#  Vortex pairs and dipoles on closed surfaces

Björn Gustafsson1

###### Abstract

We set up general equations of motion for point vortex systems on closed
Riemannian surfaces, allowing for the case that the sum of vorticities is not
zero and there hence must be counter-vorticity present. The dynamics of global
circulations which is coupled to the dynamics of the vortices is carefully
taken into account.

Much emphasis is put to the study of vortex pairs, having the Kimura
conjecture in focus. This says that vortex pairs move, in the dipole limit,
along geodesic curves, and proofs for it have previously been given by S.
Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present
paper we reach the same conclusion by following a slightly different route,
leading directly to the geodesic equation with a reparametrized time variable.

In a final section we explain how vortex motion in planar domains can be seen
as a special case of vortex motion on closed surfaces, and in two appendices
we give some necessary background on affine and projective connections.

###### Contents

  1. 1 Introduction
  2. 2 Fluid dynamics on a Riemannian surface
    1. 2.1 General notations and assumptions
  3. 3 Green functions and harmonic forms
    1. 3.1 The one point Green function
    2. 3.2 Harmonic one-forms and period relations
  4. 4 Energy renormalization and the Hamiltonian
    1. 4.1 The renormalized kinetic energy
    2. 4.2 Matrix formalism and the Hamiltonian
  5. 5 Hamilton’s equations
    1. 5.1 Phase space and symplectic form
    2. 5.2 Dynamical equations
  6. 6 Motion of a single point vortex
  7. 7 Motion of a vortex pair in the dipole limit
  8. 8 Vortex dipoles and projective connections
  9. 9 Remarks on vortex motion in planar domains
  10. 10 Appendix 1: Affine and projective connections
  11. 11 Appendix 2: Behavior of singular parts under changes of coordinates

Keywords: Point vortex, vortex pair, vortex dipole, geodesic curve, affine
connection, projective connection, Robin function, Hamiltonian function,
symplectic form, Green function.

MSC: 76B47, 53B05, 53B10, 30F30

Acknowledgements: The author has been inspired by discussions with Stefanella
Boatto, Jair Koiller, Nikolai Nadirashvili, and Jesper Göransson.

11footnotetext: Department of Mathematics, KTH, 100 44, Stockholm, Sweden.
Email<EMAIL_ADDRESS>
##  1 Introduction

This paper extends previous results in [24] on multiple point vortex motion on
closed Riemannian surfaces of arbitrary genus to cases in which the sum of the
vorticities is not zero, and therefore a counter-vorticity must be present.
This occurs for example when there is only one point vortex. For reasons
dictated by a Hodge decomposition, the counter-vorticity is naturally chosen
to be uniformly distributed on the surface.

Special for our investigations is that we, in the case of higher genus,
carefully take the circulation around the holes into account and make sure
that the Kelvin law of preservation of circulations is satisfied. This was
done already in [24], but in the case of a counter-vorticity being present one
has to take some extra steps. It is natural to choose fixed closed curves on
the surface to make up a basis for the first homology group, but the
circulation around such curves will in general not be conserved in time. In
the present paper we handle the so arising problem by considering these
circulations as free variables in the phase space, in addition to the
locations of the vortices. If there are $n$ vortices and the genus of the
surface is g the phase space will thus have real dimension $2n+2\texttt{g}$.

The Hamiltonian function will, as usual, be a renormalized kinetic energy for
the flow, see (4.26), but in the presence of a counter-vorticity the
symplectic form has to be accordingly adapted, see (5.1). With this done the
dynamical equations come out to be the expected ones (Theorem 5.1). After
having set up the vortex dynamics in general (Section 4 and 5) and after a
short discussion of the single vortex case in genus zero (Section 6), we turn
in Section 7 to the question of motion of vortex pairs, i.e. systems
consisting of two point vortices of equal strength but rotating in opposite
direction. In this case there is no counter-vorticity. The conjecture of Y.
Kimura [28], saying that a vortex pair in the dipole limit moves along a
geodesic curve, has been a major source of inspiration for the present paper.
The same applies for papers [31, 3] by S. Boatto, J. Koiller. These latter
papers actually contain a proof of Kimura’s conjecture, and in the present
paper we try to clarify the situation further by connecting the dipole
trajectory directly to the geodesic equation associated to the metric.

Throughout this paper we think of a Riemannian surface (i.e. a Riemannian
manifold of dimension two) as a Riemann surface provided with a metric which
is compatible with the conformal structure. Therefore we are lead into more or
less classical function theory on Riemann surfaces. Our treatment is based
entirely on local holomorphic coordinates. For example, the logarithmic
singularities in the stream function will look like $\log|z-w|$ near a vortex,
$z$ and $w$ being complex coordinates, $w$ representing the location of a
vortex. When such a singular term is removed from a local series expansion
(with respect to $z$) around $w$ in the necessary renormalization process and
$z$ is set equal to $w$, then the quantity which remains behaves no longer as
a function under changes of coordinates. Instead it is a kind of “connection”.
For such reasons we run into differential geometric considerations involving
different kinds of connections: affine connections (or $1$-connections) and
projective, or Schwarzian, connections ($2$-connections). The metric itself
can be viewed as the exponential of a $0$-connection. The necessary
definitions and background material are summarized in two appendices, Sections
10 and 11. Affine connections appear in many areas of mathematical physics,
and as a mathematical tool they show up in expressions for covariant
derivatives, and in particular in the equation (10.5) for geodesic curves.
Also projective connections are regularly used, for example in conformal field
theory.

The Kimura conjecture and other matters related to dipole motion has been
discussed also in [37, 38, 41, 30, 32], and from slightly different points of
view in [7, 33, 27, 6]. As mentioned, the analysis in the present paper leads
to a new confirmation of Kimura’s conjecture, although in a rather weak form:
expanding all quantities in Taylor series and keeping only the leading terms,
the dynamical equations for a vortex pair reduce to the geodesic equation in
the dipole limit. See Theorem 7.1. It seems likely that stronger forms of the
conjecture (convergence of the trajectories, for example) can be formulated
and proved, but we leave such matters for possible future investigations.

It may be remarked that Kimura’s conjecture is counter-intuitive. The reason
to think so is that vortex motion in general is governed by global laws on the
manifold, like the structure of Green functions and other harmonic functions,
whereas the geometry of geodesics is an entirely local matter. If the geometry
of the manifold is changed at one place then this will not affect what
geodesics look like at another place. It will however change the structure of
Green functions and the general vortex dynamics everywhere. The solution of
this paradox is that vortex dipoles are highly singular. A vortex pair is
partly governed by the global harmonic structure, but in the limit, when the
vortex pair becomes a dipole, this harmonic structure is completely overruled
by the differential geometric structure. In that limit, all terms in the
dynamical equations containing harmonic functions become negligible compared
to those terms depending on the metric only. See Section 7.

In Section 8 we make some attempts to treat vortex dipoles directly by
starting from the dynamical equation for a single vortex and taking the
distributional derivative with respect to the coordinate $w$ which gives the
location of the vortex. The local stream function for the flow near a single
vortex at $w$ is something like (we ignore constant factors below)

$\psi(z)=\log|z-w|+\text{regular terms},$

and the corresponding flow, when considered as a one-form (or covariant vector
field) looks like

$\nu(z)=\frac{dz}{z-w}+\text{complex conjugate}+\text{regular terms}.$

The term named “complex conjugate” is inserted to make possible for $\nu$ to
be real-valued. For the dipole we accordingly have

$d_{w}\nu(z)=\frac{dzdw}{(z-w)^{2}}+\text{complex conjugates}+\text{regular
terms}.$

The differential $dw$ shall then be thought of as containing information of
the orientation of the dipole.

The constant term (i.e. the first term in the Taylor series of the regular
terms) are in the above three expressions a $0$-connection, a $1$-connection,
and a $2$-connection, respectively (again up to constant factors). See Lemma
11.1 for a precise statement. The $0$-connection for the stream function is
the Routh’s stream function, or Robin function when $\psi$ is thought of as a
Green function. And the $1$-connection for $\nu$ is what gives the speed of
the vortex, after subtraction (in case of a curved manifold) of the affine
connection coming from the metric. The result will then be a covariant vector
field.

For $d_{w}\nu(z)$ the constant term is similarly a $2$-connection, or
projective connection, but it is more unclear what influence it has on the
motion of the dipole. Indeed, the dipole singularity is very strong and it has
a definite direction at $w$, and it seems that all finite terms should be
negligible in comparison. Therefore it is difficult to figure out any dynamics
from this picture, only that the dipole should move with infinite speed in the
direction dictated by the singularity, presumably (and as have been confirmed)
along a geodesic with respect to the metric.

In the last section (not counting the appendices) of the paper we discuss
briefly how point vortex motion in planar domains can be considered as a
special case of vortex motion on surfaces. This is done by doubling the domain
to a compact Riemann surface (the Schottky double), which is a standard tool
as far as the conformal structure is concerned. What is special in our case is
that we have to take the metric structure into account, and this becomes non-
smooth in the doubling procedure. An interesting observation is that the
boundary of the domain becomes a geodesic curve with respect to the natural
planar metric (on each of the halves) of the Schottky double.

In general the present paper is, in addition to the papers by S. Boatto and J.
Koiller already mentioned, much in spirit of papers [37, 38, 18, 19, 49] by S.
G. Llewellyn Smith, R. J. Nagem, C. Grotta Ragazzo, H. Viglioni, and Q. Wang.
The paper [4] by A. Bogatskiy contains ideas concerning the higher genus case
which are somewhat similar to those in the present paper, see in particular
Section A.3 in [4]. We wish to mention also the work [5] by A. V. Borisov and
I. S. Mamaev, and the very early article [16] (cited in [5, 38]) by A. A.
Fridman and P. Ya. Polubarinova. That work, from 1928, may be one of the first
papers on motion of dipoles and other higher singularities. Fridman was a
famous cosmologist and Pelageya Yakolevna Polubarinova (or Polubarinova-
Kochina, after marriage) was a young (at that time) disciple of Fridman. This
remarkable woman wrote her first paper in 1924, and was scientifically active
throughout her life. Her published papers and books span a period of 75 years,
the last paper appearing the same year as she died, one hundred years of age.
See [48] for a short biography related to her work on Hele-Shaw flows.

##  2 Fluid dynamics on a Riemannian surface

###  2.1 General notations and assumptions

We consider the dynamics of a non-viscous incompressible fluid on a compact
Riemannian manifold of dimension two. We follow the treatments in [46, 1, 15]
in the respects of treating the fluid velocity field as a one-form and in
extensively using the Lie derivative. These sources also provide the standard
notions and notations for differential geometry to be used. Other treatises in
fluid dynamics, suitable for our purposes, are [39, 40].

Traditionally, non-viscous fluid dynamics is discussed in terms of the fluid
velocity vector field ${\bf v}$, the density $\rho$, and the pressure $p$ of
the fluid. The basic equations are the equation of continuity (expressing
conservation of mass) and Euler’s equation (conservation of momentum):

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho{\bf v})=0,$ (2.1)
$\frac{\partial{\bf v}}{\partial t}+({\bf{v}}\cdot\nabla){\bf
v}=-\frac{1}{\rho}\nabla p.$ (2.2)

These are to be combined with a constitutive law giving a relationship between
$p$ and $\rho$.

The above equations are already simplified versions of the general Navier-
Stokes equations (which allow for viscosity terms), but we shall simplify
further by working only in two dimensions and by taking the constitutive law
to be the simplest possible:

$\rho=1.$ (2.3)

Thus $\rho$ disappears from discussion, and the equation of continuity becomes

$\nabla\cdot{\bf v}=0.$ (2.4)

One can get rid also of $p$, because when $\rho$ is constant then $p$ appears
only as a scalar potential in Euler’s equation, this equation effectively
saying that

$\frac{\partial{\bf v}}{\partial t}+({\bf{v}}\cdot\nabla){\bf v}=\nabla({\rm
something}),$ (2.5)

where this ${\rm``something"}=p$ afterwards can be recovered, up to a (time
dependent) constant.

So everything is extremely simple (in theory), and even more so in two
dimensions. As is well-known [44], every (oriented) Riemannian manifold of
dimensions two can be made into a Riemann surface by choosing isothermal local
coordinates $(x,y)=(x^{1},x^{2})$, by which $z=x+\mathrm{i}y$ becomes a
holomorphic coordinate and the metric takes the form

$ds^{2}=\lambda(z)^{2}|dz|^{2}=\lambda(z)^{2}(dx^{2}+dy^{2})$ (2.6)

with $\lambda>0$. Thus the metric tensor $g_{ij}$, as appearing in general in
$ds^{2}=g_{ij}dx^{i}dx^{j}$, becomes $g_{ij}=\lambda^{2}\delta_{ij}$ in these
coordinates. It turns out to be convenient to work with the fluid velocity
$1$-form (or covariant vector)

$\nu=\nu_{x}\,dx+\nu_{y}\,dy$

rather than with the corresponding vector field, which then becomes

${\bf v}=\frac{1}{\lambda^{2}}(\nu_{x}\,\frac{\partial}{\partial
x}+\nu_{y}\,\frac{\partial}{\partial y}).$

The Hodge star operator takes $0$-forms into $2$-forms and vice versa, and
takes $1$-forms to $1$-forms. It acts on basic differential forms as

$*1=\lambda^{2}(dx\wedge dy)={\rm vol}=\text{the volume (area) form},$
$*dx=dy,\quad*dy=-dx,\quad*(dx\wedge dy)=\lambda^{-2}.$

Thus, for $1$-forms,

$*\nu=-\nu_{y}\,dx+\nu_{x}\,dy,$

which can be interpreted as a rotation by ninety degrees of the corresponding
vector.

In addition to the Hodge star it is useful to introduce the Lie derivative
$\mathcal{L}_{\bf v}$ of a vector field ${\bf v}$. When acting on differential
forms this is related to interior derivation (or “contraction”) $i({\bf v})$
and exterior derivation $d$ by the homotopy formula

$\mathcal{L}_{\bf{v}}=d\circ i({\bf{v}})+i({\bf{v}})\circ d.$ (2.7)

The Hodge star and interior derivation are related by

$i({\bf v}){\rm vol}=*\nu,$ (2.8)

where ${\bf v}$ and $\nu$ are linked via the metric tensor as above. Thus

$d*\nu=d(i({\bf v}){\rm vol})=\mathcal{L}_{\bf v}({\rm vol})=(\nabla\cdot{\bf
v}){\rm vol}.$

See [15] for this identity, and for further details in general. We conclude
that (2.4) is equivalent to the statement that $*\nu$ is a closed form:

$d*\nu=0.$ (2.9)

Locally we can therefore write

$*\nu=d\psi$ (2.10)

for some (locally defined) stream function $\psi$. The vorticity $2$-form for
a fluid is, in terms of the flow $1$-form $\nu,$

$\omega=d\nu=(\frac{\partial\nu_{y}}{\partial
x}-\frac{\partial\nu_{x}}{\partial y})\,dx\wedge dy.$ (2.11)

The Euler equation (2.2) can be written

$(\frac{\partial}{\partial t}+\mathcal{L}_{\bf
v})({\bf\nu})=d(\frac{1}{2}|{\bf v}|^{2}-{p}).$ (2.12)

Note that the left member involves the fluid velocity both as a vector field
and as a form. The combination $\frac{\partial}{\partial t}+\mathcal{L}_{\bf
v}$ can be viewed as a counterpart, for forms, to the more traditional
“convective derivative” (or material derivative) $\frac{\partial}{\partial
t}+{\bf{v}}\cdot\nabla$, implicitly used in (2.5), for vector fields. However,
the two derivatives are not the same. The equivalence between (2.12) and (2.2)
(when $\rho=1$) is a consequence of the identity

$\mathcal{L}_{\bf v}\nu=d(\frac{1}{2}|{\bf
v}|^{2})+({\bf{v}}\cdot\nabla){\nu},$

which can be directly verified by coordinate computations. See [46, 15] for
details.

Since the pressure $p$ does not appear in any other equation, the Euler
equation on the form (2.12) only expresses, in analogy with (2.5), that the
left member is an exact $1$-form:

$(\frac{\partial}{\partial t}+\mathcal{L}_{\bf v})({\bf\nu})={\rm
exact}=d\phi.$ (2.13)

From the scalar $\phi$ the pressure $p$ then can be recovered via

$\phi=\frac{1}{2}|{\bf v}|^{2}-{p}+{\rm constant}.$ (2.14)

On acting by $d$ on (2.13), the local form of Helmholtz-Kirchhoff-Kelvin law
of conservation of vorticity follows:

$(\frac{\partial}{\partial t}+\mathcal{L}_{\bf{v}})(\omega)=0.$ (2.15)

Since the right member of (2.13) is not only a closed differential form, but
even an exact form, a stronger, global, version of the Helmholtz-Kirchhoff-
Kelvin law actually follows. This can be expressed by saying that

$\frac{d}{dt}\oint_{\gamma(t)}\nu=0$

for any closed curve $\gamma(t)$ which moves with the fluid.

