Vortex motion and geometric function theory: the role of connections

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi–Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.


Introduction
In this paper, we study point vortex motion on closed two-dimensional Riemannian manifolds, mainly from the point of view of Riemann surfaces with additional 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited.

Fluid dynamics on Riemannian manifolds (a) General notations and assumptions
We consider the dynamics of a non-viscous incompressible fluid on a compact Riemannian manifold of dimension two. We start by setting up the basic equations in somewhat more generality, in arbitrary dimension, following treatments based on using the Lie derivative, as exposed, for example, in Frankel [8] and Schutz [11]. These sources also provide the standard notations in differential geometry to be used. Other treatises in fluid dynamics, suitable for our purposes, are [10,24,25].
Let M be a Riemannian manifold of dimension n. We consider the flow of a non-viscous fluid on M. Some general notations: ρ = ρ(x, t) = density > 0 -p = p(x, t) = pressure -ds 2 = g ij dx i dx j Riemannian metric (Einstein summation convention in force) ν = v i dx i = g ij v j dx i corresponding one-form (using the metric) -ω = dν vorticity field, as a two-form - * ω = Hodge star acting on a differential form ω - * 1 = vol n , the volume form on M -i v = i(v) = interior multiplication with respect to a vector field v -L v = Lie derivative with respect to a vector field v.
The Hodge star and interior multiplication are related by i v vol n = * ν, (2.1) where v and ν are linked via the metric tensor. The Lie derivative L v can act on all kinds of tensors. When it acts on differential forms the homotopy formula (or (Henri) Cartan formula) is very useful. It shows in addition that L v commutes with d.

(b) Basic equations
We collect here the basic equations of fluid dynamics in our context. See [8,11] for details. (iv) A constitutive relation between ρ and p, for example of the form ρ = ρ(p). (2.5) As for Euler's equation, its standard vector version is and rewriting this as an equation for differential forms gives ∂ν ∂t + i v (dν) = − 1 2 d(i v (ν)) − dp ρ (2.6) or The latter is the same as (2.3). When (2.5) holds the form dp/ρ is exact, namely equal to the exterior derivative of a primitive function of 1/ρ(p). This means that the pressure p becomes a redundant variable: Euler's equation (2.3) simply becomes the statement that ((∂/∂t) + L v )(ν) is exact, say And afterwords p can be recovered, up to an additive constant, from φ = 1 2 |v| 2 − dp ρ(p) .
On acting by d (exterior derivative) on Euler's equation and using the constitutive relation, the laws of conservation of vorticity, local and global, follow immediately (recall that ω = dν): for any closed curve γ (t) which moves with the fluid. Conversely, if (2.9) holds for some time-dependent form ν and for every closed γ moving with the flow of the corresponding vector field v, then (2.7) holds, and with (2.5) in force (2.3) can be recovered.
Proof. For (2.8) one uses that d commutes with (∂/∂t) and with L v , and for (2.9) that 'γ (t) moving with the fluid', together with (2.7), gives that For the converse statement, one simply runs the above arguments backwards.
The statement that (2.9) holds for all γ (t) as stated can be thought of as a weak formulation of the Euler equation. Helmholtz equation (2.8) alone does not imply Euler's equation (2.3) if M has non-trivial topology.
More ambitious than the above would be to treat fluid mechanics from the point of view of general relativity, and to include all kinds of thermodynamic quantities, as is done in [26]. However, we shall rather go in the other direction and simplify to the case of incompressible fluids in two dimensions. Eventually, we shall specialize to the case of having the vorticity concentrated to finitely many points. It follows from (2.8) and (2.9) that such a situation is preserved in time if it holds initially, hence the system will look like a classical Hamiltonian system having only finitely many degrees of freedom. From a Riemann surface point of view, it becomes a dynamical system for Abelian differentials of the third kind (classical terminology, see [20,27]). Such a differential starts moving once the Riemann surface has been provided with a Riemannian metric.

