cycles of the vector field X, respectively (see Appendix A for a short review on the theory of currents). We recall that afoliation current of a vector field Xis a 1-current that can be approximated arbitrarily well (in the weak topology) by tangent 1-chains. Equivalently, a foliation current can be approximated by 1-currents of the form N X i=1 ciδpi X, with N∈N,ci∈ [0,∞)and pi∈M. Recall that, for any p∈M, the 1-current δp Xis defined as δp X(α) =αp(X)for any 1-form α. Afoliation cycle is a closed foliation current, that is, a foliation current whose kernel contains the linear subspace of exact 1-forms. Throughout this paper we say that a manifold is closed if it is compact and without boundary. The following theorem is our main result. THEOREM 1.3. Let X be a non-vanishing volume-preserving vector field on a closed 3-manifold M with trivial first cohomology group, H1(M)=0. The following properties are equivalent. (i) X is Eulerizable. (ii) There exists a 1-form αsuch that α(X) > 0and ιXdαis closed. (iii) There exists a (not identically zero) vector field Y that commutes with X (i.e. [X,Y] = 0), such that no sequence of elements in FYcan approximate a nontrivial foliation cycle of X, that is, FY∩CX= {0}. The proof of Theorem 1.3 yields the following characterization (new to the best of our knowledge) of Reeb fields in contact geometry, which is of independent interest. Notice that in its statement we do not need to assume that Xis volume-preserving. THEOREM 1.4. Let M be a closed 3-manifold which is not a fibration over the circle. Then a vector field X is the Reeb field of a contact structure (possibly after rescaling) if and only if there is a smooth volume form µsuch that no sequence of elements in FXcan approximate a non-trivial foliation cycle of X, that is, FX∩CX= {0}. https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 3D steady Euler flows 2169 We prove Theorems 1.3 and 1.4 in §§2 and 3, respectively. Following Sullivan [16], the key to the proofs is the Hahn–Banach theorem, which allows us to produce functionals that separate certain subsets of 1-currents. The main application of Theorem 1.3, which is presented in §4, is to show that Eulerizable flows cannot be constructed using plugs. This proves, in particular, that an Eulerizable flow cannot contain plugs exhibiting Reeb components, as in Wilson’s construction [9,19]. The fact that steady Euler flows cannot contain Reeb components (not necessarily associated to a plug) was first observed in [4]. Finally, in §5 we prove a higher-dimensional analog of Theorem 1.3 and its application showing that the steady Euler flows cannot exhibit plugs in arbitrary dimensions. For the reader’s convenience, we have included in Appendix A a short introduction to the aspects of the theory of currents that are used in the proofs. 2. Proof of the main theorem We will first establish the equivalence between items (i) and (ii) and then between items (ii) and (iii). In the proof, the volume form preserved by Xis denoted by µ. (i)⇒(ii).This is straightforward. Let gbe a metric for which Xis a stationary solution of the Euler equations and define the dual 1-form α:= iXg. Then α(X) > 0 and, since X is an Euler flow, iXdαis exact. (ii)⇒(i).Let ξbe the 2-plane distribution defined by the kernel of α. It is standard that we can endow ξwith a smooth Riemannian metric gξ, which can be extended trivially to a smooth degenerate quadratic form on T M, satisfying gξ(X,·)=0. Then g:= 1 α(X)α⊗α+gξ is a metric on Mand it clearly verifies that iXg=α. Since iXdαis closed (and hence exact because H1(M)=0), Xis a solution to the Euler equations with the metric gand the volume form µ. Notice that the freedom on the quadratic form gξallows one to multiply it by an appropriate (non-constant) factor so that the Riemannian volume form induced by gis the same as µ. (ii)⇒(iii).Let Ybe the vector field defined as iYµ:= dα. It is easy to check that, since Xis volume-preserving, i[X,Y]µ=LXiYµ. Since diYµ=0, we have that Ypreserves the volume µas well, and also LXiYµ=diXiYµ=diXdα=0, thus implying that Xand Ycommute. It just remains to prove that FY∩CX= {0}. Suppose this is not the case, that is, that there