##  3 Green functions and harmonic forms

###  3.1 The one point Green function

In the sequel $M$ will be a closed (compact) Riemann surface provided with a
Riemannian metric on the form (2.6).

Given a $2$-form $\omega$ on $M$ one can immediately obtain a corresponding
Green potential $G^{\omega}$ by Hodge decomposition, i.e. by orthogonal
decomposition in the Hilbert space of square integrable $2$-forms. The inner
product for such forms is

$(\omega_{1},\omega_{2})_{2}=\int_{M}\omega_{1}\wedge*\omega_{2}.$ (3.1)

The given $2$-form $\omega$ then decomposes as the orthogonal decomposition of
an exact form and a harmonic form:

$\omega=d(\rm something)+{\rm harmonic}.$

This Hodge decomposition can more precisely be written as

$\omega=-d*dG^{\omega}+{\rm constant}\cdot{\rm vol},$ (3.2)

where $G^{\omega}$ is normalized to be orthogonal to all harmonic $2$-forms,
namely satisfying

$\int_{M}G^{\omega}\,{\rm vol}=0.$ (3.3)

The constant in (3.2) necessarily equals the mean value of $\omega$,

${\rm constant}=\frac{1}{V}\int_{M}\omega,$ (3.4)

where $V$ denotes the total volume (=area) of $M$:

$V=\int_{M}{\rm vol}.$

When $\omega$ is exact, as in (2.11), the second term in the right member of
(3.2) disappears, hence

$-d*dG^{\omega}=\omega$ (3.5)

in this case. In the other extreme, when $\omega$ is harmonic, i.e. is a
constant multiple of ${\rm vol}$, the first term disappears. Indeed, the whole
Green function disappears:

$G^{{\rm vol}}=0.$ (3.6)

For $1$-forms the inner product has the same expression as for $2$-forms:

$(\nu_{1},\nu_{2})_{1}=\int_{M}\nu_{1}\wedge*\nu_{2}.$ (3.7)

If $\nu$ represents a fluid velocity, then $(\nu,\nu)_{1}$ has the
interpretation of being (twice) 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.

The energy $\mathcal{E}(\omega,\omega)$ of any $2$-form $\omega$ is defined to
be the energy of its Green potential $G^{\omega}$. Thus, for the mutual
energy,

$\mathcal{E}(\omega_{1},\omega_{2})=(dG^{\omega_{1}},dG^{\omega_{2}})_{1}=\int_{M}G^{\omega_{1}}\wedge\omega_{2}.$
(3.8)

It follows, from (3.3) for example, that the volume form has no energy:

$\mathcal{E}({\rm vol},{\rm vol})=0.$

It was tacitly assumed above that the forms under discussion belong to the
$L^{2}$-space defined by the inner product. However, the mutual energy
sometimes extends to circumstances in which source distributions have infinite
energy. This applies in particular to the Dirac current $\delta_{a}$, which we
consider as a $2$-form with distributional coefficient, namely defined by the
property that

$\int_{M}\delta_{a}\wedge\varphi=\varphi(a)$

for every smooth function $\varphi$. Certainly $\delta_{a}$ has infinite
energy, but if $a\neq b$, then $\mathcal{E}(\delta_{a},\delta_{b})$ is still
finite and has a natural interpretation: it is the (one-point) Green function,
or “monopole” Green function:

$G(a,b)=G^{\delta_{a}}(b)=\mathcal{E}(\delta_{a},\delta_{b}).$ (3.9)

Here the first equality can be taken as a definition, and then the second
equality follows on using (3.2), (3.3):

$\displaystyle\mathcal{E}(\delta_{a},\delta_{b})$
$\displaystyle=\int_{M}dG^{\delta_{a}}\wedge*dG^{\delta_{b}}=-\int_{M}G^{\delta_{a}}\wedge
d*dG^{\delta_{b}}$
$\displaystyle=\int_{M}G^{\delta_{a}}\wedge\left(\delta_{b}-\frac{1}{V}\,{\rm
vol}\right)=G^{\delta_{a}}(b)=G(a,b).$

This shows in addition that $G(a,b)$ is symmetric.

Changing now notations from $a,b$ to $z,w$, where later $w$ will have the role
of being the location of a point vortex, the Green function $G(z,w)$, as a
function of $z$, has just one pole (at $z=w$). Its Laplacian, as a $2$-form,
is then

$-d*dG(\cdot,w)=\delta_{w}-\frac{1}{V}{\rm{vol}}.$ (3.10)

It is interesting to notice that among the two terms in the right member of
(3.10), one has infinite energy and one has zero energy
($\mathcal{E}(\delta_{w},\delta_{w})=+\infty$, $\mathcal{E}({\rm vol},{\rm
vol})=0$).

######  Remark 3.1.

It is more common to treat the Dirac delta and the Laplacian (denoted
$\Delta$) as “densities” with respect to the volume form. However, we find our
usage convenient. In any case, the relationships are

$\delta_{a}=({\rm delta\,\,^{\prime\prime}function^{\prime\prime}})\,{\rm
vol},\quad d*d\varphi=(\Delta\varphi)\,{\rm vol}.$

######  Remark 3.2 (Notational remark).

Letters like $z$, $w$ will have the double roles of being complex-valued local
coordinates on parts of the Riemann surface and of denoting the points on the
surface for which the coordinates have the values in question. A more formal
treatment could use, for example, $P\in M$ for a point and
$z(P)\in{\mathbb{C}}$ for the corresponding coordinate value.

######  Remark 3.3 (Real versus complex notation).

For future needs we wish to clarify the relationship between real and complex
coordinates in the context of tangent and cotangent vectors.

Let $z=x+\mathrm{i}y$ be a complex coordinate on $M$ and consider a curve
$t\mapsto z(t)$ in $M$ (for example the trajectory of a vortex). Set
$\dot{z}={dz}/{dt}$, and similarly for $\dot{x}$ and $\dot{y}$, so that
$\dot{z}=\dot{x}+\mathrm{i}\dot{y}$. The velocity of this moving point is
primarily to be considered as a vector in the (real) tangent space of $M$ (at
the point under consideration). This gives, in the picture of viewing tangent
vectors as derivations, the velocity vector

${\bf V}=\dot{x}\frac{\partial}{\partial x}+\dot{y}\frac{\partial}{\partial
y}=\dot{z}\frac{\partial}{\partial
z}+\dot{\bar{z}}\frac{\partial}{\partial\bar{z}},$ (3.11)

where it is understood that $\dot{\bar{z}}=\overline{\dot{z}}$.

The real tangent space used above can in a next step be complexified, which
means that one breaks the connection between $\dot{z}$ and $\dot{\bar{z}}$ and
consider them as independent complex variables. Equivalently, one allows
$\dot{x}$ and $\dot{y}$ to be complex-valued. The so obtained complex tangent
space can be decomposed as a direct sum of its holomorphic and anti-
holomorphic subspaces, and then

${\rm proj}:\quad{\bf V}=\dot{z}\frac{\partial}{\partial
z}+\dot{\bar{z}}\frac{\partial}{\partial\bar{z}}\mapsto\dot{z}\frac{\partial}{\partial
z}$

is a natural and useful identification of the real tangent space with the
holomorphic part of the complex tangent space. See [17], in particular Section
2 of Chapter 0, for further discussions. With this identification $\dot{z}$
represents the velocity of $z(t)$. Still one need to keep in mind that
$\dot{z}$ is just a complex number and that it is rather the preimage under
${\rm proj}$ above that is the true velocity, as a real tangent vector.

The vector ${\bf V}$ above correspond, via the metric, to the covector

${\lambda^{2}}(\dot{x}dx+\dot{y}dy)=\frac{\lambda^{2}}{2}(\dot{\bar{z}}dz+\dot{{z}}d\bar{z}),$
(3.12)

where $\lambda=\lambda(x,y)=\lambda(z)$ (depending on context). Note that

$\frac{1}{2}\dot{\bar{z}}=\frac{1}{2}(\dot{x}-\mathrm{i}\dot{y}),\quad\frac{1}{2}\dot{{z}}=\frac{1}{2}(\dot{x}+\mathrm{i}\dot{y}),$

as coefficients of $dz$ and $d\bar{z}$ (respectively) have similar roles (and
signs) as the Wirtinger derivatives $\partial/\partial z$,
$\partial/\partial\bar{z}$. For a covector $\nu$ in general we therefore
define

$\nu_{z}=\frac{1}{2}(\nu_{x}-\mathrm{i}\nu_{y}),\quad\nu_{\bar{z}}=\frac{1}{2}(\nu_{x}+\mathrm{i}\nu_{y}),$

so that

$\nu=\nu_{x}dx+\nu_{y}dy=\nu_{z}dz+\nu_{\bar{z}}d\bar{z}.$

The corresponding (contravariant) vector is then, as in our fluid dynamical
contexts,

${\bf v}=\frac{1}{\lambda^{2}}\big{(}\nu_{x}\frac{\partial}{\partial
x}+\nu_{y}\frac{\partial}{\partial
y}\big{)}=\frac{1}{\lambda^{2}}\big{(}\nu_{z}\frac{\partial}{\partial
z}+\nu_{\bar{z}}\frac{\partial}{\partial\bar{z}}\big{)}.$

###  3.2 Harmonic one-forms and period relations

For later use we record the formulas (differentiation with respect to $z$)

$dG(z,w)=\frac{\partial G(z,w)}{\partial z}dz+\frac{\partial
G(z,w)}{\partial\bar{z}}d\bar{z}=2\operatorname{Re}\big{(}\frac{\partial
G(z,w)}{\partial z}dz\big{)},\qquad$ $*dG(z,w)=-\mathrm{i}\frac{\partial
G(z,w)}{\partial z}dz+\mathrm{i}\frac{\partial
G(z,w)}{\partial\bar{z}}d\bar{z}=2\operatorname{Im}\big{(}\frac{\partial
G(z,w)}{\partial z}dz\big{)}.$ (3.13)

If $\gamma$ is any closed oriented curve in $M$ then clearly
$\oint_{\gamma}dG(\cdot,w)=0$, while the conjugate period defines a function

$U_{\gamma}(w)=\oint_{\gamma}*dG(\cdot,w)=\oint_{\gamma}\big{(}-\mathrm{i}\frac{\partial
G(z,w)}{\partial z}dz+\mathrm{i}\frac{\partial
G(z,w)}{\partial\bar{z}}d\bar{z}\big{)}$ (3.14)
$=-2\mathrm{i}\oint_{\gamma}\frac{\partial G(z,w)}{\partial
z}dz=2\mathrm{i}\oint_{\gamma}\frac{\partial
G(z,w)}{\partial\bar{z}}d\bar{z},$ (3.15)

which, away from $\gamma$, is harmonic in $w$ and makes a unit additive jump
as $w$ crosses $\gamma$ from the left to the right. The harmonicity of
$U_{\gamma}(w$) is perhaps not obvious from outset since $G(\cdot,w)$ is not
itself harmonic, but the deviation from harmonicity, namely the extra term in
(3.10), is independent of $w$ and therefore disappears under differentiation.

Differentiating (3.14) with respect to $w$ gives

$dU_{\gamma}(w)=-2\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial
w}d{z}\otimes
dw-2\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial\bar{w}}d{z}\otimes
d\bar{w},$
$*dU_{\gamma}(w)=-2\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial
w}d{z}\otimes
dw+2\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial\bar{w}}d{z}\otimes
d\bar{w}\quad$

(integration with respect ot $z$, Hodge star and $d$ with respect to $w$). We
have written out a tensor product sign to emphasize that the product between
the $dz$ and $dw$ is not a wedge product. Adding and subtracting we obtain the
analytic (respectively anti-analytic) differentials

$dU_{\gamma}(w)+\mathrm{i}*dU_{\gamma}(w)=2\frac{\partial
U_{\gamma}(w)}{\partial
w}dw=-4\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial
w}d{z}\otimes dw,$ (3.16)
$dU_{\gamma}(w)-\mathrm{i}*dU_{\gamma}(w)=2\frac{\partial
U_{\gamma}(w)}{\partial\bar{w}}d\bar{w}=-4\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial\bar{w}}d{z}\otimes
d\bar{w}.$ (3.17)

Let now
$\\{\alpha_{1},\dots,\alpha_{\texttt{g}},\beta_{1},\dots,\beta_{\texttt{g}}\\}$
be representing cycles for a canonical homology basis for $M$ such that each
$\beta_{j}$ intersects $\alpha_{j}$ once from the right to the left and no
other intersections occur (see [12] for details). Then there are corresponding
harmonic differentials $dU_{\alpha_{j}}$, $dU_{\beta_{j}}$ obtained on
choosing $\gamma=\alpha_{j},\beta_{j}$ in the above construction, and these
constitute a basis of harmonic differentials associated with the chosen
homology basis. Precisely we have, for $k,j=1,\dots,\texttt{g}$,

$\oint_{\alpha_{k}}(-dU_{\beta_{j}})=\delta_{kj},\quad\oint_{\beta_{k}}(-dU_{\beta_{j}})=0,$
(3.18)
$\oint_{\alpha_{k}}dU_{\alpha_{j}}=0,\qquad\oint_{\beta_{k}}dU_{\alpha_{j}}=\delta_{kj}.$
(3.19)

The Kronecker deltas, $\delta_{kj}$, come from the discontinuity (jump)
properties of $U_{\gamma}$ mentioned after (3.14).

The integrals of the basic harmonic differentials along non-closed curves
become periods of two point Green functions:

$\int_{b}^{a}dU_{\alpha_{j}}=\oint_{\alpha_{j}}*dG^{\delta_{a}-\delta_{b}},$
(3.20)
$\int_{b}^{a}dU_{\beta_{j}}=\oint_{\beta_{j}}*dG^{\delta_{a}-\delta_{b}}.$
(3.21)

Here the integration in the left member is to be performed along a path that
does not intersect the curve in the right member. These formulas are immediate
from the definition (3.14) of $U_{\gamma}$.

##  4 Energy renormalization and the Hamiltonian

###  4.1 The renormalized kinetic energy

When $\nu$ is the flow $1$-form of an incompressible fluid, the equation of
continuity says that $d*\nu=0$ (see (2.9)). The vorticity $2$-form is
$\omega=d\nu$, and (3.5) holds. Thus $d(\nu+*dG^{\omega})=0$, and setting

$\eta=\nu+*dG^{\omega}$ (4.1)

it follows that $\eta$ is harmonic: $d\eta=0=d*\eta$. Locally we can write
$*\eta=d\psi_{0}$ for some harmonic function $\psi_{0}$. Then

$\psi=G^{\omega}+\psi_{0}$ (4.2)

becomes a locally defined stream function for the flow, so that

$*\nu=d\psi=dG^{\omega}+d\psi_{0}.$

Different local choices of $\psi$ may differ by additive time dependent
constants. The roles of the additional terms $\eta$ and $\psi_{0}$,
complementing the contribution from the Green function, will be somewhat
clarified in the context of planar vortex motion and the hydrodynamic Green
function in Section 9.