Hodge theory and Green functions (a) Some rudimentary Hodge theory
Green functions and differentials with prescribed singularities and periods can most conveniently be introduced via Hodge theory. We briefly recall the necessary concepts here, confining ourselves to the case of two dimensions. For more details and general notational conventions, see [8,28]. We assume that M is a compact (closed) oriented Riemannian manifold of dimension n = 2. We may think of M as a Riemann surface provided with a Riemannian metric which is compatible with the conformal structure, namely on the form ds 2 = λ(z) 2 (dx 2 + dy 2 ) = λ(z) 2 |dz| 2 .
Here z = x + iy is any local holomorphic coordinate on M, and λ > 0 is assumed to be a smooth.
The Laplacian operator acting on a form ω of any degree p = 0, 1, 2 is ω = * d * dω + d * d * ω, where the star is the Hodge star operator. When acting on 1-forms the Hodge star does not depend on the metric (only on the conformal structure), in fact * dx = dy, * dy = −dx. On 0-forms it does depend on the metric, for example * 1 = λ(z) 2 dxdy = vol 2 , the volume form on M (invariant area). Similarly for 2-forms: * vol 2 = 1.
A p-form ω is harmonic if ω = 0. The only 0-forms (functions) which are harmonic on all of M are the constant functions, and the only global harmonic 2-forms are the constant multiples of the volume form. The space of harmonic 1-forms on M has dimension 2g, where g is the genus of M, and by the Hodge-deRham theory this space is dual in a natural way to the first homology group of M. Similarly for harmonic 0-and 2-forms. Having ω = 0 on all of M is equivalent to that the two equations dω = 0, d * ω = 0 hold.
The natural inner product on the space of p-forms is We denote by L 2 (M) p the Hilbert space of p-forms with (ω, ω) p < ∞ and inner product (3.1). The Hodge theorem [28] says that any ω ∈ L 2 (M) p has an orthogonal decomposition where δ = − * d * is the coexterior derivative, η is a harmonic p-form, μ is a coexact (p − 1)-form, and ν is an exact (p + 1)-form. In (3.2), the forms μ and ν are not uniquely determined, only dμ, δν and η are. The decomposition can, however, be made more precise as where the p-form τ becomes unique on requiring that it shall be orthogonal to all harmonic forms. We shall need the Hodge theorem only for 2-forms, in which case it makes the one-point Green function appear naturally. Indeed, if ω is a 2-form then one term in the Hodge decomposition disappears immediately, and the rest can be written Here the first term represents the harmonic 2-form in the decomposition, with the constant c ω necessarily being the mean value of ω with respect to volume: The second term represents what is in the range of d, and the function G ω there is to be normalized (as for the additive level) so that it is orthogonal to all harmonic 2-forms, i.e. so that We then call G ω the Green potential of ω. We see that −d * dG ω = ω provided ω has net mass zero; otherwise, there will be a compensating countermass (multiple of vol 2 ), namely the first term in the right member of (3.4).
With ν the fluid velocity 1-form, the inner product (ν, ν) 1 has the interpretation of being the kinetic energy of the flow. For a function (potential) u, we consider the Dirichlet integral (du, du) 1 to be its energy. Thus, constant functions have no energy. Similarly, for 2-forms we consider vol 2 to have no energy, and the energy of ω is then defined to be the energy (dG ω , dG ω ) 1 of its Green potential G ω . For the corresponding quadratic form, E(ω 1 , ω 2 ) = (dG ω 1 , dG ω 2 ) 1 , we get, after a partial integration and on using (3.4) and (3.6) (3.7)

(b) The one-point Green function
The mutual energy and many other concepts in Hodge theory extend in a straightforward manner to circumstances in which some source distributions have infinite energy. This applies in particular to the Dirac current δ a , which we consider as a 2-form with distributional coefficient, namely defined by the property that for every smooth function (0-form) ϕ. Certainly δ a has infinite energy, but if a = b, then E(δ a , δ b ) is still finite and has a natural interpretation: it is the Green function: Here the first equality can be taken as a definition, and then the second equality follows on using (3.4) and (3.6): This shows in addition that G(a, b) is symmetric.
