Vortex on surfaces and Brownian-motion in higher dimensions: special metrics

Clodoaldo Grotta-Ragazzo∗.

A single hydrodynamic vortex on a surface will in general moves unless its
Riemannian metric is a special “Steady Vortex Metric” (SVM). Metrics of
constant curvature are SVM only in surfaces of genus zero and one.

In this paper:

  1. 1.

I show that K. Okikiolu’s work on the regularization of the spectral zeta
function leads to the conclusion that each conformal class of every compact
surface with a genus of two or more possesses at least one Steady Vortex
Metric (SVM).

  2. 2.

I apply a probabilistic interpretation of the regularized zeta function for
surfaces, as developed by P. G. Doyle and J. Steiner, to extend the concept of
SVM to higher dimensions.

The new special metric, which aligns with the Steady Vortex Metric (SVM) in
two dimensions, has been termed the “Uniform Drainage Metric” for the
following reason: For a compact Riemannian manifold $M$, the “narrow escape
time” (NET) is defined as the expected time for a Brownian motion starting at
a point $p$ in $M\setminus B_{\epsilon}(q)$ to remain within this region
before escaping through the small ball $B_{\epsilon}(q)$, which is centered at
$q$ with radius $\epsilon$ and acts as the escape window. The manifold is said
to possess a uniform drainage metric if, and only if, the spatial average of
NET, calculated across a uniformly distributed set of initial points $p$,
remains invariant regardless of the position of the escape window
$B_{\epsilon}(q)$, as $\epsilon$ approaches $0$.

Key words: point vortex, Riemann surfaces, diffusion process, Brownian motion,
special metrics, spectral zeta function.

AMS Classification: 76B47, 30F30, 58J65, 31C12, 60J45, 53C25

Abbreviated title: SVM and Brownian-motion special metrics

∗ Instituto de Matemática e Estatística da Universidade de São Paulo,
Rua do Matão 1010, 05508-090, São Paulo, SP, Brazil.
Partially supported by FAPESP grant 2016/25053-8.
email<EMAIL_ADDRESS>
ORCID: 0000-0002-4277-4173

##  1 Introduction

The motion of point vortices on the plane is a classical subject in fluid
mechanics that goes back to Helmholtz, Kelvin, and Kirchhoff. The first to
consider the motion of point vortices on a curved surface, the sphere embedded
in ${\mathbb{R}}^{3}$, was Zermello in 1902. The paper [7] has a historical
review on the early research on hydrodynamic vortices on surfaces. An
intrinsic definition of the motion of vortices on a surface, which is
independent of the embedding of the surface in ${\mathbb{R}}^{3}$ and on
coordinates, started with Boatto and Koiller [5] (see also [6] [12] [40]) and
was recently completed by Björn Gustafsson [22] [23].

A single vortex in the Euclidean plane, or in the round sphere, or in a flat
torus does not move, and this motivated the definition of “Steady Vortex
Metric” [40]: a Riemannian metric for which a single vortex does not move
regardless of its position. J. Koiller conjectured that a single vortex in a
compact surface of constant curvature and of a genus greater than one does
move. In [39], [24] Koiller’s conjecture was numerically verified for a
particular surface of constant curvature of genus two: the Bolza surface. This
result motivated the first main question to be answered in this work: Does a
steady vortex metric exist on any orientable compact surface of a genus
greater than one?

K. Okikiolu proved that a certain functional on the space of Riemannian
metrics, which is an analog for closed surfaces of the ADM mass from general
relativity, has a minimizing metric on each conformal class. It turns out that
the special metrics of Okikiolu are steady vortex metrics, which gives a
positive answer to the question in the paragraph above. This raises the
question about the “meaning” (or properties) of this special metric. The
steady vortex metric minimizes a certain functional [37] and has the property
in its name, but does it have any other interesting geometrical property
besides those? This question was the second motivation for this work.

The special metric found by Okikiolu is a critical point of a functional
related to the regularized Green’s function of the Laplacian: the “Robin
function”. P. G. Doyle and J. Steiner [11] gave a probabilistic interpretation
to the Robin function that is related to the concept of “Narrow-Escape-
Time”(NET) [26]. The NET is defined as the expected time for a Brownian motion
starting at $p$ in $M\backslash B_{\epsilon}(q)$ to remain within this region
before escaping through the small ball $B_{\epsilon}(q)$, which is centered at
$q$ with radius $\epsilon$ and acts as the escape window.

The NET is an important abstraction in science, as argued by Holcman and
Schuss in the Introduction of [26]: “The narrow escape problem in diffusion
theory, which goes back to Helmholtz (Helmholtz (1860)) and Lord Rayleigh
(Rayleigh (1945)) in the context of the theory of sound, is to calculate the
mean first passage time of Brownian motion to a small absorbing window…. The
renewed interest in the problem is due to the emergence of the narrow escape
time (NET) as a key to the determination of biological cell function from its
geometrical structure. The NET is ubiquitous in molecular and cellular biology
and is manifested in stochastic models of chemical reactions…”

The average NET, with respect to a uniform distribution of initial positions
(volume measure), that a particle takes to escape from $S\backslash
B_{\epsilon}(q)$ through the small window $B_{\epsilon}(q)$ is proportional to
$-\log\epsilon+R(q)+{\cal O}(\epsilon)$, where $R$ is the Robin function. So,
for small $\epsilon$ the Robin function indicates the drainage capacity of
different points $q$ in $S$. The Robin function is constant if, and only if,
the metric is a steady vortex metric (SVM). Therefore, in a surface with a
$SVM$ the drainage capacity of different points is the same and this lead to
the alternative name “uniform drainage metric”, a property that makes sense in
dimensions larger than two. Note: the notion of hydrodynamic point vortex
cannot be generalized to dimensions greater than two.

The main contribution in this paper is the definition of uniform drainage
metric in dimensions greater than two and its geometric characterization in
dimensions 3 and 4.

Following the same steps given in this paper, a characterization of a uniform
drainage metric in higher dimensions can be accomplished by means of certain
coefficients that appear in the so-called Minakshisundaram-Plejel asymptotic
expansion of the heat kernel. I prefer not to state any results in this
direction because, in higher dimensions, it is necessary to compute more of
these coefficients, which can be expressed in terms of powers of the Laplacian
and the distance function $\ell$, and they become very complicated [38].

The existence of uniform drainage surfaces of arbitrary finite genus in any
conformal class is guaranteed by the theorem of Okikiolu. In higher dimensions
any compact Riemannian manifold that is a homogeneous space is a uniform
drainage manifold.111 There is a special class of Riemannian metrics on closed
manifolds that are critical metrics of the trace of the heat kernel under
conformal variations of the metric [13]. A metric in this special class is
always a uniform drainage metric (a consequence of Theorem 4.1 (ii) in [13]).
The metric of any Riemannian homogeneous space is critical for the trace of
the heat kernel. Does there exist a closed (compact and boundaryless) manifold
that does not admit a uniform drainage metric?

This paper is organized as follows.

In Section 2 I give a precise definition of the steady vortex metric and
present two fundamental theorems that stem from Okikiolu’s work. I then use
these theorems to compare the steady vortex metric with other natural
Riemannian metrics: of constant curvature, canonical or Bergman, and Arakelov.
The proofs of the two theorems are presented in Appendix A in a slightly
different way than those given by Okikiolu. These theorems plus some simple
arguments imply: “No orientable surface of genus 2 and of constant curvature
is a Steady Vortex Surface.”

In Section 3, I present a regularization of the Green’s function in dimensions
greater than two using the Minakshisundaram-Plejel asymptotic expansion of the
heat kernel. This provides a definition of the Robin function in higher
dimensions. In the Appendix B I show that the Robin function can be written in
terms of the analytic extension of the Minakshisundaram–Pleijel zeta function,
and therefore uniform drainage manifolds have a special spectral property
derived from this relation. The relation between the Robin function and the
Minakshisundaram–Pleijel zeta function appeared in [43], for surfaces, and in
[4], in a more general context and in dimension greater than two.

In Section 4 I give a characterization of uniform drainage metric in
dimensions 2, 3 and 4. In dimension 4 a uniform drainage metric has constant
Robin function (a global property) and constant scalar curvature (a local
property).