is a sequence of 2-chains cn, with cn(iYµ) =0 and a non-zero foliation cycle bsatisfying lim n→∞ ∂cn(β) =b(β) for any 1-form β. Now, for the 1-form αwe have on the one hand that ∂cn(α) =cn(dα) =cn(iYµ) =0, https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 2170 D. Peralta-Salas, A. Rechtman and F. Torres de Lizaur and on the other hand that b(α) =Zb α > 0 because α(X) > 0 and bis a foliation cycle of X. We arrive at a contradiction. Accordingly, FY∩CX= {0}. (iii)⇒(ii).Let Z1denote the vector space of 1-currents on M, that is, the continuous dual of the space 1of smooth 1-forms on M(see Appendix A). It is well known [15] that the set of foliation 1-currents ZX⊂Z1is a closed convex cone with compact convex base, that is, there is a compact convex set K⊂ZX\ {0}such that ZX= {λK, λ ∈ [0,∞)}. By item (iii), since ZX∩FY=CX∩FY, we conclude that Kcannot intersect FY. Since FYis a closed vector subspace of Z1, a standard application of the Hahn–Banach theorem (see, for example, [14, Ch. 4, Theorem 4.5]) ensures the existence of a continuous linear functional F:Z1→Rthat is strictly positive in K(thus strictly positive in ZX\ {0}as well) and vanishes in FY. The continuous dual of Z1being 1, we can identify the continuous linear functional Fwith a 1-form α; this form verifies, on the one hand, that α(X) > 0,(1) because b(α) > 0 for any b∈ZX\ {0}, and, on the other hand, that ∂c(α) =c(dα) =0 (2) for any 2-current csuch that ∂c∈FY. In particular, iYdα=0 because any 2-current c0 tangent to Ycan be approximated by a sequence of 2-chains tangent to Y(and hence of zero flux), thus implying that ∂c0∈FYand so c0(dα) =0. Finally, let us show that iXdαis closed and hence exact because Mis assumed to have trivial first cohomology group. Consider the linear subspace of 2-forms that are proportional to iYµ: Y:= {ω∈2, ω =tiYµ, t∈R}. We claim that dα=T iYµfor some constant T6= 0. This implies that Yis volumepreserving and, since Xand Ycommute, it readily follows that d(iXdα) =0. Indeed, first notice that dαcannot be identically zero; otherwise, since H1(M)=0, the 1-form αwould be non-degenerate and exact, which is not possible on a closed manifold. Now let Z2be the vector space of 2-currents on M. Suppose there is no such T, thus dα /∈Y. Again, by a standard application of the Hahn–Banach theorem, there exists a 2-current c∈Z2that is positive on dαand whose kernel contains Y. Accordingly, we have that c(iYµ) =0. We claim that ∂c∈FY. Indeed, let {ck}be a sequence of 2-chains that converge (in the weak topology) to the 2-current c. By continuity, it follows that RckiYµ=: kwith k→0 as k→ ∞. Take a 2-chain bsuch that RbiYµ6= 0 (this obviously exists because Yis not identically zero). Then the sequence of 2-chains defined as eck:= ck−k RbiYµb https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 3D steady Euler flows 2171 has zero flux (i.e. ReckiYµ=0) and converges in the weak topology to c. The continuity of the boundary operator implies that ∂eckconverges to ∂c, thus proving the claim. Finally, by equation (2) we have that ∂c(α) =c(dα) =0, which contradicts the fact that cis positive on dα. So dα∈Y, as we wanted to show. This completes the proof of the theorem. Remark 2.1. The implication (i)⇒(iii)holds on any closed 3-manifold (there is no need to assume that H1(M)=0). Remark 2.2. From the proof of (ii)⇔(iii)we also obtain a characterization of the set of vorticities of a given vector field X. More precisely, let Xbe a non-vanishing vector field on a closed 3-manifold Mwith volume form µand assume that Mis not a fibration over the circle (so that, by Tischler’s theorem [18], there do not exist non-degenerate closed 1-forms). Then a vector field Ycan be written as Y=Tcurl Xfor some metric gand non-zero constant Tif and only if FY∩CX= {0}. Remark 2.3. It follows from the proof of the implication (iii)⇒(ii)that if cis a zero-flux 2-current for a vector field Y(i.e. c(iYµ) =0) then ∂c∈FY. Remark 2.4. In the particular case where Y=X, it is enough to assume that BX∩CX= {0}, where BXis the set of boundaries of tangent 2-chains: BX= {∂c|cis a 2-chain tangent to X}. Indeed, the Hahn–Banach theorem implies that there exists a 1-form αsuch that