(c) The two-point Green function (fundamental potential) The background homogenous sink c ω vol 2 in §3b disappears in the case of several poles with net mass zero. The simplest case is the two-point fundamental potential, with one source and one sink, equally strong. This potential can be traced back to Riemann's original ideas, and it is explicitly used, for example, in [29]. In contrast to the one-point Green function, the two-point Green function does not depend on the Riemannian metric.
To conform with some previous usage, we define the fundamental potential without the factor 1/2π in front of the logarithm. We may define it in terms of the one-point Green function as where a, b, w ∈ M are distinct points. It is mainly considered as a function of z, normalized so that it vanishes for z = w, and it is useful for discussing harmonic and holomorphic differentials, in general. Indeed, is the unique Abelian differential of the third kind with poles of residues ±1 at z = a, b and with purely imaginary periods. It does not depend on w, as can be seen from the decomposition (3.9). By definition dV is exact, while * dV has certain periods around closed curves. To be precise, if α is a closed curve then the function is, away from α, harmonic in each of a and b, and it makes a unit additive jump as one of these variables crosses α. The jump with respect to the variable a is −1 when a crosses α from the right to the left. Assuming that a moves along another closed curve β, which has no further intersections with α, it follows that U α (a, b), as a compensation for the jump, has to increase by +1 as a runs through the rest of β. This is what lies behind the period relations (3.12) and (3.13).
Differentiating U α (a, b) with respect to a and ignoring the distributional contribution from the jump one obtains a differential which is harmonic (with respect to a) in all M. It is certainly harmonic in b too, but it can be seen from (3.9) that it actually does not depend on b. For its integral we have if we integrate along a path that does not intersect α.
Let {α 1 , . . . , α g , β 1 , . . . , β g } be representing cycles for a canonical homology basis for M such that each β j intersects α j once from the right to the left and no other intersections occur (see [27] for details). The harmonic differentials dU α j , dU β j obtained by the above construction then constitute (when taken in appropriate order and with appropriate signs) the canonical basis of harmonic differentials associated with the chosen homology basis. Precisely we have, for k, j = 1, . . . , g, In terms of the above harmonic differentials, the conjugate periods of V can by (3.10) and (3.11) be explicitly expressed as and Here the integration in the right member is to be performed along a path that does not intersect the curve in the left member. One point with the above formulae is that they explicitly exhibit the dependence of the conjugate periods of V on the locations a, b of the poles. This is essential because, in our applications, V will have the role of being (part of) a stream function, and the conjugate periods of V will enter into the circulations of the flow, which are to be conserved in time by Kelvin's Law (2.9).

Affine and projective connections
Besides differential forms, and tensor fields in general, affine and projective connections are quantities on Riemann surfaces which are relevant for point vortex motion, so we expand somewhat on these notions below. The affine connections have the same meanings as in ordinary differential geometry, used to define covariant derivatives, for example, and they play an important role in many areas of mathematical physics. Some general references for the kind of connections we are going to consider are [30][31][32][33][34][35]. The ideas go back at least to E. Cartan. We shall define them in the simplest possible manner, namely as quantities defined in terms of local holomorphic coordinates and transforming in specified ways when changing from one coordinate to another. Letz = ϕ(z) represent a holomorphic local change of complex coordinate on M and define three nonlinear differential expressions {·, ·} k , k = 0, 1, 2, by The last expression is the Schwarzian derivative of ϕ. For {z, z} 0 , there is an additive indetermincy of 2π i, so actually only its real part, or exponential, is completely well defined.