In Section 5, I present a family of non-flat tori found by Okikiolu [36], and
which will be called Okikiolu’s tori, that are uniform drainage surfaces.
These tori are the only non-constant curvature uniform drainage surface that
are explicitly known. For any $a>\sqrt{\pi/2}$ there is an Okikiolu’s torus
that is conformally equivalent to the flat torus
${\mathbb{R}}^{2}/(a{\mathbb{Z}}\times a^{-1}{\mathbb{Z}})$. Therefore,
uniform drainage metrics may not be unique in a conformal class. The curvature
of the Okikiolu’s tori was computed in ibid., where it was realized that in
the limit as $a\to\infty$ the curvature at almost every point of the torus
tends to $1/\sqrt{4\pi}$. In Section 5 I embed a cylinder in
${\mathbb{R}}^{3}$ whose quotient under the group of translations along the
cylinder axis is an Okikiolu’s torus. In this way, I can visualize the
deformation of a flat torus into a pinched torus that is isometric to a round
sphere with two opposite points being identified. The deformation is done
along an interesting family of uniform drainage surfaces.

I finally remark about a possible upshot of the relation between the Robin
function and the drainage capacity of different points. The importance of the
NET in cellular biology is partially due to diffusion processes that occur in
membranes towards special exit gates (escape windows). The minimum of the
Robin function is an equilibrium position of a single vortex [24] and also a
point where the drainage capacity of the surface, as defined above, is
maximum. Equilibrium positions of systems of point vortices, an issue that has
been extensively studied, also have a probabilistic interpretation. If the
position of a vortex is related to an entrance or exit gate, depending on the
vortex sign, then some equilibrium configurations will certainly be more
efficient in connecting different gates by means of diffusion than others. If
this idea is correct, then the the importance of equilibrium configurations on
surfaces of spheres, including those which are not round, will be greatly
enhanced.

##  2 Steady vortex metrics on orientable closed surfaces.

The definition of a hydrodynamic vortex requires some preliminaries (see
[40]). The fundamental equations of hydrodynamics on $S$, Euler’s equations,
necessitate that $S$ be endowed with a Riemannian metric $g$. Here, $g$
represents a smooth family of inner products on the tangent spaces of $S$. In
local coordinates, the Riemannian metric is given by
$g=\sum_{jk}g_{jk}dx_{j}\otimes dx_{k}$. The associated volume form is
$\mu=\sqrt{|g|}dx_{1}\wedge dx_{2}$, where $|g|$ denotes the absolute value of
the determinant of the matrix $g_{jk}$

In a neighborhood of each point of $S$, there exist coordinates (sometimes
called isothermal coordinates) in which
$g=\lambda^{2}(x)(dx_{1}^{2}+dx_{2}^{2})$ and $\mu=\lambda^{2}(x)dx_{1}\wedge
dx_{2}$. The existence of isothermal coordinates is a manifestation of the
fact that any surface is locally conformal to the Euclidean plane. In this
paper, I will also use $\lambda^{2}$ to denote the conformal factor between
arbitrary given metrics $g_{0}$ and $g_{1}$. This will be explicitly stated
when used.

The one-forms $\theta_{1}=\lambda dx_{1}$ and $\theta_{2}=\lambda dx_{2}$
constitute an orthonormal moving coframe. The Hodge-star operator acts
linearly on forms and is defined by

$\ast
1=\theta_{1}\wedge\theta_{2}=\mu,\qquad\ast\theta_{1}=\theta_{2},\qquad\ast\theta_{2}=-\theta_{1},\qquad\ast\mu=1.$

The Laplace operator acting on functions is given by $\Delta=\ast d\ast
d=\frac{1}{\lambda^{2}}(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})$ and the
Gaussian curvature by $K=-\frac{1}{\lambda^{2}}\Delta\log\lambda$.

Let $V=\int_{S}\mu$ be the total area of $S$. The Green’s function of $(S,g)$
is the unique solution in distribution sense to the equation

$-\Delta_{q}G(q,p)=\delta_{p}(q)-V^{-1},$ (2.1)

that has the following properties (see [2], theorem 4.13):

  * •

for all functions $\phi\in C^{2}$

$\phi(p)=\frac{1}{V}\int_{S}\phi\mu-\int_{S}G(q,p)\Delta\phi(q)\mu(q),$ (2.2)

  * •

$G(q,p)$ is $\hbox{\rm C}^{\infty}$ on $S\times S$ minus the diagonal,

  * •

$G$ is symmetric $G(q,p)=G(p,q)$,

  * •

$G$ is bounded from below and $\int_{S}G(q,p)\mu(q)=0$.

A point vortex of intensity $\Gamma\in{\mathbb{R}}$ at the point $p$ is the
fluid velocity field defined on $S-\\{p\\}$ given by $q\to\ast\nabla\Gamma
G(q,p)$, where $\nabla$ is the gradient operator and $\ast$ is the operator
that rotates a vector by $\pi/2$.

The Robin function (the regularization of $G$) is a $\hbox{\rm C}^{\infty}$
function on $S$ ([40] Theorem 5.1) defined as

$R(p)=\lim_{\ell(q,p)\to 0}\left[G(q,p)+\frac{1}{2\pi}\log\ell(q,p)\right],$
(2.3)

where $\ell(q,p)$ is the Riemannian distance between $p$ and $q$.

The motion of a single vortex depends not only on its initial position but
also on the initial value of a harmonic velocity field (a background flow)
[23]. In the following statement [40] [24] the initial background flow is
assumed to be equal to zero:

A vortex initially placed at any point on a surface $S$ with Riemannian metric
$g$ remains at rest if, and only if, the Robin function $R$ associated with
$g$ is constant. A Riemannian metric with this property is called a “Steady
Vortex Metric”.

The first main result in this paper is the following.

######  Theorem 2.1 (Steady Vortex Metric).

Let $S$ be a compact Riemann surface. There exists at least one steady vortex
metric $g$ compatible with the conformal structure of $S$. There are examples
where $g$ is not unique.

The theorem effectively says that there always exist a metric for which the
Robin function is constant.

This theorem is a direct consequence of a theorem proven by K. Okikiolu [36]
[37] and its proof is given in Appendix A.

The second theorem in this Section requires some definitions. A one-form
$\theta$ on $S$ is harmonic if $d\theta=0$ and $d\ast\theta=0$. Since $\ast$
rotates one-forms by $\pi/2$, harmonic forms are conformal invariants. The
vector space of harmonic forms on $S$ is finite and has dimension $2{\cal G}$
[9], where ${\cal G}$ is the genus of $S$. Let
$\\{\theta_{1},\ldots\theta_{2{\cal G}}\\}$ be an arbitrary orthonormal basis
of harmonic one-forms in the sense that

$(\theta_{j},\theta_{k})=\int_{S}\theta_{j}\wedge\ast\theta_{k}=\delta_{jk}.$
(2.4)

Note: this definition of orthonormality depends only on the conformal
structure.

######  Theorem 2.2.

Let $(S,g)$ be a compact oriented Riemannian surface and $\sigma$ be the two-
form

$\sigma=\sum_{k=1}^{2{\cal G}}\theta_{k}\wedge*\theta_{k}\,.$

Then the Robin function $R$ is the only solution, up to an additive constant,
of the equation

$\left(\Delta R+\frac{K}{2\pi}-\frac{2}{V}\right)\mu=-\sigma\,.$ (2.5)

If the genus ${\cal G}$ of $S$ is zero, then $\sigma=0$. So a metric on the
sphere is a Steady Vortex Metric if, and only if, it is of constant curvature.