The relation (4.1) written on the form $\nu=\eta-*dG^{\omega}$ is an
orthogonal decomposition of the flow $1$-form $\nu$ with respect to the inner
product (3.7). Indeed,

$(\eta,-*dG^{\omega})_{1}=\int_{M}\eta\wedge dG^{\omega}=-\int_{M}d\eta\wedge
G^{\omega}=0,$

since $\eta$ is harmonic. It follows that (twice) the total (kinetic) energy
$(\nu,\nu)_{1}$ of the flow is given by

$2(\nu,\nu)_{1}=\int_{M}\nu\wedge*\nu=\frac{\mathrm{i}}{2}\int_{M}(\nu+\mathrm{i}*\nu)\wedge(\nu-\mathrm{i}*\nu)=$
$=\int_{M}dG^{\omega}\wedge*dG^{\omega}+\int_{M}\eta\wedge*\eta=\mathcal{E}(\omega,\omega)+\int_{M}\eta\wedge*\eta=$
$=\int_{M}G^{\omega}\wedge\omega+\sum_{j=1}^{\texttt{g}}(\oint_{\alpha_{j}}\eta\oint_{\beta_{j}}*\eta-\oint_{\alpha_{j}}*\eta\oint_{\beta_{j}}\eta).$

We shall now specialize on the point vortex case, with vortices of strengths
$\Gamma_{k}$ located at points $w_{k}\in M$ ($k=1,\dots,n$). However, we shall
not assume that these strengths add up to zero. Hence the sum

$\Gamma=\sum_{k=1}^{n}\Gamma_{k}$ (4.3)

may be non-zero and there will then be a compensating uniform counter
vorticity. The total vorticity $\omega=d\nu$, which satisfies
$\int_{M}\omega=0$, appears as the right member in

$-d*dG^{\omega}=\omega=\sum_{k=1}^{n}\Gamma_{k}\delta_{w_{k}}-\frac{\Gamma}{V}{\rm
vol}.$ (4.4)

Explicitly we have, on recalling (3.6),

$G^{\omega}(z)=\sum_{k=1}^{n}\Gamma_{k}{G(z,w_{k})}.$ (4.5)

On using (3.13) the conjugate $\alpha$-periods of $G^{\omega}$ can be
expressed as

$\oint_{\alpha_{j}}*d{G^{\omega}}=-\sum_{k=1}^{n}2\mathrm{i}\Gamma_{k}\oint_{\alpha_{j}}\frac{\partial
G(z,w_{k})}{\partial z}dz.$ (4.6)

By differentiation and using also (3.15), (3.16) we have

$\frac{\partial}{\partial
w_{k}}\oint_{\alpha_{j}}*d{G^{\omega}}=\Gamma_{k}\frac{\partial
U_{\alpha_{j}}(w_{k})}{\partial w_{k}}=\mathrm{i}\Gamma_{k}\frac{\partial
U^{*}_{\alpha_{j}}(w_{k})}{\partial w_{k}},$ (4.7) $\frac{\partial}{\partial
w_{k}}\big{(}\oint_{\alpha_{j}}*d{G^{\omega}}\big{)}dw_{k}=\frac{\Gamma_{k}}{2}(dU_{\alpha_{j}}(w_{k})+\mathrm{i}*dU_{\alpha_{j}}(w_{k})).$
(4.8)

Here $U^{*}_{\alpha_{j}}$ denotes a harmonic conjugate of $U_{\alpha_{j}}$,
whereby $*dU_{\alpha_{j}}=d(U^{*}_{\alpha_{j}})$. Similar relations hold for
periods around $\beta_{j}$, and for $\bar{w}_{k}$ derivatives.

The expression for the kinetic energy can in the point vortex case be written,
at least formally,

$2(\nu,\nu)_{1}=\sum_{k,j=1}^{n}\Gamma_{k}\Gamma_{j}G(w_{k},w_{j})+\sum_{j=1}^{\texttt{g}}\big{(}\oint_{\alpha_{j}}\eta\oint_{\beta_{j}}*\eta-\oint_{\alpha_{j}}*\eta\oint_{\beta_{j}}\eta\big{)}.$

However, the presence of the terms $\Gamma_{k}^{2}G(w_{k},w_{k})$ makes the
first term become infinite. In order to renormalize this singular behavior we
isolate the logarithmic pole in the Green function by writing

$G(z,w)=\frac{1}{2\pi}(-\log|z-w|+H(z,w)),$ (4.9)

and expand, for a fixed $w$, the regular part in a power series in $z$ as

$H(z,w)=h_{0}(w)+\frac{1}{2}\left(h_{1}(w)(z-w)+\overline{h_{1}(w)}(\bar{z}-\bar{w})\right)+$
(4.10)
$+\frac{1}{2}\left(h_{2}(w)(z-w)^{2}+\overline{h_{2}(w)}(\bar{z}-\bar{w})^{2}\right)+h_{11}(w)(z-w)(\bar{z}-\bar{w})+\mathcal{O}(|z-w|^{3}).$

In Appendix 2, Section 11, it is discussed how the coefficients in the
expansion (4.10) behave under conformal mapping. For example, the coefficient
$h_{11}(w)$ transforms as the density of a metric, and it is indeed
proportional to the given metric:

$h_{11}(w)=\frac{\pi}{2V}\lambda(w)^{2}.$ (4.11)

See (11.5). The coefficients $h_{0}(w)$, $h_{1}(w)$ and $h_{2}(w)$ transform,
in certain combinations, as “connections” under conformal mappings. For
$h_{0}(w)$ this amounts to saying that it too defines a metric, namely the
Robin metric, via

$ds=e^{-h_{0}(w)}|dw|.$ (4.12)

Metrics of this kind are implicit in the theory of capacity functions, as
exposed in [42]. It should be pointed out that our version of the Robin metric
depends on the given metric, since the Green function itself depends on it.
The Robin metric can also be adapted to various given circulations, and then
it becomes more intrinsically hydrodynamic in nature. The coefficient
$h_{0}(w)$ is one example of a (coordinate) Robin function.

Now, letting $w$ have the role of being a vortex point indicates that one
could renormalize the kinetic energy by simply discarding the singular term
$\log|z-w|$, as this seems at first sight to produce just a circular symmetric
flow, not affecting the speed of the vortex. However, this is not fully
correct in the case of curved surfaces. The term $\log|z-w|$ cannot be just
removed, it need be replaced by a term which counteracts the inhomogeneous
transformation law of $h_{0}(w)$ (see (11.8)). Such a term comes naturally
from the given metric $ds=\lambda(w)|dw|=e^{\log\lambda(w)}|dw|$. We see that
minus $\log\lambda(w)$ has the right properties, and it combines with
$h_{0}(w)$ into

$R_{\rm robin}(w)=\frac{1}{2\pi}(h_{0}(w)+\log\lambda(w)).$ (4.13)

This is indeed a function, the Robin function, and it appears naturally when
writing the singularity of the Green function in terms of the distance with
respect to the given Riemannian metric:

$G(z,w)=-\frac{1}{2\pi}\log{\rm dist}(z,w)+R_{\rm robin}(w)+\mathcal{O}({\rm
dist}(z,w)).$

As for the infinite kinetic energy, we conclude that it should be renormalized
by replacing the diagonal terms $G(w_{k},w_{k})$ by $R_{\rm robin}(w_{k})$.
This gives the same equations of motion as more direct approaches, or those
available in the literature, like [26, 3, 10]. Thus (twice) the renormalized
energy is

$2(\nu,\nu)_{1,{\rm renorm}}=\sum_{k=1}^{n}\Gamma_{k}^{2}R_{\rm
robin}(w_{k})+\sum_{k\neq j}\Gamma_{k}\Gamma_{j}G(w_{k},w_{j})+$
$+\sum_{j=1}^{\texttt{g}}\big{(}\oint_{\alpha_{j}}\eta\oint_{\beta_{j}}*\eta-\oint_{\alpha_{j}}*\eta\oint_{\beta_{j}}\eta\big{)}.$
(4.14)

This depends on the locations $w_{1},\dots,w_{n}$ of the vortices (even the
last term depends on these, although somewhat more indirectly).

Many authors start out directly with the Robin function (4.13), but there are
some advantages with exposing the two terms in it as individual quantities.
One is that it clarifies the structure of the vortex motion equations by
separating harmonic contributions, such as $h_{0}(w_{k})$ and
$G(w_{k},w_{j})$, from differential geometric contributions, like
$\log\lambda(w_{k})$. The first category of terms can be classified as
nonlocal, coming from solutions of elliptic partial differential equations on
the entire surface, whereas the second category are purely local in nature.
There is also the contribution from global circulations, represented by the
final term in (4.14), and this may be considered to be truly global. So the
vortex motion is in principle governed by a balance between different
categories of contributions, local, nonlocal, and global in nature. As we
shall see later this balance is drastically changed in the more singular case
of dipole motion (Sections 7 and 8). Then only the local terms survive.

We need to elaborate further the expression (4.1) and relate it to the global
circulations. The $1$-form $\eta$ is harmonic and hence can be expanded as

$\eta=-\sum_{j=1}^{\texttt{g}}A_{j}dU_{\beta_{j}}+\sum_{j=1}^{\texttt{g}}B_{j}dU_{\alpha_{j}}.$
(4.15)

The coefficients $A_{k}$, $B_{k}$ are in view of (3.18), (3.19) given by

$A_{k}=\oint_{\alpha_{k}}\eta,\quad B_{k}=\oint_{\beta_{k}}\eta.$

The circulations of the total flow $\nu=\eta-*dG^{\omega}$ then become

$a_{k}=\oint_{\alpha_{k}}\nu=A_{k}-\oint_{\alpha_{k}}*dG^{\omega},$
$b_{k}=\oint_{\beta_{k}}\nu=B_{k}-\oint_{\beta_{k}}*dG^{\omega}.$

We shall treat the circulations $a_{1},\dots,a_{\texttt{g}}$,
$b_{1},\dots,b_{\texttt{g}}$ as free (independent) variables, along with the
locations $w_{1},\dots,w_{n}$ of the vortices. These variables will be the
coordinates of the phase space, and together they determine $\eta$ and $\nu$.
The locations of the vortices are complex variables, in contrast to the
circulations which are real.

If $\gamma$ is a closed oriented curve in $M$, fixed in time and avoiding the
point vortices, then according to the Euler equation (2.13), and in the
notation used there, the circulation of $\nu$ around $\gamma$ changes with
speed

$\frac{d}{dt}\oint_{\gamma}\nu=\oint_{\gamma}\frac{\partial\nu}{\partial
t}=\oint_{\gamma}(d\phi-\mathcal{L}_{\bf v}\nu)=-\oint_{\gamma}i({\bf
v})d\nu=$ $=-\oint_{\gamma}i({\bf v})\omega=\oint_{\gamma}i({\bf
v})\big{(}\frac{\Gamma}{V}{\rm
vol}\big{)}=\frac{\Gamma}{V}\oint_{\gamma}*\nu.$

Here we used also (2.7), (2.8), (4.4). It follows in particular, on choosing
$\gamma=\alpha_{k},\beta_{k}$, that

$\frac{da_{k}}{dt}=\frac{\Gamma}{V}\oint_{\alpha_{k}}*\nu=\frac{\Gamma}{V}\oint_{\alpha_{k}}*\eta,$
(4.16)
$\frac{db_{k}}{dt}=\frac{\Gamma}{V}\oint_{\beta_{k}}*\nu=\frac{\Gamma}{V}\oint_{\beta_{k}}*\eta.$
(4.17)

###  4.2 Matrix formalism and the Hamiltonian

The period matrix (written here in block form)

$\left(\begin{array}[]{cc}P&R\\\ R^{T}&Q\\\
\end{array}\right)=\left(\begin{array}[]{cc}(-\oint_{\beta_{k}}*dU_{\beta_{j}})&(\oint_{\beta_{k}}*dU_{\alpha_{j}})\\\
(\oint_{\alpha_{k}}*dU_{\beta_{j}})&(-\oint_{\alpha_{k}}*dU_{\alpha_{j}})\\\
\end{array}\right)=$
$=\left(\begin{array}[]{cc}(\int_{M}dU_{\beta_{k}}\wedge*dU_{\beta_{j}})&(-\int_{M}dU_{\beta_{k}}\wedge*dU_{\alpha_{j}})\\\
(-\int_{M}dU_{\alpha_{k}}\wedge*dU_{\beta_{j}})&(\int_{M}dU_{\alpha_{k}}\wedge*dU_{\alpha_{j}})\\\
\end{array}\right)$ (4.18)

is symmetric and positive definite (see [12] in general). In particular, $P$
and $Q$ are themselves symmetric and positive definite. As for $R$ and $R^{T}$
we need to be explicit with what are row and column indices above: in all
entries above, $k$ is the row index, $j$ the column index. Thus, for example,
$R_{kj}=\oint_{\beta_{k}}*dU_{\alpha_{j}}$.

Next we write

$dU_{\alpha}=\left(\begin{array}[]{c}dU_{\alpha_{1}}\\\ \vdots\\\
dU_{\alpha_{\texttt{g}}}\end{array}\right),\quad
dU_{\beta}=\left(\begin{array}[]{c}dU_{\beta_{1}}\\\ \vdots\\\
dU_{\beta_{\texttt{g}}}\end{array}\right).$

As mentioned, these two column matrices together define a basis of the
harmonic forms. Another basis is provided by the corresponding Hodge starred
column vectors $*dU_{\alpha}$, $*dU_{\beta}$, in similar matrix notation. The
relation between the bases is

######  Lemma 4.1.

The two bases $\\{dU_{\alpha},dU_{\beta}\\}$ and
$\\{*dU_{\alpha},*dU_{\beta}\\}$ are related by

$\left(\begin{array}[]{cc}*dU_{\alpha}\\\ dU_{\beta}\\\
\end{array}\right)=\left(\begin{array}[]{cc}R^{T}&Q\\\ -P&-R\\\
\end{array}\right)\left(\begin{array}[]{cc}dU_{\alpha}\\\ dU_{\beta}\\\
\end{array}\right).$ (4.19)

###### Proof.

One simply checks that the two members in (4.19) have the same periods with
respect to the homology basis $\\{\alpha_{j},\beta_{j}\\}$. ∎

We arrange also the circulations of the flow $\nu$ into column vectors:

$a=\left(\begin{array}[]{c}a_{1}\\\ \vdots\\\
a_{\texttt{g}}\end{array}\right)=\left(\begin{array}[]{c}\oint_{\alpha_{1}}\nu\\\
\vdots\\\ \oint_{\alpha_{\texttt{g}}}\nu\end{array}\right),\quad
b=\left(\begin{array}[]{c}b_{1}\\\ \vdots\\\
b_{\texttt{g}}\end{array}\right)=\left(\begin{array}[]{c}\oint_{\beta_{1}}\nu\\\
\vdots\\\ \oint_{\beta_{\texttt{g}}}\nu\end{array}\right),$

briefly written as

$a=\oint_{\alpha}\nu,\quad b=\oint_{\beta}\nu.$ (4.20)

Similarly for other vectors of circulations, for example

$A=\oint_{\alpha}\eta=\oint_{\alpha}\nu+\oint_{\alpha}*dG^{\omega}=a+\oint_{\alpha}*dG^{\omega},$
(4.21)
$B=\oint_{\beta}\eta=\oint_{\beta}\nu+\oint_{\beta}*dG^{\omega}=b+\oint_{\beta}*dG^{\omega}.$
(4.22)

The $\alpha$-periods of the conjugated Green function were computed in (4.6),
and from (4.7), (4.8) it follows that

$\frac{\partial A}{\partial w_{k}}dw_{k}+\frac{\partial
A}{\partial\bar{w}_{k}}d\bar{w}_{k}={\Gamma_{k}}\,dU_{\alpha}(w_{k}).$

Taking into account also the dependence of $a_{1},\dots,a_{\texttt{g}}$, and
doing the same for $B$ and the $\beta$-periods, gives the total differentials

$dA=da+d\oint_{\alpha}*dG^{\omega}=da+\sum_{k=1}^{n}\Gamma_{k}dU_{\alpha}(w_{k}),$
(4.23)
$dB=db+d\oint_{\beta}*dG^{\omega}=db+\sum_{k=1}^{n}\Gamma_{k}dU_{\beta}(w_{k}).$
(4.24)

In matrix notation (4.15) becomes

$\eta=-A^{T}dU_{\beta}+B^{T}dU_{\alpha}=\left(\begin{array}[]{cc}B^{T},&-A^{T}\end{array}\right)\left(\begin{array}[]{c}dU_{\alpha}\\\
dU_{\beta}\end{array}\right),$ (4.25)

and the contribution from $\eta$ to the kinetic energy is the quadratic form

$\int_{M}\eta\wedge*\eta=\sum_{k=1}^{\texttt{g}}(\oint_{\alpha_{k}}\eta\oint_{\beta_{k}}*\eta-\oint_{\beta_{k}}\eta\oint_{\alpha_{k}}*\eta)=$
$=\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\ \end{array}\right)\left(\begin{array}[]{c}A\\\
B\end{array}\right).$

The last equality is based on straight-forward computations. Note that $P$,
$Q$, $R$ are fixed matrices, while $A$ and $B$ depend on $w_{1},\dots,w_{n}$,
$a_{1},\dots,a_{\texttt{g}}$, $b_{1},\dots,b_{\texttt{g}}$ via (4.21), (4.22).