Remark 4.1. Both z and a (andz andã) will be viewed as local variables, but a will be slightly 'less variable' than z in the sense that it will be used as a local base point for Taylor expansions.
The following chain rules hold, if z depends on w via an intermediate variable u: In particular, It turns out that the three operators {·, ·} k , k = 0, 1, 2, are unique in having properties as above, i.e. one cannot go on with anything similar for k = 3, 4, . . .. See [30,31] for details.

Definition 4.2.
An affine connection on M is an object which is represented by local differentials r(z) dz,r(z) dz, . . . (one in each coordinate variable, and not necessarily holomorphic) glued together according to the ruler One may also consider 0-connections, quantities defined up to multiples of 2π i and which transform according top This means exactly that e p(z) is well defined and transforms as differential of order one, and its absolute value λ(z) = e Rep(z) transforms as the coefficient a Riemannian metric when this is written as ds = λ(z)|dz|. If the Riemann surface initially is provided with a Riemannian metric ds = λ(z)|dz|, then there is a natural affine connection associated with it by This is the same as the Levi-Civita connection in general tensor analysis, and the real and imaginary parts, made explicit above, coincide (up to sign) with the components of the classical Christoffel symbols Γ k ij . Compare formulae in [3]. Independent of any metric, an affine connection r gives rise to a projective connection q by This q is sometimes called the 'curvature' of r [36]. That curvature is however not the same as the Gaussian curvature in case r(z) comes form a metric.
In the presence of an affine connection, one can define, for every half The covariance means that In order to clarify the notion of a half-order differential, recall that a differential of order k (or more exactly, bi-degree (k, 0), if also dz is allowed) on a Riemann surface is an object which in each coordinate patch (with coordinate function denoted z,z, . . . above) is represented by a function (φ,φ, . . .) in such a way that φ =φ((dz)/dz) k holds wherever the patches overlap. This certainly makes sense for any integer k, but it can be shown, see [30,37], that also half integer values of k are good enough. The crucial fact here is that it is possible to select branches of the fractional powers (dz/dz) k in a consistent way all over the surface. This can be understood in a convenient way by passing to the universal covering surface, the unit disc in most cases, on which a differential of order k is represented by a function φ which is automorphic in the sense that φ(z) = φ(T(z))(cz + d) −2k for any Möbius map T(z) = (az + b)/(cz + d), ad − bc = 1, in the covering group of the surface. In this picture, it is clear that 2k ∈ Z is the appropriate condition.
Also a projective connection allows for certain covariant derivatives: for each m = 0, 1, 2, . . . there is a linear differential operator Λ m taking differentials of order ((1 − m)/2) to differentials of order ((1 + m)/2): In case the projective connection comes from an affine connection as in (4.2), these Λ m are given by The right member here is what Λ 2 (φ) looks like even if q does not come from any affine connection. We refer to [33,34] for further details and more examples.

Behaviour of singular parts under changes of coordinates
Similarly, the expansion of the analytic completion ν + i * ν = 2i(∂ψ/∂z) dz of the flow 1-form ν will look like (again up to a constant factor) One step further, one may consider vortex dipoles, i.e. two vortices of opposite strengths infinitesimally close to each other. The corresponding flow is obtained by taking the derivative ∂/∂a of the above flow 1-form and then multiply this by a complex factor to adjust for the orientation of the dipole. Taking this factor to be one, for simplicity, we obtain something of the form and by the same argument as above the motion of the dipole should then be determined by the coefficient 2c 2 (a). Lemma 5.1 says that this coefficient is (essentially) a projective connection. However, vortex dipoles are very singular, and have to be renormalized in a special way in order to prevent them from having infinite speed (cf. [38]). Now the general statement in this context is the following.  F(z, a). Precisely, we assume the following local forms, in the z and z variables: and

Remark 5.2.
It is actually not necessary for the conclusions of the lemma that ψ, f dz, F dzda are harmonic/analytic away from the singularity, it is enough that the local forms of the singularity and constant terms are given as above.