If the genus of $S$ is greater than zero, then $\sigma$ is the area form of
the Bergman metric. The most common definition of the Bergman metric ([28]
eqs. 1.4.22 and 1.4.23) uses a basis of holomorphic differentials
$\\{\omega_{1},\ldots,\omega_{\cal G}\\}$ that satisfy the orthonormality
conditions
$\frac{i}{2}\int_{S}\omega_{j}\wedge\overline{\omega}_{k}=\delta_{jk}$ (here
the overbar denotes complex conjugation). A form $\omega_{j}$ is holomorphic
if, and only if, $\omega_{j}=\theta_{j}+\ast\theta_{j}$ for some harmonic
differential $\theta_{j}$ ([14], Theorem I.3.11). If we define
$\theta_{j+{\cal G}}=\ast\theta_{j}$, $j=1,\ldots,{\cal G}$, then the
orthogonality condition for holomorphic differentials implies the
orthogonality condition for harmonic differentials (2.4) and

$\sigma=\sum_{k=1}^{2{\cal G}}\theta_{k}\wedge*\theta_{k}=i\,\sum_{j=1}^{\cal
G}\omega_{j}\wedge\overline{\omega}_{j}\,,\quad\text{with}\quad\int_{S}\sigma=2{\cal
G}.$ (2.6)

The Bergman metric normalized as $\sigma/(2{\cal G})$ can also be defined
using the Jacobian variety associated with $S$ (see [44], [28], or equation
1.25 in [15]). In several references [44] [37] [27] the normalized Bergman
metric $\sigma/(2{\cal G})$ is called by the alternative name “canonical
metric”.

Theorem 2.2 appeared in the work of Okikiolu [37] (proposition 2.3) as a “well
known” result related to the Arakelov Green’s function (in Appendix A I give a
more self-contained proof of Theorem 2.2 than that in [37]). The Arakelov
Green’s function is used in the definition of the “Arakelov metric” that is
characterized by the equation (see [28] eq. 1.4.24):

$\frac{K_{A}}{2\pi}\mu_{A}=(2-2{\cal G})\frac{\sigma}{2{\cal G}}\,,\qquad{\cal
G}\geq 1\,,$ (2.7)

where: $\mu_{A}$ and $K_{A}$ denote the area form and the curvature of the
Arakelov metric.

Equation (2.5) implies that the several “natural” metrics considered in this
paper satisfy the following relations:

$\begin{array}[]{lll}&\left(\frac{K_{svm}}{2\pi}-\frac{2}{V}\right)\mu_{svm}=-\sigma&\quad(\text{SVM})\\\
&\left(\Delta R_{cc}-\frac{2{\cal
G}}{V}\right)\mu_{cc}=-\sigma&\quad(\text{constant curvature=CC})\\\
&\left(\Delta R_{B}+\frac{K_{B}}{2\pi}\right)\sigma=(2-2{\cal
G})\frac{\sigma}{2{\cal G}}&\quad(\text{Bergman})\\\ &\left(\Delta
R_{A}-\frac{2}{V}\right)\mu_{A}=-\frac{\sigma}{{\cal
G}}&\quad(\text{Arakelov})\end{array}$ (2.8)

From equations (2.7) and (2.8) we obtain

$\begin{split}CC=SVM&\Leftrightarrow Bergman=CC\Leftrightarrow
Bergman=SVM\,,\\\ Arakelov=SVM&\Leftrightarrow Bergman=Arakelov\Leftrightarrow
Arakelov=CC\,.\end{split}$ (2.9)

For ${\cal G}\geq 1$, therefore, a constant curvature metric is a Steady
Vortex Metric if and only if the Bergman metric has constant curvature. For
${\cal G}=1$ this is the case, since the flat metric is the Bergman metric and
also the Arakelov metric.

In any closed surface $S$ of genus ${\cal G}\geq 2$ the curvature of the
Bergman metric is non positive [29] (theorem 5.5.1). If the curvature of the
Bergman metric $K_{B}$ is non constant in every $S$, which as far as I know
has not been proved, then constant curvature metrics will never be SVM for
${\cal G}\geq 2$. The last theorem in [30] states that $K_{B}(p)=0$ if and
only if $S$ is hyperelliptic and $p$ is one of the $2{\cal G}+2$ classical
Weierstrass points on $S$. Therefore $K_{B}$ is not constant in hyperelliptic
surfaces. Since every surface of genus 2 is hyperelliptic, the following
theorem holds.

######  Theorem 2.3.

No orientable surface of genus 2 and of constant curvature is a Steady Vortex
Surface.

The Gauss-Bonnet theorem implies that the average curvature of the Bergman
metric is

$K_{Ba}:=\frac{1}{V_{B}}\int_{S}K_{B}\sigma=2\pi\frac{2-2{\cal G}}{2{\cal
G}}\,,\quad\text{where}\quad V_{B}=\int_{S}\sigma=2{\cal G}\,.$ (2.10)

We define the deviatoric part $K_{B\delta}$ of $K_{B}$ as :

$K_{B\delta}:=K_{B}-K_{Ba}\,\quad\text{with}\quad\int_{S}K_{B\delta}\,\sigma=0\,.$
(2.11)

Equation (2.8) then implies that the Robin function of the Bergman metric
satisfies the simple relation

$-\Delta R_{B}=\frac{1}{2\pi}K_{B\delta}\,.$ (2.12)

This equation implies that in any conformal coordinates,
$\\{z,\overline{z}\\}$, $R_{B}$ has a simple expression in terms of the
potentials $F_{j}(z)$ of the holomorphic differentials
$\omega_{j}=dF_{j}=F_{j}^{\prime}(z)dz=\partial_{z}F_{j}(z)dz$ that appear in
the definition of $\sigma$ in equation (2.6). Indeed: $i\,\sum_{j=1}^{\cal
G}\omega_{j}\wedge\overline{\omega}_{j}=i\,\sum_{j=1}^{\cal
G}F_{j}^{\prime}(z)\overline{F_{j}^{\prime}(z)}dz\wedge
d\overline{z}=\lambda^{2}_{B}\frac{i}{2}dz\wedge d\overline{z}$, with
$\lambda_{B}^{2}=2\sum_{j=1}^{\cal
G}F_{j}^{\prime}(z)\overline{F_{j}^{\prime}(z)}$, $\
\Delta=\frac{4}{\lambda^{2}_{B}}\partial_{z}\partial_{\,\overline{z}}$, $\
\lambda^{2}_{B}=2\partial_{z}\partial_{\,\overline{z}}\sum_{j=1}^{\cal
G}F_{j}(z)\overline{F_{j}(z)}$, and $\
K_{B}=-\frac{4}{\lambda^{2}_{B}}\partial_{z}\partial_{\,\overline{z}}\log\lambda_{B}$;
imply

$R_{B}(z,\overline{z})=\frac{1}{4\pi}\log\left[\sum_{j=1}^{\cal
G}F_{j}^{\prime}(z)\overline{F_{j}^{\prime}(z)}\right]+\frac{1-{\cal
G}}{2{\cal G}}\sum_{j=1}^{\cal G}F_{j}(z)\overline{F_{j}(z)}+constant\,.$
(2.13)

The Riemann sphere admits a six-dimensional group of conformal transformations
(the Moebius group) and a three-dimensional group of isometries. The Pull-back
metric $g_{1}$ of the round metric $g_{0}$ by a Moebius transformation that is
not an isometry satisfies $g_{1}=\lambda^{2}g_{0}$ with $\lambda^{2}\neq 1$
almost everywhere. The Robin function associated to $g_{1}$ is constant
because, although different from $g_{0}$, $g_{1}$ is isometric to $g_{0}$.
This type of “nonuniqueness” of a steady vortex metric within a conformal
class will happen whenever the group of diffeomorphisms that preserves the
conformal structure is larger than the group of isometries. Since all spheres
with constant curvature are isometric to the round sphere, we conclude that
$g_{0}$ is the only steady vortex metric modulo isometries. The question about
the uniqueness of steady vortex metrics on tori will be postponed to Section
5.

##  3 Generalization to higher dimensions.

The definition of hydrodynamic point vortex is restricted to two dimensions.
There is an analogy between vortex and electric charges in two dimensions
[40]. Since the theory of electrostatics can be generalized to higher
dimensions, electrostatics could be the physical guide to the definition of an
“eletrostactic force-free metric” in dimension $n$. The idea although
interesting leads to some difficulties, which will be discussed in the next
paragraph, and it will not be pursued any further.