α(X) > 0 and iXdα=0. Since Xis non-vanishing, it then follows that dα=FiXµfor some function F:M→R. It is easy to check that Fis a first integral of Xbecause LXµ=0. We can then define a vector field Y:= F X that commutes with X(i.e. [X,Y] = 0) and satisfies the zero-flux condition in item (iii). Applying Theorem 1.3, we then conclude that Xis an Eulerizable flow with constant Bernoulli function (because iXdα=0), so it is geodesible. This is consistent with Sullivan’s characterization [16] of geodesible volume-preserving fields. Remark 2.5. According to Remark 2.2, if Mis not a fibration over the circle, the assumption FY∩CX= {0}implies that the 1-form αconstructed in the proof of the implication (iii)⇒(ii)satisfies dα=T iYµfor some non-zero constant T; in particular, div Y=0 and BY∩CX= {0}(see Remark 2.4 for a definition of BY). We believe that, in general, the existence of a commuting vector field Ysuch that div Y=0 and BY∩CX= {0}does not imply the existence of another commuting field Y0with FY0∩CX= {0}; this would show that, contrary to Sullivan’s characterization of geodesible flows, there is no hope of characterizing the Eulerizable flows using only commuting tangent homologies (instead of commuting zero-flux homologies). This is supported by the fact that div Y=0 and BY∩CX= {0}imply that dα=FiYµfor some function F(provided that Yis nonvanishing); in the particular case where Y=X, this is a characterization of geodesible volume-preserving or stable Hamiltonian flows, which are not Reeb flows of a contact structure in general, that is, the function Fis genuinely not constant [3, §3.9]. Remark 2.6. We observe that the condition FY∩CX= {0}in item (iii) is independent of the existence of a volume-preserving vector field Ythat commutes with X. Indeed, https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 2172 D. Peralta-Salas, A. Rechtman and F. Torres de Lizaur using the C∞volume-preserving plug introduced by Kuperberg [9], one can construct (in a flow-box) a volume-preserving vector field Xthat is axisymmetric and contains a Reeb cylinder. The axisymmetry condition is equivalent to saying that there is a volumepreserving vector field Ythat commutes with X,[X,Y] = 0. Since Euler flows cannot exhibit plugs (see §4), we conclude that the zero-flux homology condition is independent of the existence of Y. Remark 2.7. Following Tao [17], we say that a non-vanishing vector field Xadmits a strongly adapted 1-form αif α(X) > 0 and iXdαis exact. If Xis volume-preserving, the proof of Theorem 1.3 shows that this is equivalent to being Eulerizable (there is no need to assume that H1(M)=0). If Xis not assumed to preserve a volume form, a simple variation of the proof of Theorem 1.3 allows one to prove the following result. Let Mbe a closed 3-manifold with H1(M)=0 endowed with a volume form µ. Then Xadmits a strongly adapted 1-form αif and only if there exists a (not identically zero) vector field Y satisfying [X,Y]=−(div X)Y(i.e. LXiYµ=0) and such that no sequence of elements in FYcan approximate a non-trivial foliation cycle of X, that is, FY∩CX= {0}. Given Xand a strongly adapted 1-form α, the vector field Yis simply defined as iYµ=dα. We refer to Remark 5.3 in §5 for a generalization of this result to arbitrary dimensions. 3. Proof of Theorem 1.4 We recall that a contact form on a 3-manifold Mis a 1-form αwhose associated 3-form α∧dαis a volume form (the 2-plane distribution ξ:= ker αis then everywhere nonintegrable, and is called a contact structure). The unique vector field Xsatisfying α(X)=1 and iXdα=0 is called the Reeb field of the contact form α. Let Xbe the Reeb vector field of a contact form α. In particular, Xpreserves the volume form µ:= α∧dα, and we have that iXµ=dα. Suppose there is a non-zero element b∈FX∩CX, where we define FXusing the volume µ. Then, on the one hand b(α) > 0, because bis a foliation cycle, but on the other hand bis in FX, so b=∂cwhere cis a 2-current that satisfies c(iXµ) =c(dα) =b(α) =0, and we get a contradiction. So if Xis Reeb, then FX∩CX= {0}. Now suppose that FX∩CX= {0}. Arguing as in the proof of the implication (iii)⇒(ii) in Theorem 1.3, we construct a 1-form αsatisfying α(X) > 0 and dα=T iXµ, for some constant T. If Mdoes not fiber over the circle, we must have that T6= 0. But then α∧dα=Tα(X)µ is a volume form, so αis a contact form and the rescaled vector field X0:= 1 α(X)X is its Reeb field. 