Proof of lemma. The proof consists of straightforward substitutions, but let us for clarity give a few details. The needed ingredients are the equation dz = ϕ (z) dz and the Taylor series of ϕ, which we write asz The statement for ψ is immediate and for f dz one obtains, after multiplying the two given expressions for f dz byz −ã and dividing by dz, that as z → a. After dividing also with z − a and letting z → a one obtains which is the desired result.
As for F dzda one uses also that dzdã = ϕ (z)ϕ (a) dzda and that Inserting this into the two given expressions for Fdzda and comparing the Taylor expansions, after multiplication with (z −ã) 2 and division with dzda, gives that the constant and linear terms match automatically, while matching of the (z − a) 2 terms requires that And this is exactly the desired identity.

Remark 5.3.
For a 'non-pure' second-order pole, with expansion the new coefficient b 2 transforms as an ordinary differential: while c 2 (a) continues to transform as an projective connection (up to a constant). The proof is again straightforward.
We may adapt the above to the Green function G(z, a) = G δ a (z), despite the fact that it is not harmonic in z (because of the compensating background flow). The metric associated with the Green function becomes ds 2 = e −2h 0 (a) |da| 2 = e |da| 2 (1 + |a| 2 ) 2 , and hence equals the spherical metric, up to a factor.
The fundamental potential (3.9) on the Riemann sphere is, independently of the metric i.e. (minus) the logarithm of the modulus of the cross ratio.
The transformation properties of the coefficients h j (a) are closely related to those appearing in lemma 5.1: Compare the last equation in lemma 5.1. However, the two metrics are in general not the same: as is seen from (5.5) the metric ds 2 = h 11 (a)|da| 2 equals, up to a constant factor, the initially given metric on M, while (5.11) usually is a different metric, somewhat related to (but not identical with) what in [3] is called the 'steady vortex metric'. However, it should be noted that the metric (5.11) actually depends on the initially given metric. But if the two metrics agree up to a constant factor, as in the case of the sphere above (in example 5.5), then the situation is stable and, as will be a consequence of theorem 7.1, these metrics then will have the property that a single vortex does not move unless one adds circulations beyond those which are already present in the Green function itself.
For later reference, we point out that, by (4.1), −(∂/∂a) log λ(a)da transforms in the same way as h 1 (a)da.
Proof of lemma. The proofs of the first two equations are similar to those of the corresponding statements in lemma 5.1. The third statement comes out on combining lemma 5.1 with (5.4) in lemma 5.4, noticing that −4π where the left member behaves as the coefficient of a double differential of type dzda. It follows that 2{ (∂ 2 H(z, a))/(∂z∂a)} z=a transforms as 2c 2 (a) in lemma 5.

A weak formulation of Euler's equation
We return to fluid mechanics and specialize to incompressible fluid flow in two dimensions. Thus, M is a Riemannian manifold of dimension n = 2, and we take the constitutive equation ( give where G ω has the advantage of being single-valued. It follows that for some 1-form η satisfying dη = 0. Using (6.1), it follows that also d * η = 0, i.e. η is a harmonic 1-form. As a such, it is uniquely determined by its periods α k ν, β k ν with respect to a canonical homology basis {α k , β k : k = 1, . . . , g}. Setting for the periods it follows that there is a one-to-one correspondence between the velocity 1-form, on one hand, and the vorticity 2-form combined with given periods, on the other hand. The following three steps give an equation for ∂ω/∂t in terms of ω: v → ∂ω/∂t by Euler's equations, or conservation of vorticity: As for the first item, it may be noticed that having the data (ω, {a k , b k }) is equivalent to knowing the periods γ ν for all closed curves γ in M. Thus the first step can be viewed as a map γ ν : all closed curves γ → ν.