We now define the Hamiltonian function, $\mathcal{H}$, as the renormalized
kinetic energy of the flow considered as a function of the locations of the
point vortices and of the circulations:

$2\mathcal{H}(w_{1},\dots,w_{n};a_{1},\dots,a_{\texttt{g}},b_{1},\dots,b_{\texttt{g}})=2(\nu,\nu)_{1,{\rm
renorm}}=$ (4.26)
$\left(\begin{array}[]{cccc}\Gamma_{1},&\Gamma_{2}&\ldots&\Gamma_{n}\end{array}\right)\left(\begin{array}[]{cccc}R_{\rm
robin}(w_{1})&G(w_{1},w_{2})&\ldots&G(w_{1},w_{n})\\\ G(w_{2},w_{1})&R_{\rm
robin}(w_{2})&\ldots&G(w_{2},w_{n})\\\ \vdots&\vdots&\ddots&\vdots\\\
G(w_{n},w_{1})&G(w_{n},w_{2})&\dots&R_{\rm
robin}(w_{n})\end{array}\right)\left(\begin{array}[]{c}\Gamma_{1}\\\
\Gamma_{2}\\\ \vdots\\\ \Gamma_{n}\end{array}\right)+$
$+\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\ \end{array}\right)\left(\begin{array}[]{c}A\\\
B\end{array}\right).$

More formally one could consider the complex conjugates $\bar{w}_{j}$ of the
vortex positions as independent variables and write the Hamiltonian as

$\mathcal{H}(w_{1},\dots,w_{n},\bar{w}_{1},\dots,\bar{w}_{n};a_{1},\dots,a_{\texttt{g}},b_{1},\dots,b_{\texttt{g}}).$

##  5 Hamilton’s equations

###  5.1 Phase space and symplectic form

To formulate Hamilton’s equation one needs, besides the Hamiltonian function
itself, a phase space and a symplectic form on it. The phase space will in our
case consist of all possible configurations of the vortices, collisions not
allowed, together with all possible circulations of the flow around the curves
in the homology basis. Thus we take it to be

$\mathcal{F}=\\{(w_{1},\dots,w_{n};a_{1},\dots,a_{\texttt{g}},b_{1},\dots,b_{\texttt{g}}):w_{j}\in
M,w_{k}\neq w_{j}\text{ for }k\neq j\\}.$

Here the $w_{j}$ shall be interpreted as points on $M$, but in most equations
below they will refer to complex coordinates for such points. Compare Remark
3.2.

Assuming that $\Gamma\neq 0$ (recall (4.3)) the symplectic form on
$\mathcal{F}$ is taken to be

$\Omega=\sum_{k=1}^{n}\Gamma_{k}{\rm
vol}(w_{k})-\frac{V}{\Gamma}\sum_{j=1}^{\texttt{g}}da_{j}\wedge db_{j}=$
$=-\frac{1}{2\mathrm{i}}\sum_{k=1}^{n}\Gamma_{k}\lambda(w_{k})^{2}dw_{k}\wedge
d\bar{w}_{k}-\Gamma{V}\sum_{j=1}^{\texttt{g}}\frac{da_{j}}{\Gamma}\wedge\frac{db_{j}}{\Gamma}.$
(5.1)

The last expression makes sense also if $\Gamma=0$, because by (4.16), (4.17)
the factors $da_{j}/\Gamma$ and $db_{j}/\Gamma$ remain finite as $\Gamma\to
0$. Thus in both expressions above, the last term shall simply be removed if
$\Gamma=0$.

Let

$\xi=\sum_{k=1}^{n}(\dot{w}_{k}\frac{\partial}{\partial
w_{k}}+\dot{\bar{w}}_{k}\frac{\partial}{\partial\bar{w}_{k}})+\sum_{j=1}^{\texttt{g}}(\dot{a}_{j}\frac{\partial}{\partial
a_{j}}+\dot{b}_{j}\frac{\partial}{\partial b_{j}})$

denote a generic tangent vector of $\mathcal{F}$ viewed as a derivation. As
for the first term, recall the conventions in Remark 3.3. The Hamilton
equations in general say that

$i(\xi)\Omega=d\mathcal{H}.$ (5.2)

One main issue then is to verify that with our choices of phase space,
symplectic form and Hamiltonian function, the equations (5.2) really produce
the expected vortex dynamics. This will be accomplished in Section 5.2.

To evaluate (5.2) we first compute the left member as

$i(\xi)\Omega=-\frac{1}{2\mathrm{i}}\sum_{k=1}^{n}\Gamma_{k}\lambda(w_{k})^{2}\big{(}\dot{w}_{k}d\bar{w}_{k}-\dot{\bar{w}}_{k}d{w}_{k}\big{)}-\frac{V}{\Gamma}\sum_{j=1}^{\texttt{g}}(\dot{a}_{j}db_{j}-\dot{b}_{j}da_{j}).$
(5.3)

Explicitly (5.2) therefore says that

$\Gamma_{k}\lambda(w_{k})^{2}{\dot{w}_{k}}=-2\mathrm{i}\frac{\partial\mathcal{H}}{\partial\bar{w}_{k}},$
(5.4)
$\dot{a}_{j}=-\frac{\Gamma}{V}\frac{\partial\mathcal{H}}{\partial{b}_{j}},\quad\dot{b}_{j}=+\frac{\Gamma}{V}\frac{\partial\mathcal{H}}{\partial{a}_{j}}.$
(5.5)

Here the partial derivatives in the right members are implicit in

$d\mathcal{H}=\sum_{k=1}^{n}\frac{\Gamma_{k}^{2}}{2}\big{(}\frac{\partial
R_{\rm robin}(w_{k})}{\partial w_{k}}dw_{k}+\frac{\partial R_{\rm
robin}(w_{k})}{\partial\bar{w}_{k}}d\bar{w}_{k}\big{)}+$ (5.6)
$+\sum_{k,j=1,k\neq j}^{n}\frac{\Gamma_{k}\Gamma_{j}}{2}\big{(}\frac{\partial
G(w_{k},w_{j})}{\partial{w}_{k}}d{w}_{k}+\frac{\partial
G(w_{k},w_{j})}{\partial\bar{w}_{k}}d\bar{w}_{k}\big{)}+$ $+\sum_{k,j=1,k\neq
j}^{n}\frac{\Gamma_{k}\Gamma_{j}}{2}\big{(}\frac{\partial
G(w_{k},w_{j})}{\partial{w}_{j}}d{w}_{j}+\frac{\partial
G(w_{k},w_{j})}{\partial\bar{w}_{j}}d\bar{w}_{j}\big{)}+$
$+\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\
\end{array}\right)\left(\begin{array}[]{c}\sum_{k=1}^{n}\Gamma_{k}dU_{\alpha}(w_{k})\\\
\\\ \sum_{k=1}^{n}\Gamma_{k}dU_{\beta}(w_{k})\end{array}\right)+$
$+\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\ \end{array}\right)\left(\begin{array}[]{c}da\\\
db\end{array}\right).$

Recall here the expressions (4.23), (4.24) for $A$, $B$ in terms of the phase
space variables $a$, $b$ (as column matrices) and $w_{1},\dots,w_{n}$,
$\bar{w}_{1},\dots,\bar{w}_{n}$.

In the partial derivatives of $R_{\rm robin}$ we single out the two affine
connections

$r_{\rm metric}(w)=2\frac{\partial}{\partial w}\log\lambda(w),\qquad\qquad$
(5.7) $r_{\rm robin}(w)\ =-2h_{1}(w)=-2\frac{\partial}{\partial w}h_{0}(w),$
(5.8)

(see (11.3) for the last equality). Recalling (4.13) we then have the
following alternative expressions for (ingredients of) the first term in
$d\mathcal{H}$:

$\frac{\partial R_{\rm robin}(w_{k})}{\partial
w_{k}}=\frac{1}{4\pi}\big{(}r_{\rm metric}(w_{k})-r_{\rm
robin}(w_{k})\big{)}=$ (5.9)
$=\frac{1}{2\pi}\big{(}{h_{1}(w_{k})}+\frac{\partial}{\partial{w}_{k}}\log\lambda(w_{k})\big{)}.$

Similarly for the conjugated quantities.

###  5.2 Dynamical equations

The following theorem now makes the Hamilton equations (5.2) explicit in the
vortex dynamics case.

######  Theorem 5.1.

The dynamical equations for the vortices $w_{k}$ and the circulations $a_{j}$,
$b_{j}$ are, in matrix notation,

$\displaystyle\lambda(w_{k})^{2}{\dot{w}_{k}}=$
$\displaystyle\frac{\Gamma_{k}}{2\pi\mathrm{i}}\big{(}\overline{h_{1}(w_{k})}+\frac{\partial}{\partial\bar{w}_{k}}\log\lambda(w_{k})\big{)}-{2\mathrm{i}}\sum_{j=1,j\neq
k}^{n}\Gamma_{j}\frac{\partial G(w_{k},w_{j})}{\partial\bar{w}_{k}}-$
$\displaystyle+2\left(\begin{array}[]{cc}B^{T},&-A^{T}\end{array}\right)\left(\begin{array}[]{c}{\partial
U_{\alpha}(w_{k})}/{\partial\bar{w}_{k}}\\\ {\partial
U_{\beta}(w_{k})}/{\partial\bar{w}_{k}}\end{array}\right),$
$\displaystyle\left(\begin{array}[]{c}\dot{a}\\\ \dot{b}\end{array}\right)=$
$\displaystyle\frac{\Gamma}{V}\left(\begin{array}[]{cc}-R^{T}&-Q\\\ P&R\\\
\end{array}\right)\left(\begin{array}[]{c}A\\\ B\end{array}\right).$

Recall the column vectors (4.21), (4.22):

$A=a+\oint_{\alpha}*dG^{\omega},\quad B=b+\oint_{\beta}*dG^{\omega}.$

###### Proof.

The equations are obtained by identifying the expression (5.3) for
$i(\xi)\Omega$ with the expression (5.6) for $d\mathcal{H}$. Thus, in (5.3),
the $k$:th term in the first sum,

$-\frac{1}{2\mathrm{i}}\Gamma_{k}\lambda(w_{k})^{2}\big{(}\dot{w}_{k}d\bar{w}_{k}-\dot{\bar{w}}_{k}d{w}_{k}\big{)},$
(5.10)

is to be identified with the corresponding parts in the right member of (5.6),
namely the first three terms. Together with (5.9) this gives immediately the
first two terms in the equation for $\dot{w}_{k}$.

The third term comes from the term

$\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\
\end{array}\right)\left(\begin{array}[]{c}\sum_{k=1}^{n}\Gamma_{k}dU_{\alpha}(w_{k})\\\
\\\ \sum_{k=1}^{n}\Gamma_{k}dU_{\beta}(w_{k})\end{array}\right)$

in equation (5.6). On using (4.19) this can be rewritten as

$\left(\begin{array}[]{cc}B^{T},&-A^{T}\end{array}\right)\left(\begin{array}[]{cc}R^{T}&Q\\\
-P&-R\\\
\end{array}\right)\left(\begin{array}[]{c}\sum_{k=1}^{n}\Gamma_{k}dU_{\alpha}(w_{k})\\\
\\\ \sum_{k=1}^{n}\Gamma_{k}dU_{\beta}(w_{k})\end{array}\right)=$
$=\left(\begin{array}[]{cc}B^{T},&-A^{T}\end{array}\right)\left(\begin{array}[]{c}\sum_{k=1}^{n}\Gamma_{k}*dU_{\alpha}(w_{k})\\\
\\\ \sum_{k=1}^{n}\Gamma_{k}*dU_{\beta}(w_{k})\end{array}\right).$

Identifying here the coefficient for $d\bar{w}_{k}$ with the corresponding
coefficient in (5.10) gives

$\lambda(w_{k})^{2}\dot{w}_{k}=2\left(\begin{array}[]{cc}B^{T},&-A^{T}\end{array}\right)\left(\begin{array}[]{c}\frac{\partial
U_{\alpha}(w_{k})}{\partial\bar{w}_{k}}\\\ \frac{\partial
U_{\beta}(w_{k})}{\partial\bar{w}_{k}}\end{array}\right),$

as desired.

Finally, the term

$\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\ \end{array}\right)\left(\begin{array}[]{c}da\\\
db\end{array}\right).$

in (5.6) is to be identified with

$-\frac{V}{\Gamma}\left(\begin{array}[]{cc}-\dot{b},&\dot{a}\\\
\end{array}\right)\left(\begin{array}[]{cc}da\\\ db\\\ \end{array}\right)$

in (5.3). This gives

$\left(\begin{array}[]{c}\dot{a}\\\
\dot{b}\end{array}\right)=\frac{\Gamma}{V}\left(\begin{array}[]{cc}-R^{T}&-Q\\\
P&R\\\ \end{array}\right)\left(\begin{array}[]{c}A\\\ B\end{array}\right),$

as desired. ∎

Besides the formal proof of the theorem it is essential to show that the
dynamics given in Theorem 5.1 really is the “expected dynamics”. For the
vortices this expected dynamics is that each individual vortex, say $w_{k}$,
moves with the speed of total flow, namely the $1$-form
$\nu=\eta-*dG^{\omega}$ (see (4.1)) converted into a vector, however with the
modification that the contribution to $\nu$ from $w_{k}$ itself shall be
regularized according to standard procedures involving the Robin function.

In the notation of Remark 3.3, the velocity ${\bf V}$ of the $k$:th vortex
$w_{k}$ corresponds, with $z=w_{k}$, to the covector
$\frac{\lambda^{2}}{2}(\dot{\bar{z}}dz+\dot{{z}}d\bar{z})$, see equations
(3.11), (3.12). Thus in the notation of the Theorem 5.1 the expected dynamics
is expressed by the equation

$\eta-\\{*dG^{\omega}\\}_{\rm
renormalized}=\frac{1}{2}\lambda(w_{k})^{2}\big{(}\dot{\bar{w}}_{k}dw_{k}+{\dot{w}_{k}}d\bar{w}_{k}\big{)},$

the left member to be evaluated at $w_{k}$. Here the second term in the left
member is the well-known [37, 3] contribution from the Green function, where
(referring to (5.9), (4.5))

$\\{*dG^{\omega}\\}_{\rm renormalized}(z)=\Gamma_{k}*dR_{\rm
robin}(z)+\sum_{j=1,j\neq k}^{n}\Gamma_{j}*dG(z,w_{j}),$

evaluated at $z=w_{k}$ and the Hodge star taken with respect to $z$. This
gives

$-\\{*dG^{\omega}\\}_{\rm
renormalized}(w_{k})=\mathrm{i}\Gamma_{k}\Big{(}\frac{\partial R_{\rm
robin}(w_{k})}{\partial w_{k}}dw_{k}-\frac{\partial R_{\rm
robin}(w_{k})}{\partial\bar{w}_{k}}d\bar{w}_{k}\Big{)}+$
$+{\mathrm{i}}\sum_{j=1,j\neq k}^{n}\Gamma_{j}\Big{(}\frac{\partial
G(w_{k},w_{j})}{\partial{w}_{k}}dw_{k}-\frac{\partial
G(w_{k},w_{j})}{\partial\bar{w}_{k}}d\bar{w}_{k}\Big{)}$

The above accounts for the first two terms in the right member of Theorem 5.1.
The third term is the contribution from $\eta$, and can by (4.25) be directly
identified as $2\eta_{\bar{z}}$ evaluated at $z=w_{k}$.

Finally we verify that the periods change according to (4.16), (4.17). By
(4.25), (4.19) we have

$*\eta=\left(\begin{array}[]{cc}B^{T},&-A^{T}\\\
\end{array}\right)\left(\begin{array}[]{cc}*dU_{\alpha}\\\ dU_{\beta}\\\
\end{array}\right)=\left(\begin{array}[]{cc}B^{T},&-A^{T}\\\
\end{array}\right)\left(\begin{array}[]{cc}R^{T}&Q\\\ -P&-R\\\
\end{array}\right)\left(\begin{array}[]{cc}dU_{\alpha}\\\ dU_{\beta}\\\
\end{array}\right)=$
$=\left(\begin{array}[]{cc}(dU_{\alpha})^{T},&(dU_{\beta})^{T}\\\
\end{array}\right)\left(\begin{array}[]{cc}P&R\\\ R^{T}&Q\\\
\end{array}\right)\left(\begin{array}[]{cc}A\\\ B\\\ \end{array}\right).$

By (3.18), (3.19) integration of this gives

$\left(\begin{array}[]{cc}\oint_{\alpha}*\eta\\\ \oint_{\beta}*\eta\\\
\end{array}\right)=\left(\begin{array}[]{cc}-R^{T}&-Q\\\
P&R\end{array}\right)\left(\begin{array}[]{cc}A\\\ B\\\ \end{array}\right).$

From this we see that the last equation in the theorem is in exact agreement
with (4.16), (4.17), as claimed.