The Green’s function $G(q,p)$, solution to equation (2.1), can be understood
as the electric potential due to a positive point charge at the point $p$ plus
a uniform distribution of negative charges. The Robin function $R(p)$ defined
in equation (2.3) is the overall potential energy $G(q,p)$ minus the “singular
potential of the point charge”, $-(2\pi)^{-1}\log\ell(q,p)$, the difference
being evaluated at $p$. The force upon the point charge is $dR(p)$. The most
natural definition of Robin function in dimension $n\geq 3$ would be

$\lim_{\ell(q,p)\to 0}\left[G(q,p)-a_{n}\ell^{n-2}(q,p)\right],$ (3.14)

where $a_{n}$ is some constant that depends on $n$. Unfortunately the Robin
function defined in this way is not a smooth function unless additional
hypotheses are imposed on the Riemannian metric (see [25] for a discussion
about this definition in the context of the conformal Laplacian). Another way
to define the Robin function would be first to compute the force upon a small
Riemannian ball of radius $\epsilon$ at $p$ and then to take the limit as
$\epsilon\to 0$ to obtain $dR(p)$. This procedure may lead to quite
complicated computations as $n$ increases.

From a mathematical point of view regularity is the key property of the Robin
function, which in two dimensions is used in the definitions of vortex motion
and force upon an electric charge. In order to define the Robin function in
dimension greater than two we will regularize the $\delta$-distribution, to do
the computations in the regularized setting, and then to take the limit back
to recover the $\delta$-distribution. In order to do all these limits
independently of coordinates we use the heat equation. This procedure
naturally associates the Robin function with diffusion and Brownian motion.
This association will be further addressed in Section 4.

Let $(M,g)$ be a compact Riemannian manifold. The heat kernel $K:M\times
M\times{\mathbb{R}}_{+}\to M$ is the fundamental solution to the heat equation

$\left(\frac{\partial}{\partial
t}-\Delta_{q}\right)K(q,p,t)=0,\quad\text{with}\quad K(q,p,0)=\delta_{p}(q).$
(3.15)

The initial condition is understood as a distribution, namely, for any
$\phi\in\hbox{\rm C}^{\infty}(M)$

$\int_{M}K(q,p,t)\phi(q)\mu(q)\to\phi(p)\quad\text{as}\quad t\to 0_{+}.$

The heat kernel is a $\hbox{\rm C}^{\infty}$ symmetric, $K(q,p,t)=K(p,q,t)$,
function. Let $0<\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\ldots$ be the
nontrivial eigenvalues to the problem $\Delta\phi+\lambda\phi=0$ and
$\phi_{1},\phi_{2},\ldots$ be a corresponding $L_{2}-$orthonormal basis of
eigenfunctions for functions that integrate to zero over $M$. Then the the
spectral decomposition of the heat kernel is

$K(q,p,t)=\frac{1}{V}+\sum_{k=1}^{\infty}{\rm
e}^{-\lambda_{k}t}\phi_{k}(q)\phi_{k}(p),$

with pointwise convergence (see, for instance, [41] for basic properties of
the heat kernel). The Green’s function $G(q,p)$ is related to the heat kernel
in the following way

$G(q,p)=\int_{0}^{\infty}\left(K(q,p,t)-\frac{1}{V}\right)dt$

This is the formula that allows for the definition of the Robin function in
dimension $n>2$ by means of the regularization of the heat kernel.

As before, let $\ell(q,p)$ denote the Riemannian distance between $q$ and $p$.
There exists $\epsilon>0$ and a set of functions
$u_{0}(q,p),u_{1}(q,p),\ldots$ such that for any given integer $N\geq 0$ the
following estimate holds ( the so-called Minakshisundaram-Plejel asymptotic
expansion [32]; see Equations (7)–(9) and the accompanying text)

$\bigl{|}K(q,p,t)-\frac{{\rm e}^{\frac{-\ell^{2}(q,p)}{4t}}}{(4\pi
t)^{n/2}}\sum_{k=0}^{N}u_{k}(q,p)t^{k}\bigr{|}\leq C_{N}t^{N+1-n/2},$ (3.16)

for all $(q,p)$ with $\ell(q,p)<\epsilon$ and all $t\in(0,1)$, where $C_{N}$
is a constant that depends only on $N$ ( see, for instance, [41] exercise 5 in
Section 3.3). The functions $u_{k}$ are $\hbox{\rm C}^{\infty}$ and symmetric
$u_{k}(q,p)=u_{k}(p,q)$ [33]. If $q=p$ then the above expression implies

$K(p,p,t)=\frac{1}{(4\pi
t)^{n/2}}\left[a_{0}(p)+ta_{1}(p)+\ldots+t^{N}a_{N}(p)\right]+E_{N}(p,t)$
(3.17)

where $|E_{N}(p,t)|<C_{N}t^{N+1-n/2}$ for all $p\in M$ and $t\in(0,1)$. The
functions $a_{k}(p)=u_{k}(p,p)$ are local heat invariants of $M$ that can be
expressed in terms of powers of the Laplacian and the distance function $\ell$
([38], Theorem 1.2.1). For instance, $a_{0}(p)=1$ and $a_{1}(p)=s(p)/6$, where
$s(p)$ is the scalar curvature ([41], proof of Lemma 3.26 and Proposition
3.29, respectively).

Suppose that $n\geq 2$ is even and $N$ in equation (3.17) is chosen as
$\frac{n}{2}-1$. Then for $0<\epsilon<1$ equation (3.17) implies

$\int_{\epsilon}^{1}K(p,p,t)dt=\frac{1}{(4\pi)^{n/2}}\left[\sum_{k=0}^{\frac{n}{2}-2}a_{k}(p)\frac{\epsilon^{k+1-\frac{n}{2}}}{\frac{n}{2}-k-1}\quad-
a_{\frac{n}{2}-1}(p)\log\epsilon\right]+C(p,\epsilon)$

where $\lim_{\epsilon\to 0_{+}}C(\cdot,\epsilon)$ is a $\hbox{\rm C}^{\infty}$
function on $M$. Similarly, if $n\geq 3$ is odd and $N$ in equation (3.17) is
chosen as $\frac{n}{2}-\frac{3}{2}$ then

$\int_{\epsilon}^{1}K(p,p,t)dt=\frac{1}{(4\pi)^{n/2}}\left[\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}a_{k}(p)\frac{\epsilon^{k+1-\frac{n}{2}}}{\frac{n}{2}-k-1}\right]+\tilde{C}(p,\epsilon)$

where $\lim_{\epsilon\to 0_{+}}\tilde{C}(\cdot,\epsilon)$ is a $\hbox{\rm
C}^{\infty}$ function on $M$. These computations motivate the following
definition of the Robin function $R(p)$. If $n\geq 2$ is even then

$\begin{split}R(p)=\lim_{\epsilon\to
0_{+}}\Biggl{\\{}&\int_{\epsilon}^{\infty}\left(K(p,p,t)-\frac{1}{V}\right)dt\\\
&-\frac{1}{(4\pi)^{n/2}}\left[\sum_{k=0}^{\frac{n}{2}-2}a_{k}(p)\frac{\epsilon^{k+1-\frac{n}{2}}}{\frac{n}{2}-k-1}\
-a_{\frac{n}{2}-1}(p)\big{(}\log(4\epsilon)-\gamma\big{)}\right]\Biggr{\\}}\,,\end{split}$
(3.18)

where $\gamma=-\int_{0}^{\infty}{\rm e}^{-x}\log x\,dx=0.577215\ldots$ is the
Euler’s constant. The constant term $a_{\frac{n}{2}-1}(p)(\log
4-\gamma)/(4\pi)^{n/2}$ were added to the right-hand side of equation (3.18)
to preserve the definition of the Robin function given in equation (2.3).