Expressing (6.3) directly in terms of ν (i.e. avoiding the step via v) it becomes
Here the right member is an antisymmetric bilinear form in ν and ω, a form which is amply discussed from Lie algebra points of view in [10]. We can complete ν into a complex-valued 1-form by In regions where ω = 0 we can also write ν = dϕ, where ϕ is a locally defined velocity potential.
We emphasize that Φ makes sense only in regions with no vorticity, and that it then is an additively multi-valued analytic function. In the point vortex case, dΦ is however a well-defined meromorphic differential (Abelian differential). It is straightforward to formulate Euler's equation (2.3) directly in terms of ψ and p. Using dot for time derivative, the result is the somewhat awkward looking equation This is an equation for real-valued 1-forms, which can be expressed in terms of a local basis dx, dy and then becomes one equation for each component. Nothing prevents us from using complexvalued 1-forms, with natural basis dz = dx + idy, dz = dx − i dy. Then real-valued forms may get complex-valued coefficients. For example, dψ = (∂ψ)/(∂x) dx + (∂ψ)/(∂y) dy = (∂ψ)/(∂z) dz + (∂ψ)/(∂z) dz, with then ∂ψ/∂z and ∂ψ/∂z complex-valued, but complex conjugates of each other. The Hodge star acting on the above complex basis just multiplies with ±i : * dz = −idz, * dz = idz. If we write (6.5) in terms of the basis {dz, dz}, then the coefficient of dz contains the same information as that of dz, hence it is enough to use one of them. Choosing to work with the dz-component of (6.5) then gives the equation Here the right member can also be expressed as We summarize: We see from (6.8) that f is analytic in regions without vorticity, and in such regions it simply equals (minus)Φ: Indeed, the imaginary parts agree, and since both members are analytic also the real parts agree, after adjustment of the, possibly time-dependent, additive level in ϕ. We also see that in such regions. In other words, the time derivative of the velocity potential is a single-valued function having a direct hydrodynamical interpretation. The fact thatφ is globally single-valued can be seen as a way to express Kelvin's Law (2.9) for conservation of circulations. By contrast, the imaginary part of f need not be globally single-valued (but the derivative ∂f /∂z always is). As a further remark, Bernoulli's equation, stating that the right member of (6.9) is constant in the case of stationary flow, comes out very nicely: when the flow is stationary, so thatψ = 0, then f is analytic and real-valued in regions without vorticity, and hence equals a (real) constant there.
In the point vortex case, one may view the usage of f as one way (out of many) to get rid of the pressure p. Indeed, the pressure shows up only in the real part of f , and one can see its role there as just a way to make this real part equal to the harmonic conjugate of (minus) the imaginary part, namelyψ.
In the smooth case, (6.8) holds if and only if the integral over arbitrary subdomains D ⊂ M of the two members agree: Strictly speaking, D should be sufficiently small, since f may be multivalued, and have smooth boundary to allow for partial integration in a next step. In case λ is constant, (6.10) leads immediately to and in the point vortex limit both members can be evaluated by residues to give the well-known equations of motion for the vortices. When λ is not constant the same idea still works, but the details become slightly more involved. We assume that λ is sufficiently smooth, say real analytic for simplicity. Selecting a point a ∈ D, which when passing to the point vortex limit will be a vortex point, we expand where Q(z) simply stands for the remainder, namely In (6.10), one sees that all terms except the one with Q(z) can be pushed to the boundary, i.e. (6.10) can be written When we pass to the limit that all vorticity inside D is concentrated to the point z = a the left member in (6.12) is easily evaluated by residues, only the right member has to be handled with some care. This right member vanishes outside any (isolated) point vortex z = a, but there could be some kind of contribution exactly at z = a. One way to express the problem is to say that (∂ψ/∂z) 2 is not locally absolutely integrable, hence it does not make immediate sense as a distribution, and so it becomes difficult to make sense of the derivative ∂/∂z in front of it. The idea with the decomposition (6.12) is then that all contributions at z = a have been sorted out to lie in the left member (the boundary integrals), and the right member will not contribute at vortex points. And this is due to cancellations in the integrand (of the right member) which can be seen on writing it roughly as (one may think of smearing out the point mass of ψ a little) If the smeared out point mass is more or less symmetrically distributed around the point a, then cancellations due to the last factor in (6.13) will make its contribution to the integral negligible.