##  6 Motion of a single point vortex

In the case of a single vortex Theorem 5.1 simplifies a little. We may then
denote the vortex point $w_{1}$ simply by $w$, and the strength $\Gamma_{1}$
agrees with the total vorticity $\Gamma$ for the point vortices. If in
addition $\texttt{g}=0$ then everything simplifies considerable. There is only
one free variable, the location $w\in M$ for the vortex, and the dynamical
equation for this is

$\lambda(w)^{2}\frac{dw}{dt}=\frac{\Gamma}{2\pi\mathrm{i}}\big{(}\overline{h_{1}(w)}+\frac{\partial}{\partial\bar{w}}\log\lambda(w)\big{)}.$

The Hamiltonian and the symplectic form are

$\mathcal{H}(w)=\Gamma^{2}R_{\rm
robin}(w)=\frac{\Gamma^{2}}{2\pi}(h_{0}(w)+\log\lambda(w)),$
$\Omega=\Gamma\,{\rm
vol}(w)=-\frac{1}{2\mathrm{i}}\Gamma\lambda(w)^{2}dw\wedge d\bar{w}.$

It follows that if (and only if) the two metrics

$ds_{\rm metric}^{2}=\lambda(w)^{2}|dw|^{2}=\frac{2V}{\pi}h_{11}(w)|dw|^{2},$
(6.1) $ds_{\rm robin}^{2}\ =e^{-2h_{0}(w)}|dw|^{2}\qquad\qquad\qquad\quad$
(6.2)

are identical, up to a constant factor, then the vortex will never move,
whatever its initial position is. In [3, 19] this is referred to as $ds_{\rm
metric}$ being a “steady vortex metric”, or being “hydrodynamically neutral”.

######  Example 6.1.

An obvious example is a homogenous sphere. Indeed, for a sphere of radius one
we have, in coordinates obtained by stereographic projection into the complex
plane, well-known formulas such as

$G(z,w)=-\frac{1}{4\pi}\Big{(}\log\frac{|z-w|^{2}}{(1+|z|^{2})(1+|w|^{2})}+1\Big{)},$
$h_{0}(w)=\log(1+|w|^{2})-\frac{1}{2},\quad
h_{1}(w)=\frac{\bar{w}}{1+|w|^{2}},\quad
h_{2}(w)=\frac{\bar{w}^{2}}{2(1+|w|^{2})^{2}},$
$e^{-2h_{0}(w)}=\frac{e}{(1+|w|^{2})^{2}},\quad\lambda(w)^{2}=8h_{11}(w)=\frac{4}{(1+|w|^{2})^{2}}.$

The Hamiltonian function is constant,

$\mathcal{H}(w)=\frac{\Gamma^{2}}{2\pi}\big{(}\log(1+|w|^{2})-\frac{1}{2}+\log\frac{2}{1+|w|^{2}}\big{)}=\frac{\Gamma^{2}}{4\pi}(2\log
2-1),$

and there is no motion of the vortex.

When $\texttt{g}>0$ one has to include also the circulating flows in the
picture, so that the dynamical equations become, by Theorem 5.1,

$\displaystyle\lambda(w)^{2}\dot{w}=$
$\displaystyle\frac{\Gamma}{2\pi\mathrm{i}}\big{(}\overline{h_{1}(w)}+\frac{\partial}{\partial\bar{w}}\log\lambda(w)\big{)}+2\left(\begin{array}[]{cc}B^{T},&-A^{T}\end{array}\right)\left(\begin{array}[]{c}{\partial
U_{\alpha}(w)}/{\partial\bar{w}}\\\ {\partial
U_{\beta}(w)}/{\partial\bar{w}}\end{array}\right),$
$\displaystyle\left(\begin{array}[]{c}\dot{a}\\\ \dot{b}\end{array}\right)=$
$\displaystyle\frac{\Gamma}{V}\left(\begin{array}[]{cc}-R^{T}&-Q\\\ P&R\\\
\end{array}\right)\left(\begin{array}[]{c}A\\\ B\end{array}\right).$

##  7 Motion of a vortex pair in the dipole limit

For a vortex pair $\\{w_{1},w_{2}\\}$ with $\Gamma_{1}=-\Gamma_{2}$ we have
$\Gamma=\Gamma_{1}+\Gamma_{2}=0$, hence there is no compensating background
vorticity. The circulations $a$ and $b$ will be time independent by the last
equation in Theorem 5.1 and are not needed in phase space, which then simply
becomes

$\mathcal{F}=\\{(w_{1},w_{2}):w_{1},w_{2}\in M,w_{1}\neq w_{2}\\},$

with symplectic form

$\Omega=-\frac{1}{2\mathrm{i}}\Gamma_{1}\big{(}\lambda(w_{1})^{2}dw_{1}\wedge
d\bar{w}_{1}-\lambda(w_{2})^{2}dw_{2}\wedge d\bar{w}_{2}\big{)}.$ (7.1)

The Hamiltonian is the same quantity as before, see (4.26), but it may now be
considered as a function only of $w_{1}$ and $w_{2}$. The circulations $a$,
$b$ are fixed parameters, given in advance.

However the period vectors $A$ and $B$ still depend on time via $w_{1}$,
$w_{2}$. Indeed, using (3.20), (3.21) we have

$A=a+\Gamma_{1}\oint_{\alpha}\big{(}*dG(\cdot,w_{1})-*dG(\cdot,w_{2})\big{)}=a+\Gamma_{1}\int_{w_{2}}^{w_{1}}dU_{\alpha},$
$B=b+\Gamma_{1}\oint_{\beta}\big{(}*dG(\cdot,w_{1})-*dG(\cdot,w_{2})\big{)}=b+\Gamma_{1}\int_{w_{2}}^{w_{1}}dU_{\beta_{j}}.$

The Hamiltonian is

$2\mathcal{H}(w_{1},w_{2})=\left(\begin{array}[]{cc}\Gamma_{1},&-\Gamma_{1}\end{array}\right)\left(\begin{array}[]{cc}R_{\rm
robin}(w_{1})&G(w_{1},w_{2})\\\ G(w_{2},w_{1})&R_{\rm robin}(w_{2})\\\
\end{array}\right)\left(\begin{array}[]{c}\Gamma_{1}\\\
-\Gamma_{1}\end{array}\right)+$
$+\left(\begin{array}[]{cc}A^{T},&B^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\ \end{array}\right)\left(\begin{array}[]{c}A\\\
B\end{array}\right),$ (7.2)

and the dynamics of the vortex pair $\\{w_{1},w_{2}\\}$ becomes, by (5.4),

$\Gamma_{1}\lambda(w_{1})^{2}\,\dot{w}_{1}=-2\mathrm{i}\,\frac{\partial\mathcal{H}(w_{1},w_{2})}{\partial\bar{w}_{1}},$
$\Gamma_{1}\lambda(w_{2})^{2}\,\dot{w}_{2}=+2\mathrm{i}\,\frac{\partial\mathcal{H}(w_{1},w_{2})}{\partial\bar{w}_{2}}.$

In place of $w_{1}$ and $w_{2}$ we may turn to $w=\frac{1}{2}(w_{1}+w_{2})$
and $u=\frac{1}{2}(w_{1}-w_{2})$ as coordinates. These are similar to the
“center-arrow coordinates” used in [3, 38]. Then

$\begin{cases}w_{1}=w+u,\\\ w_{2}=w-u.\end{cases}$ (7.3)

We are interested in the limit $u\to 0$, and Taylor expansion of
$H(w_{1},w_{2})=H(w+u,w-u)$ with respect to $u$, $\bar{u}$ gives, using
relations in Appendix 2,

$H(w+u,w-u)=h_{0}(w)-\operatorname{Re}\big{(}(4h_{2}(w)-\frac{\partial
h_{1}(w)}{\partial w})u^{2}\big{)}+$ $+(4h_{11}(w)-\frac{\partial
h_{1}(w)}{\partial\bar{w}})|u|^{2}+\mathcal{O}(|u|^{3}).$ (7.4)

The linear terms vanish because of the symmetry of $H(w_{1},w_{2})$. The
second order terms will only be used briefly in Section 8, and even the
constant term $h_{0}(w)$ will eventually disappear below. In addition to (7.4)
we have the Taylor expansions

$h_{0}(w\pm
u)=h_{0}(w)\pm\big{(}h_{1}(w)u+\overline{{h}_{1}(w)u}\big{)}+\mathcal{O}(|u|^{2}),$
$\log\lambda(w\pm
u)=\log\lambda(w)\pm\frac{1}{2}\big{(}r(w)u+\overline{r(w)u}\big{)}+\mathcal{O}(|u|^{2})$

(coupled signs throughout). The latter equation uses the affine connection
$r(w)=r_{\rm metric}(w)$, see (5.7) or (10.1). For later use we record also
the expansion

$\lambda(w\pm
u)^{2}=\lambda(w)^{2}\Big{(}1\pm\big{(}r(w)u+\overline{r(w)}\bar{u}\big{)}+\mathcal{O}(|u|^{2})\Big{)}.$
(7.5)

Using these expansions we obtain, for the first matrix in the Hamiltonian
(7.2),

$2\pi\left(\begin{array}[]{cc}R_{\rm robin}(w_{1})&G(w_{1},w_{2})\\\
G(w_{2},w_{1})&R_{\rm robin}(w_{2})\\\ \end{array}\right)=$ (7.6)
$=\left(\begin{array}[]{cc}h_{0}(w+u)+\log\lambda(w+u)&-\log|2u|+H(w+u,w-u)\\\
-\log|2u|+H(w+u,w-u)&h_{0}(w-u)+\log\lambda(w-u)\\\ \end{array}\right)=$
$=\left(\begin{array}[]{cc}\log\lambda(w)&-\log|2u|\\\
-\log|2u|&\log\lambda(w)\\\
\end{array}\right)+\left(\begin{array}[]{cc}h_{0}(w)&h_{0}(w)\\\
h_{0}(w)&h_{0}(w)\\\ \end{array}\right)+$
$+\left(\begin{array}[]{cc}h_{1}(w)+\frac{1}{2}r(w)&0\\\
0&-h_{1}(w)-\frac{1}{2}r(w)\\\
\end{array}\right)u+\left(\begin{array}[]{cc}\overline{{h}_{1}(w)}+\frac{1}{2}\overline{r(w)}&0\\\
0&-\overline{{h}_{1}(w)}-\frac{1}{2}\overline{r(w)}\\\
\end{array}\right)\bar{u}+$ $+\mathcal{O}(|u|^{2}).$

When acting with $(\Gamma_{1},-\Gamma_{1})$ on both sides of the matrix (7.6),
the last three terms in the final expression disappear and the result becomes,
up to $\mathcal{O}(|u|^{2})$,

$\left(\begin{array}[]{cc}\Gamma_{1}&-\Gamma_{1}\par\end{array}\right)\left(\begin{array}[]{cc}\log\lambda(w)&-\log|2u|\\\
-\log|2u|&\log\lambda(w)\\\
\end{array}\right)\left(\begin{array}[]{cc}\Gamma_{1}\\\ -\Gamma_{1}\\\
\end{array}\right)=$ (7.7)
$=2\Gamma_{1}^{2}\big{(}\log\lambda(w)+\log|2u|\big{)}.$

The full Hamiltonian (7.2) therefore becomes, up to terms of order
$\mathcal{O}(|u|^{2})$,

$2\mathcal{H}(w+u,w-u)=\frac{\Gamma_{1}^{2}}{\pi}\big{(}\log\lambda(w)+\log|2u|\big{)}+$
$+\left(\begin{array}[]{cc}(a+\Gamma_{1}\int_{w-u}^{w+u}dU_{\alpha})^{T},&(b+\Gamma_{1}\int_{w-u}^{w+u}dU_{\beta})^{T}\end{array}\right)\left(\begin{array}[]{cc}P&R\\\
R^{T}&Q\\\
\end{array}\right)\left(\begin{array}[]{c}a+\Gamma_{1}\int_{w-u}^{w+u}dU_{\alpha}\\\
b+\Gamma_{1}\int_{w-u}^{w+u}dU_{\beta}\end{array}\right).$

Here one can see that the final term remains bounded as $u\to 0$, hence is
negligible in this limit compared to the first term. Therefore the leading
terms in this limit are given by

$\mathcal{H}(w+u,w-u)=\frac{\Gamma_{1}^{2}}{2\pi}\big{(}\log\lambda(w)+\log|2u|\big{)}+\mathcal{O}(1)\quad(u\to
0).$ (7.8)

This is essentially a constant factor times log of the distance (in the
metric) between $w_{1}=w+u$ and $w_{2}=w-u$. Indeed, we recover the simple and
beautiful formula

$\mathcal{H}(w_{1},w_{2})=\frac{\Gamma_{1}^{2}}{2\pi}\log{\rm
dist\,}(w_{1},w_{2})+\mathcal{O}(1)\quad(|w_{1}-w_{2}|\to 0)$

of Boatto and Koiller. See equations (24), (25) in [3]. Compare also [19]. The
distance is taken with respect to the Riemannian metric. The error term
$\mathcal{O}(1)$ can be identified with a what is called the “Batman function”
in [3]. In our notations the latter is given by (10.3).

Taking the differential of (7.8) gives, on using again the metric affine
connection $r(w)=r_{\rm metric}(w)$ defined by (5.7),

$d\mathcal{H}=\frac{\partial\mathcal{H}}{\partial
w}dw+\frac{\partial\mathcal{H}}{\partial\bar{w}}d\bar{w}+\frac{\partial\mathcal{H}}{\partial{u}}du+\frac{\partial\mathcal{H}}{\partial\bar{u}}d\bar{u}=$
$=\frac{\Gamma_{1}^{2}}{4\pi}\Big{(}r(w)dw+\overline{r(w)}d\bar{w}+\frac{du}{u}+\frac{d\bar{u}}{\bar{u}}\Big{)}.$
(7.9)

Recalling (7.5) we can expand the symplectic $2$-form given by (7.1) in terms
of $w$ and $u$ as

$\Omega=-\frac{\Gamma_{1}}{2\mathrm{i}}\big{(}\lambda(w+u)^{2}d(w+u)\wedge
d(\bar{w}+\bar{u})-\lambda(w-u)^{2}d(w-u)\wedge d(\bar{w}-\bar{u})=$
$=\mathrm{i}{\Gamma_{1}}\lambda(w)^{2}\Big{(}dw\wedge d\bar{u}-d\bar{w}\wedge
du+\big{(}r(w)u+\overline{r(w)u}\big{)}\big{(}dw\wedge d\bar{w}+du\wedge
d\bar{u}\big{)}\Big{)}+\mathcal{O}(|u|^{2}).$

With

$\xi=\dot{w}\frac{\partial}{\partial
w}+\dot{\bar{w}}\frac{\partial}{\partial\bar{w}}+\dot{u}\frac{\partial}{\partial
u}+\dot{\bar{u}}\frac{\partial}{\partial\bar{u}}$

this gives, as $u\to 0$,

$i(\xi)\Omega=\mathrm{i}\Gamma_{1}\lambda(w)^{2}\Big{(}-\big{(}\dot{\bar{u}}+(r(w)u+\overline{r(w)u})\dot{\bar{w}}\big{)}dw+\big{(}\dot{u}+(r(w)u+\overline{r(w)u})\dot{w}\big{)}d\bar{w}-$
$-\big{(}\dot{\bar{w}}+(r(w)u+\overline{r(w)u})\dot{\bar{u}}\big{)}du+\big{(}\dot{w}+(r(w)u+\overline{r(w)u}\big{)}\dot{u})d\bar{u}+\mathcal{O}(|u|^{2}\Big{)}.$

Comparing with (7.9) we see that the dynamics of the vortex pair is described
by the two equations

$\frac{\Gamma_{1}}{4\pi\mathrm{i}}\overline{r(w)}=\lambda(w)^{2}\big{(}\dot{{u}}+(r(w)u+\overline{r(w)u})\dot{{w}}\big{)}+\mathcal{O}(|u|^{2}),$
(7.10)
$\frac{\Gamma_{1}}{4\pi\mathrm{i}}\frac{1}{\bar{u}}=\lambda(w)^{2}\big{(}\dot{{w}}+(r(w)u+\overline{r(w)u})\dot{{u}}\big{)}+\mathcal{O}(|u|^{2}).$
(7.11)

Equation (7.11) can used to eliminate $\dot{w}$ in (7.10), which then becomes

$\frac{\Gamma_{1}}{4\pi\mathrm{i}}\overline{r(w)}=\lambda(w)^{2}\dot{u}+\big{(}r(w)u+\overline{r(w)u}\big{)}\Big{(}\frac{\Gamma_{1}}{4\pi\mathrm{i}\bar{u}}-\lambda(w)^{2}\big{(}r(w)u+\overline{r(w)u}\big{)}\dot{u}\Big{)}+\mathcal{O}(|u|^{2}).$

Here the left member cancels with one of the terms in the right member, and
the rest can be written, after division by $u$,

$0=\lambda(w)^{2}\cdot\frac{\dot{u}}{u}+r(w)\frac{\Gamma_{1}}{4\pi\mathrm{i}\bar{u}}-\lambda(w)^{2}\big{(}r(w)u+\overline{r(w)u}\big{)}^{2}\cdot\frac{\dot{u}}{u}+\mathcal{O}(|u|).$

In this equation the third term in the right member is of a smaller magnitude
than the other two terms and can be incorporated in the final
$\mathcal{O}(|u|)$. Thus we arrive at

$\lambda(w)^{2}\frac{d}{dt}\log
u+r(w)\frac{\Gamma_{1}}{4\pi\mathrm{i}\bar{u}}=0,$ (7.12)

valid with an error of at most $\mathcal{O}(|u|)$ as $u\to 0$.