If $n\geq 3$ is odd then

$\begin{split}R(p)=\lim_{\epsilon\to
0_{+}}\Biggl{\\{}&\int_{\epsilon}^{\infty}\left(K(p,p,t)-\frac{1}{V}\right)dt\\\
&-\frac{1}{(4\pi)^{n/2}}\left[\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}a_{k}(p)\frac{\epsilon^{k+1-\frac{n}{2}}}{\frac{n}{2}-k-1}\right]\Biggr{\\}}.\end{split}$
(3.19)

######  Theorem 3.1.

The Robin function can be written in the following alternative way: for $n\geq
3$ odd

$R(p)=\lim_{q\to
p}\left\\{G(q,p)-\frac{1}{(4\pi)^{n/2}}\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}u_{k}(q,p)\left(\frac{4}{\ell^{2}(q,p)}\right)^{\frac{n}{2}-k-1}\Gamma(\frac{n}{2}-k-1)\right\\}$

and for $n\geq 2$ even

$\begin{split}R(p)=&\lim_{q\to
p}\Biggl{\\{}G(q,p)-\frac{1}{(4\pi)^{n/2}}\sum_{k=0}^{\frac{n}{2}-2}u_{k}(q,p)\left(\frac{4}{\ell^{2}(q,p)}\right)^{\frac{n}{2}-k-1}\Gamma(\frac{n}{2}-k-1)\\\
&\qquad+\frac{1}{(4\pi)^{n/2}}u_{\frac{n}{2}-1}(q,p)\Bigl{[}\log\ell^{2}(q,p)\Bigr{]}\Biggr{\\}}\,.\end{split}$

Proof. At first we prove the statement for $n$ odd. From equation (3.19)

$\displaystyle R(p)$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0_{+}}\lim_{q\to
p}\Biggl{\\{}\int_{0}^{\infty}\left(K(q,p,t)-\frac{1}{V}\right)dt-\int_{0}^{\epsilon}\left(K(q,p,t)-\frac{1}{V}\right)dt$
(3.20)
$\displaystyle-\frac{1}{(4\pi)^{n/2}}\left.\left[\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}u_{k}(q,p)\frac{\epsilon^{k+1-\frac{n}{2}}}{\frac{n}{2}-k-1}\right]\right\\}$

The term $\int_{0}^{\epsilon}K(q,p,t)dt$ is estimated in the following way.
From equation (3.16)

$K(q,p,t)=\frac{{\rm e}^{\frac{-\ell^{2}(q,p)}{4t}}}{(4\pi
t)^{n/2}}\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}u_{k}(q,p)t^{k}+\Phi_{1}(q,p,t)$

such that $|\Phi_{1}(p,p,t)|\leq C_{1}t^{-1/2}$, $C_{1}>0$, for $t\in(0,1)$
(the constants $C_{1},C_{2}\ldots$ do not depend on $t$, $q$, $p$, or
$\epsilon$). Therefore

$\int_{0}^{\epsilon}K(q,p,t)dt=\frac{1}{(4\pi)^{n/2}}\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}u_{k}(q,p)\int_{0}^{\epsilon}{\rm
e}^{\frac{-\ell^{2}(q,p)}{4t}}t^{-\frac{n}{2}+k}dt+\Phi_{2}(q,p,\epsilon)$

such that $|\Phi_{2}(p,p,\epsilon)|<C_{2}\epsilon^{1/2}$, $C_{2}>0$. With the
change of variables $t=\frac{\ell^{2}}{4s}$ we obtain

$\int_{0}^{\epsilon}{\rm
e}^{\frac{-\ell^{2}}{4t}}t^{-\frac{n}{2}+k}dt=\left(\frac{\ell^{2}}{4}\right)^{k+1-\frac{n}{2}}\left[\int_{0}^{\infty}{\rm
e}^{-s}s^{\frac{n}{2}-k-2}ds-\int_{0}^{\frac{\ell^{2}}{4\epsilon}}{\rm
e}^{-s}s^{\frac{n}{2}-k-2}ds\right]$

where $\int_{0}^{\infty}{\rm
e}^{-s}s^{\frac{n}{2}-k-2}ds=\Gamma(\frac{n}{2}-k-1)$ is the Gamma function.
Using that ${\rm e}^{-s}=1-sF(s)$, where $F(s)=\int_{0}^{1}{\rm e}^{-\eta
s}d\eta$, an explicit computation gives

$\left(\frac{\ell^{2}}{4}\right)^{k+1-\frac{n}{2}}\int_{0}^{\frac{\ell^{2}}{4\epsilon}}{\rm
e}^{-s}s^{\frac{n}{2}-k-2}ds=\frac{\epsilon^{-\frac{n}{2}+k+1}}{\frac{n}{2}-k-1}-\frac{\ell^{2}}{4}\Phi_{3}\left(\frac{\ell^{2}}{4\epsilon}\right)$

where $|\Phi_{3}\left(\frac{\ell^{2}}{4\epsilon}\right)|<1$. Therefore,

$\int_{0}^{\epsilon}{\rm
e}^{\frac{-\ell^{2}}{4t}}t^{-\frac{n}{2}+k}dt=\left(\frac{\ell^{2}}{4}\right)^{k+1-\frac{n}{2}}\Gamma(\frac{n}{2}-k-1)-\frac{\epsilon^{-\frac{n}{2}+k+1}}{\frac{n}{2}-k-1}+\frac{\ell^{2}}{4}\Phi_{3}\left(\frac{\ell^{2}}{4\epsilon}\right)$

Finally, using that
$\int_{0}^{\infty}\left(K(q,p,t)-\frac{1}{V}\right)dt=G(q,p)$ and substituting
all the previous estimates into equation (3.20) we obtain

$\displaystyle R(p)$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0_{+}}\lim_{q\to
p}\left\\{G(q,p)-\frac{1}{(4\pi)^{n/2}}\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}u_{k}(q,p)\left(\frac{\ell^{2}}{4}\right)^{k+1-\frac{n}{2}}\Gamma(\frac{n}{2}-k-1)\right.$
$\displaystyle\left.+\frac{\epsilon}{V}-\Phi_{2}(q,p,\epsilon)-\frac{\ell^{2}(q,p)}{4}\frac{1}{(4\pi)^{n/2}}\sum_{k=0}^{\frac{n}{2}-\frac{3}{2}}u_{k}(q,p)\Phi_{3}\left(\frac{\ell^{2}(q,p)}{4\epsilon}\right)\right\\}.$

The limit as $q\to p$ of the second line of this equation is
$\frac{\epsilon}{V}$. Since for a fixed $\epsilon$, the limit as $q\to p$ of
the expression inside brackets exists then the limit as $q\to p$ of the sum in
the first line also exists and does not depend on $\epsilon$. So, the proof
for $n$ odd is finished.

The proof for $n$ even is similar. The only difference is that it is necessary
to estimate the additional integral

$\int_{0}^{\epsilon}{\rm
e}^{\frac{-\ell^{2}}{4t}}t^{-1}dt=\int_{\frac{\ell^{2}}{4\epsilon}}^{\infty}{\rm
e}^{-s}s^{-1}ds=-\log\left(\frac{\ell^{2}}{4\epsilon}\right)+\int_{\frac{\ell^{2}}{4\epsilon}}^{\infty}{\rm
e}^{-s}\log s\,ds.$

The last integral is equal to minus the Euler’s constant as $q\to p$. $\Box$

We remark that for $n>2$ the term of highest order in $\ell^{-2}$ is

$\frac{1}{(4\pi)^{n/2}}\left(\frac{4^{\frac{n}{2}-1}}{\ell^{n-2}(q,p)}\right)\Gamma(\frac{n}{2}-1)\,,$

where we used that $u_{0}(p,p)=a_{0}(p)=1$, is minus the “Newtonian potential”
that appears in equation (3.14).

For $n=2$, theorem 3.1 states that the Robin function defined by equation
(3.18) coincides with that given in equation (2.3).

Theorem 3.1 can also be obtained from the Hadamard parametrix, see [19]
section 5.3.

The Robin function as given in Theorem 3.1 can be written in terms of the
analytic extension of the Minakshisundaram–Pleijel zeta function, [43]
(dimension two) and [4] (dimension greater than one). The relation between the
Robin function and the zeta-function is presented in Appendix B.