Hence it is appropriate to replace the right member of (6.12) with the principal value integral lim ε→0 D\D(a,ε) On the other hand, the last factor in (6.13) has an alternating phase factor around a, and if the finite point mass is approximated, under weak convergence, by smooth vorticity distributions which lie completely in only one narrow sector from a, then the limit will depend on which sector this is. Hence the above principal value integral will not guarantee a unique limit under ordinary weak convergence. A stronger convergence concept will be needed for such a conclusion. Note however that the situation is better in the planar case, more precisely when ∂λ(a)/∂a = 0 (see (6.11)).
The above procedure will still be good enough for our purposes, and it will lead to the right point vortex dynamics, as given in [2], for example, and also to be confirmed by a Hamiltonian approach in §9. We shall not go further into the validation of the point vortex model here. It is carefully discussed (for the planar case) in section 4.4 of [24]. See also [39].
We summarize below the weak formulation we have arrived at. The main point is that the equation, (6.14), agrees with the usual Euler equation in the smooth case and that it simultaneously makes sense when the vorticity is concentrated as a Dirac mass at the point a. are as explicit as they can possibly be in the general framework we are working with. We start by summarizing the given data and assumptions in the point vortex case.
-M is a compact Riemann surface of genus g ≥ 0.
-The vorticity is concentrated to finitely many points z 1 , . . . , z n ∈ M, so that: where Γ j ∈ R is the strength of the vortex at z j . -We have prescribed periods a k , b k for the velocity 1-form ν around a canonical homology basis, α k , β k , which we take to be fixed curves avoiding the vortex points for the time interval under consideration (taken to be short enough).
Under these assumptions is a meromorphic differential on M determined by the data In other words, ν + i * ν = dΦ is an Abelian differential of the third kind on M, and it starts evolving in time once M has been provided with a Riemannian metric. Similarly, Φ is an Abelian integral (additively multi-valued in general) on M. Near a point vortex z k = z k (t) of strength Γ k we expand, like in lemma 5.1, where the coefficients c j = c j (z k ) actually depend on all data, and also on the choice of the local coordinate at the vortex point. By lemma 5.1, the first few coefficients c j behave as connections under changes of coordinates for z k . The Green function takes care of the singularities, but not of the periods, when considered as part of the stream function ψ. One needs an extra term to absorb the difference between ψ and G ω , and this is the harmonic form η in (6.2). Since this η is locally exact it can be written η = dU with U a harmonic function (multivalued if considered on all M). Then modulo additive local constants, and with U * denoting the harmonic conjugate of U (this notation is compatible with the Hodge star: d(U * ) = * dU). Since G ω is single-valued one can think of U * as representing the multi-valuedness of ψ. The Green potential G ω is harmonic away from the vortex points, while each individual G(·, z k ) = G δ z k appearing in (7.4) is not, because its Laplacian contains compensating volume terms. However, the G(·, z k ) are good enough to single out individual Robin functions at the vortices. Recalling (5.1) and (5.2), we have For the full stream function, we have similarly, by (7.3)-(7.5), The constant term c 0 (z k ) in ψ can therefore be identified as Similarly for c 1 (z k ): On using (7.3), the analytic completion (7.2) of ν becomes This gives the expansions Now we have only to insert all this into (6.14), with a = z k and D chosen such that z k is the only vortex inside D. The left member is evaluated by straightforward residue computations, and the integrand in right member vanishes because ∂ψ/∂z is analytic in D\D(z k , ε). Out of this immediately comes the vortex dynamics: Theorem 7.1. The velocity of the vortex z k (t) is given by where c 1 (z k ) is given by (7.7) and where we for the second equation have set r metric (z) = 2 ∂ ∂z log λ(z) and r robin (z) = −2c 1 (z).