The above equation, (7.12), essentially comes from (7.10), and it is to be
combined again with (7.11). For the latter it is enough to use the simplified
form

$\frac{\Gamma_{1}}{4\pi\mathrm{i}\bar{u}}=\lambda(w)^{2}\frac{{dw}}{dt},$
(7.13)

which only uses the leading terms, and for which the error is still at most
$\mathcal{O}(|u|)$. Inserting (7.13) in (7.12) results in the master equation

$\frac{d}{dt}\log u+r(w)\frac{dw}{dt}=0.$ (7.14)

One problem with (7.14) is that the speed $dw/dt$ becomes infinite, along with
the first term, in the dipole limit. However, this problem only affects the
real part of the equation. For the imaginary part one can replace true time
$t$ by an arbitrary parameter, which is scaled with $u$ so that $dw/dt$
remains finite as $|u|\to 0$. Alternatively one may scale $\Gamma_{1}$ with
$u$ so that the left member of (7.13) remains finite. Then one can still think
of any new parameter $t$ as a time variable. In any case, we take imaginary
parts of (7.14) and obtain

$\frac{d}{dt}\arg u+\operatorname{Im}(r(w)\frac{d{w}}{dt})=0.$ (7.15)

Equation (7.13) shows that the directions of $u$ and $dw/dt$ are related as

$\arg u=\arg\frac{dw}{dt}\pm\frac{\pi}{2},$ (7.16)

where the plus sign is to be chosen if $\Gamma_{1}>0$, the minus sign if
$\Gamma_{1}<0$. Now (7.15) and (7.16) taken together give the final law for
the motion of the center $w$ of the vortex pair in the dipole limit:

$\frac{d}{dt}\arg\frac{dw}{dt}+\operatorname{Im}(r(w)\frac{d{w}}{dt})=0.$
(7.17)

As explained in Appendix 1, Section 10 (see in particular equation (10.5)
there), (7.19) is exactly the equation for a geodesic curve when expressed in
an arbitrary parameter $t$. One may notice that (7.15) (and similarly for
(7.17)) can be written in the parameter-free form

${d}\arg u+\operatorname{Im}(r(w){d{w}})=0,$

confirming again the fact that the geometry of the dipole trajectory has a
meaning independent of any choice of parameter.

The real part of (7.14) says, in view of (10.1) (or (5.7)), that

$\frac{d}{dt}\big{(}\log|u|+\log\lambda(w)\big{)}=0,$

the error term $\mathcal{O}(|u|)$ being disregarded. In other words that
${|u|}{\lambda(w)}=C$ (constant) along each trajectory. By (7.13) this also
gives

$\Big{|}\frac{dw}{dt}\Big{|}=\frac{C}{\lambda(w)}.$ (7.18)

Thus along each trajectory $\lambda(w)$ has, being proportional to one over
the velocity, the same role as the refraction index in optics.

We summarize the most essential parts of the above discussion as

######  Theorem 7.1.

The dynamical equations for a vortex pair in the dipole limit is identical
with the geodesic equation for the metric $ds=\lambda(w)|dw|$, namely

$\frac{d}{dt}\arg\frac{dw}{dt}+\operatorname{Im}(r(w)\frac{d{w}}{dt})=0.$
(7.19)

Here $w=w(t)$ is the location of the dipole, $t$ is an arbitrarily scaled time
parameter chosen such that $dw/dt$ is finite. The orientation of the dipole is
perpendicular to $dw/dt$.

######  Remark 7.1.

It is possible to understand why dipole move along geodesics by thinking of
vortex pair as a “wave front”, in an optical analogy. Equation (7.16) says
that the motion is perpendicular to the wave front (the line segment from
$w-u$ to $w+u$). Equation (7.15) then expresses that if the front of a vortex
pair is not aligned with the level line of $\lambda(w)$ then the direction of
$u$ (representing the wave front) changes in such a way that the curve $w(t)$
bends towards higher values of $\lambda$.

Being slightly more direct and exact, on taking $t$ to be Euclidean arc length
the first term in (7.19) is the ordinary curvature for the curve traced out by
$w(t)$. The second term can be viewed as the inner product between the
gradient of $\lambda(w)$, which can be identified with $\overline{r(w)}$, and
$dw/dt$ rotated $90$ degrees to the right. Letting $\theta$ denote the angle
between the gradient of $\lambda(w)$ and the velocity vector $dw/dt$ we can
write

$\operatorname{Im}(r(w)\frac{d{w}}{dt})=|\overline{r(w)}||\frac{dw}{dt}|\sin{\theta}.$

The above remarks are compatible with the laws of optics, for example Fermat’s
law, and also with “Snell’s law” (see for example [20, 6]) in the somewhat
singular case that $\lambda(w)$ jumps between two constant values.

##  8 Vortex dipoles and projective connections

A vortex dipole constructed as a vortex pair melting together is a result of
two regularizations: First, a single vortex is already regularized in itself
by removal of a singularity of type $\log|z-w|$ in the stream function,
together with a possible adjustment with a term related to the curvature of
the surface. This is not so severe, and everyone agrees on the result. The
motion of the vortex then comes from the regular terms in the Taylor
expansion, and it is easy to handle. In particular the speed of the vortex is
finite.

When the two vortices in a vortex pair approach each, then a second
regularization becomes necessary. If one does not want the two vortices to
just annihilate each other in the limit, then one has to let the strengths of
the vortices tend to infinity. Even without that, the speed of the vortex pair
will be infinite in the limit, so with the vortex strengths blowing up the
speed will become infinite to an even higher degree. But in any case one can
speak of a residual trajectory, which will be a geodesic for the metric of the
manifold, as was conjectured in [28], proved in [3], and further clarified in
Theorem 7.1 above.

There is also the possibility of treating the flow directly as a dipole flow,
not going via vortex pairs, and this touches on the theory of projective (or
Schwarzian) connections. Below we shall sketch upon such a procedure, but it
seems difficult to read off the trajectory of the dipole from this approach.

We start from the expression, using (4.25),

$\nu=\eta-*dG^{\omega}=\left(\begin{array}[]{cc}B^{T},&-A^{T}\\\
\end{array}\right)\left(\begin{array}[]{cc}dU_{\alpha}\\\ dU_{\beta}\\\
\end{array}\right)-\Gamma*dG(\cdot,w)$
$=\left(\begin{array}[]{cc}b^{T}+\oint_{\beta}*dG(\cdot,w),&-a^{T}-\oint_{\alpha}*dG(\cdot,w)\\\
\end{array}\right)\left(\begin{array}[]{cc}dU_{\alpha}\\\ dU_{\beta}\\\
\end{array}\right)-\Gamma*dG(\cdot,w)$

for the flow $1$-form $\nu$ in the single vortex case, and we shall to pass to
the dipole limit by differentiation with respect to $w$. Recall that $\nu$ is
a real-valued form: on writing $\nu=\nu_{z}dz+\nu_{\bar{z}}d\bar{z}$ we have
$\nu_{\bar{z}}=\overline{\nu_{z}}$. When differentiating with respect to $w$
we get a covariant tensor, denoted $d_{w}\nu$, which is real valued in a
similar sense. In $d_{w}\nu$ there will appear two kinds of differentials:
$dz$, $d\bar{z}$ are differentials for the flow itself (as a $1$-form) at a
general point $z$. The dipole is located at $w$ and the differentials $dw$,
$d\bar{w}$ represent its orientation. These two kinds of differentials are
combined only via a tensor product (not an antisymmetric wedge product).

On using (3.13), (3.16), (3.17) one finds that differentiation of $\nu$ with
respect to $w$ gives, writing for simplicity $dzdw$ in place of $dz\otimes dw$
(etc),

$d_{w}\nu(z)=\frac{1}{2}\sum_{j=1}^{\texttt{g}}\big{(}dU_{\beta_{j}}(w)\otimes
dU_{\alpha_{j}}(z)-dU_{\alpha_{j}}(w)\otimes dU_{\beta_{j}}(z)\big{)}+$
$+\mathrm{i}\Gamma\big{(}\frac{\partial^{2}G(z,w)}{\partial z\partial
w}dzdw+\frac{\partial^{2}G(z,w)}{\partial
z\partial\bar{w}}dzd\bar{w}-\frac{\partial^{2}G(z,w)}{\partial\bar{z}\partial{w}}d\bar{z}d{w}-\frac{\partial^{2}G(z,w)}{\partial\bar{z}\partial\bar{w}}d\bar{z}d\bar{w}\big{)}.$

Here the first term vanishes in the limit $z\to w$. In the remaining terms we
single out the poles as separate terms and let $z\to w$ in what remains. This
gives, on using (11.6), (11.7), (11.5) and representing the first term simply
by $\mathcal{O}(|z-w|)$ (at the end),

$d_{w}\nu(z)=-\frac{\Gamma}{4\pi\mathrm{i}}\Big{(}\big{(}\frac{1}{(z-w)^{2}}+2\frac{\partial^{2}H(z,w)}{\partial
z\partial w}\big{)}dzdw+2\frac{\partial^{2}H(z,w)}{\partial
z\partial\bar{w}}dzd\bar{w}-$
$-2\frac{\partial^{2}H(z,w)}{\partial\bar{z}\partial{w}}d\bar{z}d{w}-\big{(}\frac{1}{(\bar{z}-\bar{w})^{2}}+2\frac{\partial^{2}H(z,w)}{\partial\bar{z}\partial\bar{w}}\big{)}d\bar{z}d\bar{w}\Big{)}+\mathcal{O}(|z-w|)=$
$=-\frac{\Gamma}{2\pi}\operatorname{Im}\Big{(}\big{(}\frac{1}{(z-w)^{2}}+\frac{\partial
h_{1}(w)}{\partial w}-2h_{2}(w)\big{)}dzdw-\big{(}\frac{\partial
h_{1}(w)}{\partial\bar{w}}-2h_{11}(w)\big{)})dzd\bar{w}\Big{)}$
$+\mathcal{O}(|z-w|).$

To get the speed of dipole one would like to insert $z=w$ above, but clearly
this does not give any sensible result, just a flow $1$-form in $z$ that
becomes infinite at $z=w$. Still this singular flow is associated with a
certain direction determined by $dw$, or more precisely by the action of $dw$
on a vector m representing the orientation of the dipole. See Example 8.1
below. In any case, this singular term overrules all other terms.

Up to a real factor, we thus have above the singularity

$\operatorname{Im}\frac{dzdw}{(z-w)^{2}}$

plus the finite quantity

$\operatorname{Im}\Big{(}\big{(}\frac{\partial h_{1}(w)}{\partial
w}-2h_{2}(w)\big{)}dzdw-\big{(}\frac{\partial
h_{1}(w)}{\partial\bar{w}}-2h_{11}(w)\big{)}dzd\bar{w}\Big{)}.$ (8.1)

Certainly this finite term will be of minor importance compared with the
singular term. The singular term actually determines the entire flow (at a
given instance) once the orientation of the dipole is given and, for example,
the imaginary parts of the periods are given, say are set to zero. Indeed, on
letting $dw$ act on a vector m representing the orientation, $\langle
dw,{\texttt{m}}\rangle$ becomes a complex number and there remains the
differentials $dz$ and $d\bar{z}$ representing the flow, and this can then be
associated with with a unique meromorphic differential on $M$.

To elaborate the above statements a little, let $\texttt{m}\in{\mathbb{C}}$ be
given. Then there exists a unique meromorphic differential $\mu=\mu_{z}dz$ on
$M$ of the form

$\mu=\frac{{\texttt{m}}\,dz}{(z-w)^{2}}+{\rm regular}$

and having periods

$\operatorname{Im}\oint_{\alpha_{k}}\mu=0,\quad\operatorname{Im}\oint_{\beta_{k}}\mu=0\quad(k=1,\dots,\texttt{g}).$

Such a $\mu$ represents the dipole flow via

$\langle d_{w}\nu,\cdot\otimes{\texttt{m}}\rangle=\operatorname{Im}\mu,$

where the left member means that the differentials $dw$ and $d\bar{w}$ in
$d_{w}\nu$ shall act on m. In practice this just means that $dw$ and
$d\bar{w}$ shall be replaced by m and $\bar{\texttt{m}}$, respectively,
regarded as complex numbers

However, this purely conformal picture (we have not yet used the metric) gives
no information of how the dipole is to move. That must be determined by the
metric, and there seems to be no other reasonable possibility than that it
shall move along the geodesic perpendicular to m. In particular, the regular
part (8.1) seems not to influence the dipole dynamics. What one can say from a
mathematical point of view is that the coefficient in the leading term of
(8.1),

$-\frac{1}{6}q_{\rm robin}(w)=\frac{\partial h_{1}(w)}{\partial w}-2h_{2}(w)$

is a projective connection, up to a factor, see Lemma 11.2. The second term in
(8.1) is directly linked to the two metrics involved: by (6.2), (6.1), (11.5)
we have

$\frac{\partial h_{1}(w)}{\partial\bar{w}}-2h_{11}(w)=\frac{1}{4}\Delta
h_{0}(w)-\frac{\pi}{V}\lambda(w)^{2}.$

This entire quantity transforms as the density of a metric. But it is not
necessarily positive. In the case of a sphere, for example, it vanishes (see
Example 6.1). Also $q_{\rm robin}$ vanishes in the case.

######  Example 8.1.

As an example, let the initial condition be that $w=0$ and
${\texttt{m}}=\frac{\partial}{\partial v}$, where $w=u+\mathrm{i}v$. Then
$\langle dw,{\texttt{m}}\rangle=\mathrm{i}$, and the flow $1$-form from the
singularity becomes

$\operatorname{Im}\frac{dz\,\langle
dw,{\texttt{m}}\rangle}{(z-w)^{2}}=\operatorname{Re}\frac{dz}{z^{2}}=\frac{(x^{2}-y^{2})dx-2xydy}{(x^{2}+y^{2})^{2}}.$

On the $x$-axis ($y=0$) this is ${dx}/{x^{2}}$, in other words a flow along
the axis with a polar singularity of order two, as expected. Indeed, the
entire flow is the well-known dipole picture which appears in many
applications of conformal mapping.

##  9 Remarks on vortex motion in planar domains

Vortex motion in a planar domain can easily be treated as a special case of
vortex motion on Riemann surfaces by turning to the Schottky double of the
planar domain. For simplicity we shall only discuss the case of one single
vortex in the planar domain. The case of several vortices will be similar in
an obvious way. The ideas in this section extend to cases of vortex motion on
general open Riemannian surfaces with analytic boundary.

Let $\Omega\subset{\mathbb{C}}$ be the planar domain, assumed to be bounded by
finitely many real analytic curves. The Schottky double, first described in
[45], is the compact Riemann surface $M=\hat{\Omega}$ obtained by completing
$\Omega$ with a “backside” $\tilde{\Omega}$, having the opposite conformal
structure, and glueing the two along the common boundary. Thus
$\hat{\Omega}=\Omega\cup\partial\Omega\cup\tilde{\Omega}$ in a set theoretic
sense, and the conformal structure becomes smooth over $\partial\Omega$, as
can be seen from well-known reflection principles. If $z$ is a point in
$\Omega\subset M$, then $\tilde{z}$ will denote the corresponding (reflected)
point on $\tilde{\Omega}\subset M$. Both $z$ and $\tilde{z}$ can also be
considered as points in ${\mathbb{C}}$, then serving as coordinates of the
mentioned points in $M$ (holomorphic respectively anti-holomorphic
coordinates), and as such they are the same: $z=\tilde{z}\in{\mathbb{C}}$.

In our case we need also a Riemannian structure, with a metric. This is to be
the Euclidean metric on each of $\Omega$ and $\tilde{\Omega}$, but these do
not fit smoothly across curved parts of $\partial\Omega$, it will only be
Lipschitz continuous. But that is good enough for our purposes because the
vortex will anyway never approach the boundary. (In the case of several
vortices it is however possible to make up situations in which vortices do
reach the boundary).