##  4 The “Narrow Escape Time (NET)”.

In the context of a compact boundaryless manifold $M$, the narrow escape
problem can be described in the following way. Consider a Brownian motion on
$M$, whose infinitesimal generator is the Laplace-Beltrami operator $\Delta$.
Let $B_{\epsilon}(q)\subset M$ be a geodesic ball of small radius
$\epsilon>0$. This ball will be the absorbing set or the small window through
which a particle can escape. The amount of time that a particle initially at
$p$ is expected to spend in $M\backslash B_{\epsilon}(q)$ (the mean sojourn
time) will be denoted as $v_{\epsilon}(p,q)$. This function is the “narrow
escape time” (NET) since it measures the mean time it takes for a particle
initially at $p$ to escape through the narrow window $B_{\epsilon}(q)$. The
NET is the solution to the problem (see [26], equation 3.1):

$\begin{split}&\nu\Delta_{p}v_{\epsilon}(p,q)=-1\,,\quad p\in M\backslash
B_{\epsilon}(q)\,,\\\ &\text{with}\quad
v_{\epsilon}(p,q)=0\quad\text{for}\quad p\in\partial
B_{\epsilon}(q)\,,\end{split}$ (4.21)

where $\nu$ is a diffusion coefficient with dimensional units
length${}^{2}/$time. The NET averaged against a uniform distribution of
initial points in $M\backslash B_{\epsilon}(q)$,

$\overline{v}_{\epsilon}(q)=\frac{1}{V-|B_{\epsilon}(q)|}\int_{M\backslash
B_{\epsilon}(q)}v_{\epsilon}(p,q)\mu(p),$ (4.22)

gives the expected time a particle randomly placed in the manifold remains in
it until it scapes through $B_{\epsilon}(q)$.

In dimension 2 the following theorem was proved in [11] (Lemma 4.1 and Theorem
4.2 part 2).

######  Theorem 4.1.

In dimensions 2, 3, and 4, the “Narrow Escape Time” (NET) is given by

$v_{\epsilon}(p,q)=-\frac{V}{\nu}\,G(p,q)+\overline{v}_{\epsilon}(q)+E(p,q,\epsilon)\,,$
(4.23)

where $\lim_{\epsilon\to 0}E(p,q,\epsilon)=0$. The average NET, equation
(4.22), is given by

$\begin{array}[]{rll}\overline{v}_{\epsilon}(q)&=\frac{V}{\nu}\left\\{-\frac{1}{2\pi}\log\epsilon+R(q)+E_{2}(q,\epsilon)\right\\}&n=2\\\
\overline{v}_{\epsilon}(q)&=\frac{V}{\nu}\left\\{\frac{1}{4\pi}\epsilon^{-1}+R(q)+E_{3}(q,\epsilon)\right\\}&n=3\\\
\overline{v}_{\epsilon}(q)&=\frac{V}{\nu}\left\\{\frac{1}{4\pi^{2}}\epsilon^{-2}-\frac{1}{48\pi^{2}}S(q)\log\epsilon+\frac{1}{192\pi^{2}}S(q)+R(q)+E_{4}(q,\epsilon)\right\\}&n=4\,,\end{array}$

where $S(q)$ is the scalar curvature at $q$ and $\lim_{\epsilon\to
0}E_{n}(p,q,\epsilon)=0$.

Remarks:

$\bullet$ The normalization $\int_{M}G(p,q)\mu(p)=0$ (equation (2.2)) ensures
the compatibility of both sides of equation (4.23).

$\bullet$ The NET increases as $\epsilon$ decreases in the same way as the
Newtonian potential in ${\mathbb{R}}^{n}$ increases as the distance to the
singularity decreases (see e.g. [26], section 3, for the same result for
surfaces). This is true in all dimensions, not only $n=2,3,4$.

$\bullet$ In dimensions 2 and 3, the divergent terms of
$\overline{v}_{\epsilon}(q)$ with respect to $\epsilon$ do not depend on $q$.
For $n=4$ this is no longer true, since $\overline{v}_{\epsilon}(q)$ contains
a logarithmic divergent term that is proportional to the mean curvature
$S(q)$. If the mean curvature is constant on $M$, then the dependence of
$\overline{v}_{\epsilon}(q)$ on $q$ as $\epsilon\to 0$ is determined by the
Robin function, as it is in dimensions 2 and 3.

Proof. We will prove only the case n=4. The proof of the cases n=2 and n=3 is
simpler and goes along the same lines.

We write
$v_{\epsilon}(p,q)=-\frac{V}{\nu}\,G(p,q)+\frac{V}{\nu}h_{\epsilon}(p,q)$ and
from equations (2.1) and (4.21) we obtain

$\Delta_{p}h_{\epsilon}(p,q)=0\,,\quad p\in M\backslash
B_{\epsilon}(q)\,,\quad\text{with}\quad
h_{\epsilon}(p,q)=G(p,q)\quad\text{for}\quad p\in\partial B_{\epsilon}(q)\,.$
(4.24)

Let $x\in{\mathbb{R}}^{n}$ be an orthonormal coordinate system on the tangent
space of $M$ at $q$. Let $x(p):=\exp^{-1}_{q}p$ be geodesic normal coordinates
in $M$ defined in a neighborhood of $q$. The metric tensor in this coordinates
is given by $g_{ij}(x)=\delta_{ij}-\frac{1}{3}R_{ikjl}x^{k}x^{l}+{\cal
O}(|x|^{3})$. Theorem 3.1 implies that for $p$ sufficiently close to $q$

$G(p,q)=\frac{u_{0}(p,q)}{(4\pi)^{2}}\left(\frac{4}{|x|^{2}}\right)-\frac{u_{1}(p,q)}{(4\pi)^{2}}\log|x|^{2}+R(q)+{\cal
R}_{1}(p,q)\,.$

From $u_{0}(p,q)=1/\sqrt{{\rm det}(\exp_{q}p)}$ ([41] equation (3.11)) we
obtain

$\begin{split}u_{0}(q,p)&=\frac{1}{({\rm
det}\,g)^{1/4}}=1-\frac{1}{12}R_{kil}^{i}(q)x^{k}x^{l}+{\cal O}(|x|^{3})\\\
&=1-\frac{1}{48}S(q)|x|^{2}-\frac{1}{12}Z_{kl}(q)x^{k}x^{l}+{\cal
O}(|x|^{3})\end{split}$

where: $R_{jkl}^{i}$ is the Riemman curvature tensor and $Z_{kl}$ is the
traceless Ricci tensor. From [41] Proposition 3.29,

$u_{1}(p,q)=u_{1}(q,q)+{\cal O}(|x|)=\frac{1}{6}S(q)+{\cal O}(|x|)\,.$

The expressions in the previous paragraph imply that for $p$ sufficiently
close to $q$

$G(p,q)=\underbrace{\frac{1}{4\pi^{2}|x|^{2}}-\frac{S(q)}{192\pi^{2}}-\frac{S(q)}{48\pi^{2}}\log|x|+R(q)}_{\text{term
a}}-\underbrace{\frac{Z_{kl}(q)}{48\pi^{2}}\frac{x^{k}x^{l}}{|x|^{2}}}_{\text{term
b}}+\underbrace{{\cal R}_{2}(p,q)}_{\text{term c}}\,,$ (4.25)

where $\lim_{|x|\to 0}{\cal R}_{2}(p,q)=0$.

The solution $h_{\epsilon}$ to the problem in equation (4.24) can be split
into three terms a, b, and c, according to the decomposition of the boundary
conditions as given in equation (4.25). The term a is constant for
$|x|=\epsilon$. This term appears in the expression for
$\overline{v}_{\epsilon}(q)$ in the statement of the theorem.

The maximum principle dictates that the maximum of the function
$p\mapsto|h_{c}(p,q,\epsilon)|$, where $h_{c}(p,q,\epsilon)$ is the solution
to

$\Delta_{p}h_{c,\epsilon}(p,q)=0,\quad p\in M\setminus
B_{\epsilon}(q),\quad\text{with}\quad h_{c,\epsilon}(p,q)={\cal
R}_{2}(p,q)\quad\text{for}\quad p\in\partial B_{\epsilon}(q),$

is attained on $\partial B_{\epsilon}(q)$. Given that
$h_{c,\epsilon}(p,q)={\cal R}_{2}(p,q)$ for $p\in\partial B_{\epsilon}(q)$,
the limit $\epsilon\to 0$ implies $p\to q$, and since $\lim_{p\to q}{\cal
R}_{2}(p,q)=0$, it follows that $\lim_{\epsilon\to 0}|h_{c}(p,q,\epsilon)|\to
0$ for $p\in M\setminus B_{\epsilon}(q)$. Consequently, the component of
$h_{\epsilon}$ corresponding to term (c) in equation (4.25) contributes to the
function $E(q,p,\epsilon)$ as stated in the theorem.