4. An application: vector fields constructed with plugs are not Eulerizable 4.1. Plugs and geodesible flows. In this subsection we introduce the notion of a plug and we show that vector fields constructed with plugs are not geodesible. This implies, in particular, that vector fields with plugs cannot be Beltrami flows; the general case of https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 3D steady Euler flows 2173 steady Euler flows will be considered in the next subsection. Plugs were introduced by Wilson [19] in the context of the Seifert conjecture. We start with the definition of a plug. Definition 4.1. A plug is a 3-manifold Pwith boundary of the form D× [−1,1], where Dis a compact surface with boundary (usually a disk). Pis endowed with a non-vanishing vector field X, such that the following statements hold. (1) Xis vertical in a neighborhood of ∂P, that is, X=∂/∂z,z∈ [−1,1]. Thus Xis inward transverse along D× {−1}, outward transverse at D× {1}and tangent to the rest of ∂P. (2) There is a point p∈D× {−1}whose positive orbit is trapped in P. The set D−1:= D× {−1}is called the entry region of the plug. (3) If the orbit of q=(x,−1)∈D× {−1}is not trapped, then it intersects D× {1}at the point q=(x,1). We say that qis the point facing q, and D1:= D× {1}denotes the exit region of the plug. (4) There is an embedding of Pinto R3preserving the vertical direction. A plug allows one to change a vector field on a 3-manifold locally: given a flow-box, the interior can be replaced by the plug, thus changing the dynamics. For example, the trapped orbits will now be limited to an invariant set contained inside the plug. Our main result in this subsection is the proof that a plug cannot be geodesible, and hence any vector field constructed using a plug is not geodesible. For this we use Sullivan’s characterization of geodesible fields [16]. The following proof is taken from [12]. We remark that in this subsection, the vector field Xis not assumed to be volume-preserving. PROPOSITION 4.2. The vector field inside a plug (P,X)is not geodesible. Proof. Let x∈D−1be a point with trapped forward orbit, and assume there is a finitelength curve σ⊂D−1from xto ∂D−1such that the orbits of the points σ\ {x}are not trapped by the plug. Such a point always exists, since the points in ∂D−1are not trapped. Let σ: [0,1] → D−1be a parametrization such that σ(1)=xand σ (0)∈∂D−1. We want to show that Xis not geodesible. Using Sullivan’s theorem [16] we know that it is enough to find a sequence of tangent 2-chains whose boundaries are arbitrarily close to a foliation cycle. Consider the curve σt=σ ([0,t]), for t∈ [0,1]. For t<1, the orbits of the points in σtunder the flow of Xhit D1after a finite time. Let Atbe the tangent surface defined by the union of the flow-lines of the points in σt, which lie between D−1 and D1, At:= t [ s=0 γ (σ(s)), where γ (σ(s)) is the X-orbit of σ (s)inside the plug. Observe that eσt:= At∩D1is the curve facing σt, by the exit–entry condition on plugs (item (3) in Definition 4.1). Hence we can define eσ1⊂D1and for every t∈ [0,1]we have that |σt|=|eσt|, where by |·|we will denote the length of the curves and, more generally, the mass of currents (see, for example, [10]). Consider now a sequence {tn}n∈Nthat converges to 1. Since the orbit of σ(1)=x is trapped, and Xis non-vanishing, the length of the curve γ (σ(tn)) goes to infinity https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 2174 D. Peralta-Salas, A. Rechtman and F. Torres de Lizaur as n→ ∞, and we can assume without loss of generality that |γ (σ (tm))|≤|γ (σ(tn))| for m≤n. Define the sequence of 2-currents 1 |γ (σ(tn))|Atn(λ) := 1 |γ (σ(tn))|ZAtn λ, where λis any 2-form. We first observe that this sequence of 2-currents has finite mass. Indeed, 1 |γ (σ(tn))|ZAtn λ≤|Atn| |γ (σ(tn))|kλkL∞≤CkλkL∞, where we have used that |Atn|≤|γ (σ(tn))|·|σtn| ≤ C|γ (σ (tn))|, for some constant that does not depend on n. This last inequality comes from the assumption that the curve σhas finite length. Moreover, it is clear that the 2-currents (1/|γ (σ (tn))|)Atnform a sequence of tangent 2-chains. In Lemma 4.3 we prove that the boundaries of these 2-chains approach a foliation cycle, thus implying that Xcannot be geodesible. The proposition then follows.  