The metric on $M=\hat{\Omega}$ is thus to be

$ds=\begin{cases}|dz|,\quad z\in\Omega,\\\
|d\tilde{z}|,\quad\tilde{z}\in\tilde{\Omega}.\end{cases}$ (9.1)

To see how this behaves across $\partial\Omega$ we need a holomorphic
coordinate defined in a full neighborhood of this curve in $M$. A natural
candidate can be defined in terms of the Schwarz function $S(z)$ for
$\partial\Omega$, a function which is defined by its properties of being
holomorphic in a neighborhood of $\partial\Omega$ in ${\mathbb{C}}$ and by
satisfying

$S(z)=\bar{z}\quad\text{on }\partial\Omega.$ (9.2)

See [9, 47] for details about $S(z)$. We remark that $z\mapsto\overline{S(z)}$
is the (local) anti-conformal reflection map in $\partial\Omega$ and that
$S^{\prime}(z)=T(z)^{-2}$, where $T(z)$ is the positively oriented and
holomorphically extended unit tangent vector on $\partial\Omega$.

The complex coordinate $z$ in $\Omega$ extends, as a holomorphic function, to
a full neighborhood of $\Omega\cup\partial\Omega$, both when this neighborhood
is considered as a subset of ${\mathbb{C}}$ and when it is considered as a
subset of $M$. The first case is trivial, and the second case depends on
$\partial\Omega$ being analytic. In terms of the Schwarz function this second
extension is given by

$z=\begin{cases}z\quad&\text{for }z\in\Omega\cup\partial\Omega,\\\
\overline{S({\tilde{z}})}\quad&\text{for }\tilde{z}\in\tilde{\Omega},\text{
close to }\partial\Omega.\end{cases}$ (9.3)

In the latter expression, $\overline{S(\tilde{z})}$, $\tilde{z}$ is to be
interpreted as a complex number. When the metric on $M$ is expressed in the
coordinate (9.3) it becomes

$ds=\begin{cases}|dz|\quad&\text{for }z\in\Omega\cup\partial\Omega,\\\
|S^{\prime}({z})||d{z}|\quad&\text{for
}{z}\in{\mathbb{C}}\setminus\overline{\Omega},\text{ close to
}\partial\Omega.\end{cases}$ (9.4)

In the second case, $z=\overline{S(\tilde{z})}$, $\tilde{z}\in\tilde{\Omega}$,
whereby $\tilde{z}=\overline{S(z)}$ and so
$|d\tilde{z}|=|S^{\prime}({z})||d{z}|$. Thus (9.4) is consistent with (9.1).
We see from the coordinate representation (9.4) that the metric is only
Lipschitz continuous across $\partial\Omega$. This is the best one can expect.

The associated affine connection (5.7) (or (10.1)) is in the coordinate (9.3)
given by

$r(z)=\begin{cases}0\quad&\text{for }z\in\Omega\cup\partial\Omega,\\\
\\{S({z}),{z}\\}_{1}\quad&\text{for
}{z}\in{\mathbb{C}}\setminus\overline{\Omega},\text{ close to
}\partial\Omega,\end{cases}$

where (see Appendix 1, Section 10, for notations)

$\\{S(z),z\\}_{1}=\frac{S^{\prime\prime}(z)}{S^{\prime}(z)}=-2\frac{T^{\prime}(z)}{T(z)}.$
(9.5)

Thus $r(z)$ is discontinuous across $\partial\Omega$ and it should on this
curve be represented by its mean-value

$r_{\rm
MV}(z)=\frac{1}{2}\\{S(z),z\\}_{1}=-\frac{T^{\prime}(z)}{T(z)}\quad(z\in\partial\Omega).$
(9.6)

######  Example 9.1.

Let $\Omega={\mathbb{D}}$. Then $S(z)=\frac{1}{z}$ and the coordinate $z$ in
(9.3) extends to the entire complex plane, thus representing all of
$M={\mathbb{D}}\cup\partial{\mathbb{D}}\cup\tilde{{\mathbb{D}}}$ except for
the point $\tilde{0}\in\tilde{{\mathbb{D}}}$. And the metric expressed in this
coordinate becomes

$ds=\begin{cases}|dz|,\quad&|z|\leq 1,\\\
|z|^{-2}{|d{z}|},\quad&|z|>1.\end{cases}$

The affine connection similarly becomes, including the mean-value on the
boundary,

$r(z)=\begin{cases}0,\quad&|z|<1,\\\ -z,\quad&|z|=1,\\\
-2z,\quad&|z|>1.\end{cases}$

The geodesic curves in $\Omega$ are of course the (Euclidean) straight lines
(similarly in $\tilde{\Omega}$), geodesic curves crossing $\partial\Omega$ are
straight lines reflecting into the other side under equal angles on
$\partial\Omega$ (just as ordinary optical reflection), while $\partial\Omega$
is in itself a geodesic curve. The latter is intuitively obvious since at any
point on $\partial\Omega$ there should be one geodesic in the tangential
direction, and this has no other way to go than to follow the boundary.

To confirm the last statement analytically, let $t$ be an arc length (with
respect to the Euclidean metric) parameter along $\partial\Omega$, so that
$T(z)=\frac{dz}{dt}$ on $\partial\Omega$. The curvature $\kappa$ of
$\partial\Omega$ is

$\kappa=\frac{d}{dt}\arg\frac{dz}{dt}\quad z\in\partial\Omega,$

and using that $T(z)\overline{T(z)}=1$ on $\partial\Omega$ one finds that

$T^{\prime}(z)=\mathrm{i}\kappa\quad(z\in\partial\Omega).$

In particular $T^{\prime}(z)$, and so $r_{\rm MV}(z)T(z)$ (by (9.6)), is
purely imaginary on $\partial\Omega$. Combining with (9.6) it follows that

$\frac{d}{dt}\arg\frac{dz}{dt}={\mathrm{i}}r_{\rm MV}(z)T(z),$

hence

$\frac{d}{dt}\arg\frac{dz}{dt}+{\operatorname{Im}}\big{(}r_{\rm
MV}(z)T(z)\big{)}=0\quad(z\in\partial\Omega).$

Thus the geodesic equation (7.19) holds for the curve $\partial\Omega$, as
claimed.

We remark also that the curvature $\kappa$ of the boundary curve
$\partial\Omega$ appears also in the expression for the Gaussian curvature for
the metric on $M$. That curvature vanishes on $\Omega$ and on
$\tilde{\Omega}$, whereas it on $\partial\Omega$ has a singular contribution,
with density $2\kappa$ with respect arc-length measure on $\partial\Omega$.

A single vortex in a planar domain $\Omega$ moves along a level line of the
appropriate Robin function, or Routh’s stream function [34, 35, 36]). If the
vortex is close to the boundary then it follows the boundary closely, with
high speed. From the perspective of the Schottky double the boundary
conditions for planar fluid motion are such that there is automatically a
mirror vortex on the other side in the double, thus we really have a vortex
pair close to $\partial\Omega$ on the double. In the limit this becomes a
vortex dipole, moving with infinite speed along the geodesic $\partial\Omega$.

Considering in some more detail such a symmetric vortex pair, with vortex
locations $w\in\Omega$ and $\tilde{w}\in\tilde{\Omega}$, we first notice that
the Green function $G^{\omega}(z)$ for $\omega=\delta_{w}-\delta_{\tilde{w}}$
simply is the anti-symmetric extension to the Schottky double of the ordinary
Green function $G_{\Omega}(z,w)$ for $\Omega$:

$G^{\delta_{w}-\delta_{\tilde{w}}}(z)=G_{\Omega}(z,w)\quad(z\in\Omega).$

Then the stream function $\psi$ in (4.2) becomes what is sometimes called the
hydrodynamic Green function, which depends on the prescribed periods. This
function, which can be traced back (at least in special cases) to [29, 36],
has more recently been discussed in for example [8, 14, 13, 25].

We wish to clarify how this hydrodynamic Green function is related to the
modification, in the beginning of Section 4, of the general Green function
flow $-*dG^{\omega}$ by an additional term $\eta$. To do this we fix, in the
case of a Schottky double $M=\hat{\Omega}$, the homology basis in such a way
that the curves $\beta_{j}$, $j=1,\dots,\texttt{g}$, closely follow the inner
components of $\partial\Omega$, and each curve $\alpha_{j}$ goes from the
$j$:th inner component of $\partial\Omega$ through $\Omega$ to the outer
component, and then back again on the backside.

We also introduce the harmonic measures $u_{j}$, $j=1,\dots,\texttt{g}$, here
defined to be those harmonic functions in $\Omega$ which takes the boundary
value one on the designated (number $j$) inner component of $\partial\Omega$
and vanishes on the rest of $\partial\Omega$. Their differentials $du_{j}$
extend harmonically to the Schottky double with
$\oint_{\alpha_{k}}du_{j}=-2\delta_{kj}$, $\oint_{\alpha_{k}}du_{j}=0$. Thus
$du_{j}=-2dU_{\beta_{j}}$ in terms of our general notations (as in (3.18),
(3.19)).

In the block matrix notation of (4.19) we have

$-*dU_{\beta}=P\,dU_{\alpha}+R\,dU_{\beta},$

where $P=(P_{kj})$, $R=(R_{kj})$ and (see (3.18), (3.19))

$P_{kj}=-\oint_{\beta_{j}}*dU_{\beta_{k}},\quad
R_{kj}=\oint_{\alpha_{j}}*dU_{\beta_{k}}.$

The last integral can be written

$R_{kj}=-\frac{1}{2}\oint_{\alpha_{j}}*du_{k}=-\frac{1}{2}\int_{\alpha_{j}\cap\Omega}\frac{\partial
u_{k}}{\partial
n}ds-\frac{1}{2}\int_{\alpha_{j}\cap\tilde{\Omega}}\frac{\partial
u_{k}}{\partial n}ds,$

and it is easy to see that it is zero because of cancelling contributions from
$\Omega$ and $\tilde{\Omega}$ due to the symmetry of $du_{k}$ and $\alpha_{j}$
going the opposite way on the backside.

As a general conclusion we therefore have that $R=0$ in the matrix (4.19) when
$M$ is the Schottky double of a planar domain. As a consequence,

$-*dU_{\beta}=P\,dU_{\alpha}.$

Similarly, the other equation contained in (4.19) becomes

$*dU_{\alpha}=Q\,dU_{\beta}.$ (9.7)

Turning now to flow $\eta$ in (4.15), this is necessarily symmetric on
$M=\hat{\Omega}$, hence

$\oint_{\alpha_{j}}\eta=0.$

It follows that the coefficients $A_{j}$ in (4.15) vanish, whereby that
equation becomes

$\eta=\sum_{j=1}^{\texttt{g}}B_{j}dU_{\alpha_{j}}.$

In terms of the stream function $\psi=G^{\omega}+\psi_{0}$ (see again (4.2))
this gives, inserting also (9.7),

$d\psi_{0}=*\eta=\sum_{j=1}^{\texttt{g}}B_{j}*dU_{\alpha_{j}}=\sum_{j=1}^{\texttt{g}}C_{j}dU_{\beta_{j}}=d\big{(}-\frac{1}{2}\sum_{j=1}^{\texttt{g}}C_{j}u_{j}\big{)},$

with $C_{j}=\sum_{i=1}^{\texttt{g}}B_{i}Q_{ij}$. It follows in particular that
$\psi_{0}$, and hence all of $\psi$, is single-valued on $\Omega$.

In summary, the total stream function is

$\psi(z)=G_{\Omega}({z,w})+\sum_{j=1}^{\texttt{g}}C_{j}U_{\beta_{j}}(z),$

and it is single-valued when restricted to $\Omega$. It is clear that the
$C_{j}$ will actually depend on $w$, and for symmetry reasons the above
formula eventually takes the form

$\psi(z)=G_{\Omega}({z,w})+\sum_{i,j=1}^{\texttt{g}}C_{ij}U_{\beta_{i}}(z)U_{\beta_{j}}(w)$

for some constants $C_{ij}$ subject to $C_{ij}={C_{ji}}$.

##  10 Appendix 1: 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. Therefore we give below a short introduction to these
notions. 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. Projective
connections have more recently become important, in for example conformal
field theory.

Some general references for the kind of connections we are going to consider
are [43, 21, 22, 23, 25]. We 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.

Let $\tilde{z}=\varphi(z)$ represent a holomorphic local change of complex
coordinate on a Riemann surface $M$ and define three nonlinear differential
expressions $\\{\cdot,\cdot\\}_{k}$, $k=0,1,2$, by

$\displaystyle\\{\tilde{z},z\\}_{0}$ $\displaystyle=\log\varphi^{\prime}(z)$
$\displaystyle=-2\log\frac{1}{\sqrt{\varphi^{\prime}}}$
$\displaystyle\\{\tilde{z},z\\}_{1}$
$\displaystyle=(\log\varphi^{\prime}(z))^{\prime}=\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}$
$\displaystyle=-2\sqrt{\varphi^{\prime}}\,\,(\frac{1}{\sqrt{\varphi^{\prime}}})^{\prime}$
$\displaystyle\\{\tilde{z},z\\}_{2}$
$\displaystyle=(\log\varphi^{\prime}(z))^{\prime\prime}-\frac{1}{2}((\log\varphi^{\prime}(z))^{\prime})^{2}=\frac{\varphi^{\prime\prime\prime}}{\varphi^{\prime}}-\frac{3}{2}(\frac{\varphi^{\prime\prime}}{\varphi^{\prime}})^{2}$
$\displaystyle=-2\sqrt{\varphi^{\prime}}\,\,(\frac{1}{\sqrt{\varphi^{\prime}}})^{\prime\prime}$

The last expression is the _Schwarzian derivative_ of $\varphi$. For
$\\{\tilde{z},z\\}_{0}$ there is an additive indetermincy of $2\pi\mathrm{i}$,
so actually only its real part, or exponential, is completely well-defined.

As alternative definitions we have, with $\tilde{z}=\varphi(z)$,
$\tilde{w}=\varphi(w)$,

$\displaystyle\\{\tilde{w},w\\}_{0}$ $\displaystyle=\lim_{z\to
w}\log\frac{\tilde{z}-\tilde{w}}{z-w},$ $\displaystyle\\{\tilde{w},w\\}_{1}$
$\displaystyle=2\lim_{z\to w}\frac{\partial}{\partial
z}\log\frac{\tilde{z}-\tilde{w}}{z-w},$ $\displaystyle\\{\tilde{w},w\\}_{2}$
$\displaystyle=6\lim_{z\to w}\frac{\partial^{2}}{\partial z\partial
w}\log\frac{\tilde{z}-\tilde{w}}{z-w}.$

The following chain rules hold, if $z$ depends on $w$ via an intermediate
variable $u$:

$\\{z,w\\}_{k}(dw)^{k}=\\{z,u\\}_{k}(du)^{k}+\\{u,w\\}_{k}(dw)^{k}\quad(k=0,1,2).$

In particular,

$\\{z,w\\}_{k}(dw)^{k}=-\\{w,z\\}_{k}(dz)^{k}\quad(k=0,1,2).$

It turns out that the three operators $\\{\cdot,\cdot\\}_{k}$, $k=0,1,2$, are
unique in having properties as above, i.e., one cannot go on with anything
similar for $k\geq 3$. See [21, 22] for details.

######  Definition 10.1.

An affine connection (or $1$-connection) on $M$ is an object which is
represented by local differentials $r(z)dz$,
$\tilde{r}(\tilde{z})d\tilde{z}$,…(one in each coordinate variable, and not
necessarily holomorphic) glued together according to the rule

$\tilde{r}({\tilde{z}}){d\tilde{z}}=r(z){dz}-\\{\tilde{z},z\\}_{1}\,{dz}.$

######  Definition 10.2.

A projective connection (or Schwarzian connection, or $2$-connection) on $M$
consists of local quadratic differentials $q(z)(dz)^{2}$,
$\tilde{q}(\tilde{z})(d\tilde{z})^{2}$, …, glued together according to

$\tilde{q}({\tilde{z}})({d\tilde{z}})^{2}=q(z)({dz})^{2}-\\{\tilde{z},z\\}_{2}\,({dz})^{2}.$

One may also consider $0$-connections, assumed here to be real-valued. Such a
connection $p(z)$ transforms according to

$\tilde{p}(\tilde{z})=p(z)-\operatorname{Re}\\{\tilde{z},z\\}_{0}.$

This means exactly that

$ds=e^{p(z)}{|dz|}.$

is a Riemannian metric.