The part of $h_{\epsilon}$ associated the term b in equation (4.25), will be
denoted as $H_{\epsilon}(p,q)$. It satisfies the problem

$\Delta_{p}H_{\epsilon}(p,q)=0\quad\text{with}\quad
H_{\epsilon}(p,q)=-\frac{Z_{kl}(q)}{192\pi^{2}}\frac{x^{k}x^{l}}{\epsilon^{2}}\quad\text{for}\quad|x|=\epsilon\,.$
(4.26)

In order to finish the proof we must show that $|H_{\epsilon}(p,q)|\to 0$ as
$\epsilon\to 0$ $|H_{\epsilon}(p,q)|\to 0$ as $\epsilon\to 0$Y. The proof has
several steps.

######  Proposition 4.1.

$\int_{|x|=\epsilon}Z_{kl}(q)\frac{x^{k}x^{l}}{\epsilon^{2}}d\sigma(x)={\cal
O}(\epsilon^{5})\,,$

where $d\sigma(x)$ is the “area form” on the geodesic sphere $\partial
B_{\epsilon}(q)$.

Proof. The area form on $\partial B_{\epsilon}(q)$ satisfies
$d\sigma(x)=[1-\frac{1}{6}R^{i}_{kil}x^{k}x^{l}+{\cal
O}(|x|^{3})]d\sigma_{E}(x)$, where $d\sigma_{E}(x)$ is the Euclidean area form
on the sphere $|x|=\epsilon$. The function $Z_{kl}(q)x^{k}x^{l}$ is harmonic
with respect to the Euclidean Laplacian, since the trace of $Z$ is zero, and
therefore its integral over $|x|=\epsilon$ with respect to $d\sigma_{E}(x)$ is
zero. The proposition follows from the expression for $d\sigma(x)$ and
$|d\sigma_{E}|={\cal O}(\epsilon^{3})$ on $|x|=\epsilon$. $\Box$

The identity $\Delta_{z}[G(z,q)-G(z,p)]=\delta_{p}(z)$, for $z$ and $p$ in
$M\backslash B_{2\epsilon}(q)$, and Green’s second identity imply that for
$\epsilon$ sufficiently small

$\begin{split}H_{\epsilon}(p,q)=&\int_{\partial
B_{2\epsilon}}H_{\epsilon}(x,q)\nabla_{x}[G(x,q)-G(x,p)]\cdot\frac{x}{|x|}d\sigma(x)\\\
&-\int_{\partial
B_{2\epsilon}}[G(x,q)-G(x,p)]\nabla_{x}H_{\epsilon}(x,q)\cdot\frac{x}{|x|}d\sigma(x)\,.\end{split}$
(4.27)

We will first estimate the integral in the second line of equation (4.27).
Equation (4.25) implies that $G(x,q)-G(x,p)$ with $|x|=2\epsilon$ can be
written as a term $A_{1}={\cal O}(\epsilon^{2})$ that does not depend on $x$
and a term $A_{2}$ that is bounded by a constant $C_{1}(q)$ that is
independent of $x$ and $\epsilon$. The integral $A_{1}\int_{\partial
B_{2\epsilon}}\nabla_{x}H_{\epsilon}(x,q)\cdot\frac{x}{|x|}d\sigma(x)=0$
because $\int_{M\backslash
B_{2\epsilon}(q)}\Delta_{p}H_{\epsilon}(p,q)\mu(p)=0$. In order to estimate
the integral that contains $A_{2}$ we will use one of the Schauder interior
estimates [20] (Corollary 6.3)222Here is the reason for having integrated over
the domain $M\backslash B_{2\epsilon}(q)$ and not $M\backslash
B_{\epsilon}(q)$.

$\epsilon\max_{|x|=2\epsilon}\left|\nabla_{x}H_{\epsilon}(z,q)\cdot\frac{x}{|x|}\right|\leq\max_{p\in
M\backslash
B_{\epsilon}(q)}\left|H_{\epsilon}(p,q)\right|\leq\max_{|x|=\epsilon}\left|H_{\epsilon}(x,q)\right|=C_{2}(q)\,,$

where the second inequality follows from the maximum principle and the
constant $C_{2}(q)$ does not depend on $\epsilon$. So
$\left|\int_{|x|=2\epsilon}A_{2}\nabla_{x}H_{\epsilon}(x,q)\cdot\frac{x}{|x|}d\sigma(x)\right|={\cal
O}(\epsilon^{2})$ and the integral in the second line of equation (4.27) is at
most of the order of $\epsilon^{2}$. It remains to estimate the integral in
the first line of equation (4.27).

For a fixed $p\in M\backslash B_{2\epsilon}(q)$ the function
$\nabla_{x}[G(x,p)]\cdot\frac{x}{|x|}$ restricted to $|x|=2\epsilon$ is
uniformly bounded with respect to $\epsilon$. Therefore, using that
$|H_{\epsilon}(x,q)|<C_{2}(q)$, we obtain
$\int_{|x|=2\epsilon}H_{\epsilon}(x,q)\nabla_{x}[G(x,p)]\cdot\frac{x}{|x|}d\sigma(x)={\cal
O}(\epsilon^{3})$ and it remains to estimate $\int_{\partial
B_{2\epsilon}}H_{\epsilon}(x,q)\nabla_{x}[G(x,q)]\cdot\frac{x}{|x|}d\sigma(x)$.

The term $\frac{1}{4\pi^{2}|x|^{2}}$ in equation (4.25) is the leading order
term of a parametrix for the Laplace equation (see [18], equation (5.79), or
[2], theorem 4.13 equation (17)). This implies that
$G(x,q)-\frac{1}{4\pi^{2}|x|^{2}}$, where $G(x,q)$ is given in equation
(4.25), can be differentiated for $x\neq 0$ and the derivative of ${\cal
R}_{2}(x,q)$ is dominated by those of the other terms, so that
$|\nabla_{x}[G(x,q)-\frac{1}{4\pi^{2}|x|^{2}}]\cdot\frac{x}{|x|}|={\cal
O}(1/|x|)$. This and $|H_{\epsilon}(x,q)|<C_{2}(q)$ imply

$\int_{|x|=2\epsilon}H_{\epsilon}(x,q)\nabla_{x}G(x,q)\cdot\frac{x}{|x|}d\sigma=\int_{|x|=2\epsilon}H_{\epsilon}(x,q)\nabla_{x}\left[\frac{1}{4\pi^{2}|x|^{2}}\right]\cdot\frac{x}{|x|}d\sigma+{\cal
O}(\epsilon^{2})\,.$

It remains to estimate the integral in the right-hand side of this equation.

Green’s second identity with $\Delta_{x}H_{\epsilon}(x,q)=0$ and
$\nabla_{x}\left[\frac{1}{4\pi^{2}|x|^{2}}\right]\cdot\frac{x}{|x|}=-\frac{1}{2\pi^{2}|x|^{3}}$,
which is valid because $x$ are normal coordinates, imply

$\begin{split}&\int_{B_{2\epsilon}(q)\backslash
B_{\epsilon}(q)}H_{\epsilon}(x,q)\Delta\left[\frac{1}{4\pi^{2}|x|^{2}}\right]dx^{4}=\\\
&-\frac{1}{16\pi^{2}\epsilon^{3}}\int_{|x|=2\epsilon}H_{\epsilon}(x,q)d\sigma+\frac{1}{2\pi^{2}\epsilon^{3}}\int_{|x|=\epsilon}H_{\epsilon}(x,q)d\sigma\\\
&-\frac{1}{16\pi^{2}|\epsilon|^{2}}\int_{|x|=2\epsilon}\nabla
H_{\epsilon}(x,q)\cdot\frac{x}{|x|}d\sigma+\frac{1}{4\pi^{2}|\epsilon|^{2}}\int_{|x|=\epsilon}\nabla
H_{\epsilon}(x,q)\cdot\frac{x}{|x|}d\sigma\,.\end{split}$

The integrals in the last line are zero because $\int_{M\backslash
B_{s\epsilon}(q)}\Delta_{p}H_{\epsilon}(p,q)\mu(p)=0$, for $s=1,2$. Due to
equation (4.26) and Proposition 4.1,
$\frac{1}{2\pi^{2}\epsilon^{3}}\int_{|x|=\epsilon}H_{\epsilon}(x,q)d\sigma={\cal
O}(\epsilon^{2})$. A computation using the expression for the Laplacian in
geodesic normal coordinates [41] (Theorem 2.63) gives
$\Delta\left[\frac{1}{4\pi^{2}|x|^{2}}\right]={\cal O}(|x|^{-2})$. This and
$|H_{\epsilon}(x,q)|<C_{2}(q)$ imply $\int_{B_{2\epsilon}(q)\backslash
B_{\epsilon}(q)}H_{\epsilon}(x,q)\Delta\left[\frac{1}{4\pi^{2}|x|^{2}}\right]dx^{4}={\cal
O}(\epsilon^{2})$. In conclusion, all these estimates imply