We need to introduce some notation for Lemma 4.3. First, observe that the 2-currents we are considering are normal currents, that is, compactly supported currents which have finite mass and boundaries of finite mass as well. In the set of normal currents we consider the flat norm F(S):= inf{|A|+|B| : S=A+∂B}, where S,Aand Bare normal currents (for more details, see [10]) and |·|denotes the mass (as defined at the end of Appendix A). The set of normal currents is not closed under this norm; however, it is easy to check that the flat convergence of normal currents implies the usual weak convergence of currents. LEMMA 4.3. limn→∞ (1/|γ (σ(tn))|)∂ Atnis a non-trivial foliation cycle. Proof. Consider the sequence of foliation 1-currents (1/|γ (σ (tn))|)γ (σ(tn)). Then  1 |γ (σ(tn))|∂Atn−1 |γ (σ(tn))|γ (σ(tn))≤1 |γ (σ(tn))|[|σ1|+|eσ1|+|γ (σ (0))|], so the difference converges in the flat norm (and in the weak topology) to zero as n→ ∞. Additionally, the flat norm of the 1-currents (1/|γ (σ (tn))|)γ (σ(tn)) is less than or equal to one, because they have mass one. Since the space of 1-currents is Montel, there is a convergent subsequence (1/|γ (σ(tnk))|)γ (σ(tnk)). Hence, the limit defines the 1-current with mass one (and hence non-trivial): S:= lim k→∞ 1 |γ (σ(tnk))|∂Atnk=lim k→∞ 1 |γ (σ(tnk))|γ (σ(tnk)). Since the boundary operator ∂is continuous, it follows that Sis a cycle. Moreover, since the space of foliation currents is a closed convex cone CXcontaining the sequence (1/|γ (σ(tnk))|)γ (σ(tnk)), it contains its limit. Thus Sis a foliation cycle, as we wanted to prove.  Remark 4.4. The two important properties of a plug that are used in the proof above is that there are trapped orbits and that the map from the entry to the exit is absolutely continuous, thus mapping curves of bounded length onto curves of bounded length. https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press 3D steady Euler flows 2181 [3] K. Cieliebak and E. Volkov. First steps in stable Hamiltonian topology. J. Eur. Math. Soc. (JEMS) 17 (2015), 321–404. [4] K. Cieliebak and E. Volkov. A note on the stationary Euler equations of hydrodynamics. Ergod. Th. & Dynam. Sys. 37 (2017), 454–480. [5] J. Etnyre and R. Ghrist. Contact topology and hydrodynamics I. Nonlinearity 13 (2000), 441–458. [6] J. Etnyre and R. Ghrist. Contact topology and hydrodynamics II. Ergod. Th. & Dynam. Sys. 22 (2002), 819–833. [7] V. Ginzburg and B. Khesin. Steady fluid flows and symplectic geometry. J. Geom. Phys. 14 (1994), 195–210. [8] A. Izosimov and B. Khesin. Characterization of steady solutions to the 2D Euler equation. Int. Math. Res. Not. IMRN 2017 (2017), 7459–7503. [9] G. Kuperberg. A volume-preserving counterexample to the Seifert conjecture. Comment. Math. Helv. 71 (1996), 70–97. [10] F. Morgan. Geometric Measure Theory. Academic Press, Boston, 1988. [11] D. Peralta-Salas. Selected topics on the topology of ideal fluid flows. Int. J. Geom. Methods Mod. Phys. 13 (2016), 1630012 (1–23). [12] A. Rechtman. Use and disuse of plugs in foliations. PhD Thesis, ENS Lyon, 2009. [13] A. Rechtman. Existence of periodic orbits for geodesible vector fields on closed 3-manifolds. Ergod. Th. & Dynam. Sys. 30 (2010), 1817–1841. [14] B. Simon. Convexity: An Analytic Viewpoint. Cambridge University Press, Cambridge, 2011. [15] D. Sullivan. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36 (1976), 225–255. [16] D. Sullivan. A foliation of geodesics is characterized by having no tangent homologies. J. Pure Appl. Algebra 13 (1978), 101–104. [17] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017), 219–238. [18] D. Tischler. On fibering certain foliated manifolds over S1.Topology 9(1970), 153–154. [19] F. W. Wilson. On the minimal sets of non-singular vector fields. Ann. of Math. (2) 84 (1966), 529–536. https://doi.org/10.1017/etds.2020.25 Published online by Cambridge University Press