For a metric in general, $ds=\lambda(z){|dz|}=e^{p(z)}|dz|$, there is a
natural affine connection associated to it by

$r(z)=2\frac{\partial}{\partial z}\log\lambda(z)=2\frac{\partial p}{\partial
z}=\frac{\partial p}{\partial x}-\mathrm{i}\frac{\partial p}{\partial y}.$
(10.1)

This can be identified with the Levi-Civita connection in general tensor
analysis. The real and imaginary parts coincide (up to sign) with the
components of the classical Christoffel symbols $\Gamma_{ij}^{k}$. The
Gaussian curvature of the metric is

$\kappa(z)=-4\lambda(z)^{-2}\frac{\partial^{2}}{\partial
z\partial\bar{z}}\log\lambda(z)=-2{\lambda(z)^{-2}}\frac{\partial
r(z)}{\partial\bar{z}}.$

Under conformal changes of coordinates $\kappa$ transforms as a scalar:
$\tilde{\kappa}(\tilde{z})=\kappa(z)$ in previous notation.

Independent of any metric, an affine connection $r$ gives rise to a projective
connection $q$ by

$q(z)=\frac{\partial r(z)}{\partial z}-\frac{1}{2}r(z)^{2}.$ (10.2)

This $q$ is sometimes called the “curvature” of $r$ (see [11]). That curvature
is however not the same as the Gaussian curvature in case $r(z)$ comes form a
metric. A polarized version of $q$ is

$q(z,w)=\frac{1}{2}\big{(}\frac{\partial r(z)}{\partial z}+\frac{\partial
r(w)}{\partial w}-r(z)r(w)\big{)},$ (10.3)

related to the Batman function in [3].

The equation for geodesic curves $z=z(t)$ is, in terms of an arc-length
parameter $t$,

$\frac{d^{2}z}{dt^{2}}+r(z)(\frac{dz}{dt})^{2}=0$

or, written in another way,

$\frac{d}{dt}\log\frac{dz}{dt}+r(z)\frac{dz}{dt}=0.$ (10.4)

The first version is just a reformulation of the usual equation in terms of
Christoffel functions in ordinary differential geometry (see [15]).

The real part of (10.4) only contains information about the parametrization,
while the imaginary part, namely

$\frac{d}{dt}\arg\frac{dz}{dt}+\operatorname{Im}\big{(}r(z)\frac{dz}{dt}\big{)}=0.$
(10.5)

describes the geodesic curve in terms of an arbitrary real parameter $t$. The
latter property is useful in the context of the motion of vortex pairs since
the speed of such a pair tends to infinity as the distance between the two
vortices goes zero, and time therefore has to be successively reparametrized.

##  11 Appendix 2: Behavior of singular parts under changes of coordinates

The stream function $\psi$ for a flow in a neighborhood of a point vortex with
location $w$ has an expansion starting, in local coordinates and up to a
constant factor depending on the strength of the vortex,

$\psi(z)=\log|z-w|-c_{0}(w)-\mathcal{O}(|z-w|).$

Similarly, the expansion of the analytic completion
$\nu+\mathrm{i}*\nu=2\mathrm{i}\frac{\partial\psi}{\partial{z}}dz$ of the flow
$1$-form $\nu$ is (again up to a constant factor)

$f(z)dz=\frac{dz}{z-w}-c_{1}(w)dz-\mathcal{O}(z-w).$

One step further, one may consider a pure vortex dipole. The corresponding
flow is obtained by differentiating the above
$\nu=\operatorname{Re}\big{(}f(z)dz\big{)}$ with respect to $a$. The analytic
completion $F(z)dzdw$ of the so obtained flow $1$-form $d_{w}\nu$ has an
expansion

$F(z)dzdw=\frac{dzdw}{(z-w)^{2}}-2c_{2}(w)dzdw+\mathcal{O}(z-w).$

Above, and always in similar situations, $dzdw$ should be interpreted as the
tensor product $dz\otimes dw$, i.e. it is not a wedge product.

The coefficients $c_{0}$, $c_{1}$, $c_{2}$ above transforms, under conformal
changes of coordinates, as different kinds of connections as defined in
Appendix, Section 10. This is made precise in the following elementary lemma,
which we here cite without proof from [24]. Compare Lemma 3.6 and subsequent
discussions in [2].

######  Lemma 11.1.

Let $\tilde{z}=\varphi(z)$ be a local conformal map representing a change of
coordinates near a point $z=w$, and set $\tilde{w}=\varphi(w)$. Let $\psi(z)$
($=\psi(z,w)$) be a locally defined real-valued harmonic function with a
logarithmic pole at $z=w$, similarly $f(z)dz$ a meromorphic differential with
a simple pole at the same point, and $F(z)dzdw$ a double differential with a
pure (residue free) second order pole. Precisely, we assume the following
local forms, in the $z$ and $\tilde{z}$ variables:

$\displaystyle\psi(z)$
$\displaystyle=\log|z-w|-c_{0}(w)+\mathcal{O}(|{z}-{w}|)$
$\displaystyle=\log|\tilde{z}-\tilde{w}|-\tilde{c}_{0}(\tilde{w})+\mathcal{O}(|\tilde{z}-\tilde{w}|),$
$\displaystyle f(z)dz$
$\displaystyle=\frac{dz}{z-w}-c_{1}(w)dz+\mathcal{O}(z-w)$
$\displaystyle=\frac{d\tilde{z}}{\tilde{z}-\tilde{w}}-\tilde{c}_{1}(\tilde{w})d\tilde{z}+\mathcal{O}(\tilde{z}-\tilde{w}),$
$\displaystyle F(z)dzdw$
$\displaystyle=\frac{dzdw}{(z-w)^{2}}-2c_{2}(w)dzdw+\mathcal{O}(z-w)$
$\displaystyle=\frac{d\tilde{z}d\tilde{w}}{(\tilde{z}-\tilde{w})^{2}}-2\tilde{c}_{2}(\tilde{w})d\tilde{z}d\tilde{w}+\mathcal{O}(\tilde{z}-\tilde{w}).$

Then, as functions of the location of the singularity and up to constant
factors, $c_{0}$ transforms as a $0$-connection, $c_{1}$ as an affine
connection and $c_{2}$ as a projective connection:

$\tilde{c}_{0}(\tilde{w})=c_{0}(w)+\operatorname{Re}\\{\tilde{w},w\\}_{0},$
$\tilde{c}_{1}(\tilde{w})d\tilde{w}=c_{1}(w)dw+\frac{1}{2}\\{\tilde{w},w\\}_{1}dw,$
$2\tilde{c}_{2}(\tilde{w})d\tilde{w}^{2}=2c_{2}(w)dw^{2}+\frac{1}{6}\\{\tilde{w},w\\}_{2}dw^{2}.$

The first statement is most conveniently expressed by saying that

$ds=e^{-\tilde{c}_{0}(\tilde{w})}|d\tilde{w}|=e^{-c_{0}(w)}|dw|$

defines a conformally invariant metric.

######  Remark 11.1.

It is actually not necessary for the conclusions of the lemma that $\psi$,
$fdz$, $Fdzdw$ are harmonic/analytic away from the singularity, it is enough
that the local forms of the singularity and constant terms given as above
hold.

We may adapt the above lemma to the Green function $G(z,w)=G^{\delta_{w}}(z)$,
despite it is not harmonic in $z$ (because of the compensating background
flow). First we write

$G(z,w)=\frac{1}{2\pi}(-\log|z-w|+H(z,w))$ (11.1)

and then expand the regular part as

$H(z,w)=h_{0}(w)+\frac{1}{2}\left(h_{1}(w)(z-w)+\overline{h_{1}(w)}(\bar{z}-\bar{w})\right)+$
(11.2)
$+\frac{1}{2}\left(h_{2}(w)(z-w)^{2}+\overline{h_{2}(w)}(\bar{z}-\bar{w})^{2}\right)+h_{11}(w)(z-w)(\bar{z}-\bar{w})+\mathcal{O}(|z-w|^{3}).$

We note from (4.10) that

$H(w,w)=h_{0}(w),\quad\\{\frac{\partial H(z,w)}{\partial
z}\\}_{z=w}=\frac{1}{2}h_{1}(w).$

Thus the symmetry of $H(z,w)$ gives

$h_{1}(w)=\frac{\partial h_{0}(w)}{\partial w},$ (11.3)

For the second order derivatives we have

$\\{\frac{\partial^{2}H(z,w)}{\partial
z^{2}}\\}_{z=w}=h_{2}(w),\qquad\qquad\quad$ (11.4)
$\\{\frac{\partial^{2}H(z,w)}{\partial
z\partial\bar{z}}\\}_{z=w}=h_{11}(w)=\frac{\pi}{2V}\lambda(w)^{2},$ (11.5)
$\\{\frac{\partial^{2}H(z,w)}{\partial z\partial
w}\\}_{z=w}=\frac{1}{2}\frac{\partial h_{1}(w)}{\partial w}-h_{2}(w),$ (11.6)
$\\{\frac{\partial^{2}H(z,w)}{\partial
z\partial\bar{w}}\\}_{z=w}=\frac{1}{2}\frac{\partial
h_{1}(w)}{\partial\bar{w}}-h_{11}(w).$ (11.7)

The transformation properties of the coefficients $h_{j}(w)$ are closely
related to those appearing in Lemma 11.1. Citing from [24] we have

######  Lemma 11.2.

Under a local holomorphic change $\tilde{z}=\varphi(z)$ of coordinates, with
$\tilde{w}=\varphi(w)$, we have

$\tilde{h}_{0}(\tilde{w})=h_{0}(w)+\operatorname{Re}\\{\tilde{w},w\\}_{0},$
(11.8)
$\tilde{h}_{1}(\tilde{w})d\tilde{w}=h_{1}(w)dw+\frac{1}{2}\\{\tilde{w},w\\}_{1}dw,$
(11.9)
$\big{(}\frac{\partial\tilde{h}_{1}(\tilde{w})}{\partial\tilde{w}}-2\tilde{h}_{2}(\tilde{w})\big{)}d\tilde{w}^{2}=\big{(}\frac{\partial
h_{1}(w)}{\partial
w}-2{h}_{2}(w)\big{)}d{w}^{2}+\frac{1}{6}\\{\tilde{w},w\\}_{2}dw^{2},$ (11.10)
$\tilde{h}_{11}(\tilde{w})|d\tilde{w}|^{2}=h_{11}(w)|dw|^{2}.$ (11.11)

## References

  * [1] V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics, vol. 125 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998.
  * [2] I. Biswas and A. K. Raina, Projective structures on a Riemann surface, Internat. Math. Res. Notices, (1996), pp. 753–768.
  * [3] S. Boatto and J. Koiller, Vortices on closed surfaces, in Geometry, mechanics, and dynamics, vol. 73 of Fields Inst. Commun., Springer, New York, 2015, pp. 185–237.
  * [4] A. Bogatskiy, Vortex flows on closed surfaces, J. Phys. A, 52 (2019), pp. 475501, 23.
  * [5] A. V. Borisov and I. S. Mamaev, On the problem of motion of vortex sources on a plane, Regul. Chaotic Dyn., 11 (2006), pp. 455–466.
  * [6] M. M. Cawte, X. Yu, B. P. Anderson, and A. S. Bradley, Snell’s law for a vortex dipole in a Bose-Einstein condensate, SciPost Phys., 6 (2019), pp. Paper No. 032, 18.
  * [7] A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), pp. 785–796.
  * [8] H. Cohn, Conformal mapping on Riemann surfaces, Dover Publications, Inc., New York, 1980.  Reprint of the 1967 edition, Dover Books on Advanced Mathematics.
  * [9] P. J. Davis, The Schwarz function and its applications, The Mathematical Association of America, Buffalo, N. Y., 1974.  The Carus Mathematical Monographs, No. 17.
  * [10] D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. A., 471 (2015), pp. 20140890, 25.
  * [11] B. Dubrovin, Geometry of $2$D topological field theories, in Integrable systems and quantum groups (Montecatini Terme, 1993), vol. 1620 of Lecture Notes in Math., Springer, Berlin, 1996, pp. 120–348.
  * [12] H. M. Farkas and I. Kra, Riemann surfaces, vol. 71 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1992.
  * [13] M. Flucher, Variational problems with concentration, vol. 36 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 1999.
  * [14] M. Flucher and B. Gustafsson, Vortex motion in two-dimensional hydrodynamics, Royal Instituste of Technology Research Bulletins, TRITA-MAT-1997-MA-02 (1979), pp. 1–24.
  * [15] T. Frankel, The Geometry of Physics, Cambridge University Press, Cambridge, third ed., 2012.  An introduction.
  * [16] A. Fridman and P. Polubarinova, On moving singularities of a flat motion of an incompressible fluid (in Russian), Geofizicheskii Sbornik, (1928), pp. 9–23.
  * [17] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978.  Pure and Applied Mathematics.
  * [18] C. Grotta Ragazzo, The motion of a vortex on a closed surface of constant negative curvature, Proc. A., 473 (2017), pp. 20170447, 17.
  * [19] C. Grotta Ragazzo and H. H. d. B. Viglioni, Hydrodynamic vortex on surfaces, J. Nonlinear Sci., 27 (2017), pp. 1609–1640.
  * [20] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984.
  * [21] R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966.
  * [22] , Special coordinate coverings of Riemann surfaces, Math. Ann., 170 (1967), pp. 67–86.
  * [23] , On uniformization of complex manifolds: the role of connections, vol. 22 of Mathematical Notes, Princeton University Press, Princeton, N.J., 1978.
  * [24] B. Gustafsson, Vortex motion and geometric function theory: the role of connections, Philos. Trans. Roy. Soc. A, 377 (2019), pp. 20180341, 27.
  * [25] B. Gustafsson and A. Sebbar, Critical points of Green’s function and geometric function theory, Indiana Univ. Math. J., 61 (2012), pp. 939–1017.
  * [26] D. Hally, Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J. Math. Phys., 21 (1980), pp. 211–217.
  * [27] D. D. Holm and H. O. Jacobs, Multipole vortex blobs (MVB): symplectic geometry and dynamics, J. Nonlinear Sci., 27 (2017), pp. 973–1006.
  * [28] Y. Kimura, Vortex motion on surfaces with constant curvature, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), pp. 245–259.
  * [29] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung, Acta Math., 41 (1916), pp. 305–344.  IV. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche.
  * [30] J. Koiller, Getting into the vortex: on the contributions of James Montaldi, J. Geom. Mech., 12 (2020), pp. 507–523.
  * [31] J. Koiller and S. Boatto, Vortex paires on surfaces, AIP Conference Proceedings 1130; https://doi.org/10.1063/1.3146241, 77 (2009).
  * [32] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy, and A. Constantin, Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation, J. Fluid Mech., 921 (2021), pp. Paper No. A1, 35.
  * [33] K. N. Kulik, A. V. Tur, and V. V. Yanovskiĭ, Interaction of point and dipole vortices in an incompressible fluid, Teoret. Mat. Fiz., 162 (2010), pp. 459–480.
  * [34] C. C. Lin, On the motion of vortices in two dimensions. I. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), pp. 570–575.
  * [35] , On the motion of vortices in two dimensions. II. Some further investigations on the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), pp. 575–577.
  * [36] , On the Motion of Vortices in Two Dimensions, University of Toronto Studies, Applied Mathematics Series, no. 5, University of Toronto Press, Toronto, Ont., 1943.
  * [37] S. G. Llewellyn Smith, How do singularities move in potential flow?, Phys. D, 240 (2011), pp. 1644–1651.
  * [38] S. G. Llewellyn Smith and R. J. Nagem, Vortex pairs and dipoles, Regul. Chaotic Dyn., 18 (2013), pp. 194–201.
  * [39] C. Marchioro and M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994.
  * [40] P. K. Newton, The $N$-vortex problem, vol. 145 of Applied Mathematical Sciences, Springer-Verlag, New York, 2001.  Analytical techniques.
  * [41] A. Regis Rodrigues, C. Castilho, and J. Koiller, Vortex pairs on a triaxial ellipsoid and Kimura’s conjecture, J. Geom. Mech., 10 (2018), pp. 189–208.
  * [42] L. Sario and K. Oikawa, Capacity functions, Die Grundlehren der mathematischen Wissenschaften, Band 149, Springer-Verlag New York Inc., New York, 1969.
  * [43] M. Schiffer and N. S. Hawley, Connections and conformal mapping, Acta Math., 107 (1962), pp. 175–274.
  * [44] M. Schiffer and D. C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954.
  * [45] F. Schottky, Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen, J. Reine Angew. Math., 83 (1877), pp. 300–351.
  * [46] B. F. Schutz, Jr., Geometrical methods in mathematical physics, Cambridge University Press, Cambridge, 1980.
  * [47] H. S. Shapiro, The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, 9, John Wiley & Sons Inc., New York, 1992.  A Wiley-Interscience Publication.
  * [48] A. Vasil′ev, From the Hele-Shaw experiment to integrable systems: a historical overview, Complex Anal. Oper. Theory, 3 (2009), pp. 551–585.
  * [49] Q. Wang, The $n$-vortex problem on a Riemann sphere, Comm. Math. Phys., 385 (2021), pp. 565–593.