It remains to clarify in more detail how r robin depends on the data z j , Γ j , a j , b j , namely on how the function U in (7.7) depends on these data. This missing information will be provided in the next section, more precisely by equations (8.8)-(8.10) there.
We see from the theorem that the motion of a vortex is due to two different sources: the presence of other vortices and circulating flows, summarized in r robin , and the variation of the metric, summarized in r metric . Each of these is an affine connection in the sense §4. The difference between two affine connections is a differential, or covariant vector. This makes the equation consistent from a tensor analysis point of view since dz k /dt is to be considered as a contravariant vector and the factor λ 2 = g 11 = g 22 (with g 12 = g 21 = 0) transforms it into a covariant vector. take γ = n j=1 Γ j γ j , where each γ j is an arc from some base point b to z j . These (time-dependent) arcs can be chosen so that they, under a small time interval under consideration, do not intersect the homology basis {α k , β k }.
Using γ the periods of * dG ω can be exhibited more clearly as Relations (8.3) and (8.4) generalize the corresponding vortex pair equations (3.14) and (3.15), in fact they can be obtained by taking linear combinations of relations of the latter form.
The differentials dU α k and dU β k are fixed (independent of z 1 , . . . , z n ), so all dependence on vortex locations now lies in the 1-chain γ : Further on, we may use (8.2) to express (8.3) and (8.4) in terms of the integrals dU α k , dU β k as in (3.10) and (3.11): Here b may be taken to be the base point mentioned above (after (8.2)), but actually the right members in (8.5) and (8.6) do not depend on b. The functions U α k , U β k have unlimited harmonic continuations all over M, then become additively multivalued, but the assumption Γ j = 0 guarantees that whatever branch one chooses on supp γ , the final result in the right members above will always be the same. Now η being a harmonic 1-form it can be expanded as where A j = A j (z 1 , . . . , z n ), B j = B j (z 1 , . . . , z n ) are coefficients which depend on the locations of the poles. This can also be written (locally) as by which we have identified the function U appearing in (7.5) as The coefficients A k , B k also depend on the prescribed circulations a k , b k of the flow, so that, using (3.12), (3.13), (8.5)-(8.7), for what is the same as the Robin function defined in terms of invariant distance. Clearly the first term in H agrees with the Hamiltonian given in [2].

Hamilton's equations
The equations of motion for the vortices should be obtained by differentiating H(z 1 , . . . , z n ) with respect to thez k . More accurately, the formulation of Hamilton's equation requires, besides the Hamiltonian function itself, the introduction of a phase space provided with a symplectic form. The phase space consists in our case of all possible configurations of the vortices, collisions not allowed, i.e. we take it to be P = {(z 1 , . . . , z n ) : z j ∈ M, z k = z j for k = j}, and the symplectic form on P is to be These are standard choices, known from [1,4,12], for example. The Hamilton equations are, in general terms [8][9][10] and with a for us suitable choice of sign Proof. Choosing k = 1, for example, in (9.1) gives, on using (5.3) together with the symmetries of G(z k , z j ) and of the quadratic form representing the energy of η, −2i ∂H(z 1 , . . . , z n ) A k ∂U α j (z 1 ) A k ∂U β j (z 1 ) We see that we have agreement between the two versions of dynamics provided just the terms containing A k and B k match individually, i.e. (choosing A k , for example) g k,j=1 ∂U α j (z 1 ) ∂U β j (z 1 ) This is a special case of Such a linear relationship between harmonic forms dU α j , dU β j , * dU β k holds if and only if both members have the same integrals around all curves α , β in the homology basis. And that this is the case follows on using the period properties (3.12) and (3.13) of the differentials above together with the symmetry of the matrix (8.11). This finishes the proof.
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