$\int_{|x|=2\epsilon}H_{\epsilon}(x,q)\nabla_{x}\left[\frac{1}{4\pi^{2}|x|^{2}}\right]\cdot\frac{x}{|x|}d\sigma=-\frac{1}{16\pi^{2}\epsilon^{3}}\int_{|x|=2\epsilon}H_{\epsilon}(x,q)d\sigma={\cal
O}(\epsilon^{2})\,,$

which finishes the proof. $\Box$

##  5  Examples of non-constant curvature uniform drainage surfaces:
Okikiolu’s tori.

The flat metric $g_{0}$ on any two-dimensional torus is a steady vortex metric
(SVM). Equation (2.5) implies that there exists a second SVM $g_{1}$ conformal
to $g_{0}$, $g_{1}=\lambda^{2}g_{0}$, if and only if

$\left(\frac{K_{1}}{2\pi}-\frac{2}{V_{1}}\right)\mu_{1}=-\frac{2}{V_{0}}\mu_{0}$
(5.28)

Normalizing the volumes $\mu_{0}$ and $\mu_{1}$ such that $V_{0}=V_{1}=1$,
using $-\Delta_{0}\log\lambda=\lambda^{2}K_{1}$ and
$\mu_{1}=\lambda^{2}\mu_{0}$, and defining $f=\log\lambda^{2}$ we get the
following equation for $f$

$\Delta_{0}f=8\pi-8\pi{\rm e}^{f}$ (5.29)

To each nontrivial solution to this equation corresponds a SVM $g_{1}$
conformal to $g_{0}$.

In the following we present a family of examples due to Okikiolu [36] of non
flat 2-dimensional tori that have constant Robin function, and so are uniform
drainage surfaces. Each non flat torus in the family is conformal to a flat
torus, which is also a uniform drainage surface. The Robin function of the non
flat tori are smaller than those of the conformally equivalent flat tori, and
so the narrow escape time of the non flat tori are smaller than those of the
conformally equivalent flat tori. There are two differences between our
presentation and that of Okikiolu. We simplify the proof that the Robin
functions of the non flat tori are smaller than those of the flat tori and we
represent the non flat tori in ${\mathbb{R}}^{3}$ as the quotient of an
isometrically embedded cylinder.

Consider the torus ${\mathbb{R}}^{2}/(a{\mathbb{Z}}\times
a^{-1}{\mathbb{Z}})$, $a\geq 1$, with the conformal structure induced by the
flat metric $g_{0}$. If $a\leq 2/\sqrt{\pi}$, then $g_{0}$ is the unique
uniform drainage metric [35], and if $a>\sqrt{\pi/2}$, then $g_{0}$ is not
unique [31]. When $a>\sqrt{\pi/2}$ a second natural vortex metric can be
constructed in the following way [36]. Let $(x,y)$ be Cartesian coordinates on
${\mathbb{R}}^{2}$. We will look for a nontrivial solution to equation (5.29)
that depends only on the variable $x$, $\partial_{y}f=0$, with $f(x+a)=f(x)$.
Then $f$ must satisfy $\ddot{f}:=\frac{d^{2}f}{dx^{2}}=8\pi(1-{\rm e}^{f})$.
This ordinary differential equation has a single equilibrium and a first
integral

$H(f,p)=p^{2}/2+8\pi({\rm e}^{f}-f-1)\,,\quad p=\dot{f}\,.$ (5.30)

This shows that all solutions $f$ are periodic with a period $T(E)$, where $E$
is the value of the first integral associated to the solution. The linearized
period at $(f,\dot{f})=(0,0)$ is $T(0)=\sqrt{\pi/2}$.

The period function $E\to T(E)$ of equation $\ddot{f}=8\pi(1-{\rm e}^{f})$ was
studied in [8] (p. 315), where it is shown that $\frac{d}{dE}T(E)>0$. We will
additionally show that $\lim_{E\to\infty}T(E)=\infty$. Consider the solution
associated to the initial condition $f(0)=0$, $\dot{f}(0)=-\sqrt{2E}$ and
integrate the equation $\ddot{f}=8\pi(1-{\rm e}^{f})$ on the interval
$[0,\beta]$, where $\beta>0$ is the smallest value such that $f(\beta)=0$.
Since $\dot{f}(\beta)=\sqrt{2E}$, the result is

$\sqrt{2E}/(4\pi)=\beta-\int_{0}^{\beta}{\rm e}^{f}dt<\beta<T(E)\,,$ (5.31)

and therefore $\lim_{E\to\infty}T(E)=\infty$. As a result, equation (5.29) has
nontrivial solutions for all $a>\sqrt{\pi/2}$ such that $f(x+a)=f(x)$ (indeed
as many different solutions as we wish provided $a$ is sufficiently large).

For a given $a>\sqrt{\pi/2}$, let $g_{1}={\rm e}^{f(x)}(dx^{2}+dy^{2})$ be the
metric associated to a periodic solution to $\ddot{f}=8\pi(1-{\rm e}^{f})$
with minimal period $a$. We will use lemma A.1 to show that the Robin function
$R_{1}$ associated to $g_{1}$ has a smaller value than the Robin function
$R_{0}$ of the flat metric. The area form associated to $g_{1}$ is given by
$\mu_{1}={\rm e}^{f}\mu_{0}=\left[1-\frac{\ddot{f}}{8\pi}\right]dx\wedge dy$
and the equation that determines the function $\phi$ in lemma A.1 becomes

$\Delta_{0}\phi\ \ dx\wedge dy=\mu_{1}-\mu_{0}=-\frac{\ddot{f}}{8\pi}dx\wedge
dy\,,\qquad\int_{S}\phi\mu_{0}=0$

that implies

$\phi(x)=-\frac{f(x)}{8\pi}+\frac{1}{8\pi a}\int_{0}^{a}f(x)dx\,.$

The constant $c=-\frac{1}{V}\int_{S}\phi(p)\mu_{1}(p)$ in lemma A.1 can be
easily computed and is equal to
$c=\frac{1}{(8\pi)^{2}a}\int_{0}^{a}\dot{f}^{2}dx$. These computations and
equation (A.38) imply

$R_{1}-R_{0}=\frac{1}{4\pi
a}\int_{0}^{a}fdx+\frac{1}{(8\pi)^{2}a}\int_{0}^{a}\dot{f}^{2}dx$ (5.32)

If we use the first integral $H$ in equation (5.30) to eliminate $f$ in the
right-hand side of this equation and then use $\frac{1}{a}\int_{0}^{a}{\rm
e}^{f}dx=1$, which we obtain integrating $\ddot{f}=8\pi(1-{\rm e}^{f})$ over
the interval $[0,a]$, then

$R_{1}-R_{0}=-\frac{H}{32\pi^{2}}+\frac{1}{32\pi^{2}a}\int_{0}^{a}\dot{f}^{2}dx\,.$
(5.33)

The equation $\ddot{f}=8\pi(1-{\rm e}^{f})$ can be written in Hamiltonian form
with Hamiltonian function $H$. Using the definition of the action
$I(E)=\frac{1}{2\pi}\oint pdf$ from Hamiltonian mechanics [1] we can write

$\frac{1}{32\pi^{2}a}\int_{0}^{a}\dot{f}^{2}dx=\frac{1}{32\pi^{2}a}\int_{0}^{a}p\dot{f}dx=\frac{1}{16\pi
a}\frac{1}{2\pi}\oint pdf=\frac{I(E)}{16\pi a}$

In this expression $a$ is the period of $f$, and therefore $a=T(E)$ where $E$
is the value of $H$ associated to $f$. The Hamiltonian function can be written
as a function of the action $E=H(I)$ with $H^{\prime}(I)=2\pi/T(I)$. All these
results imply that equation (5.33) can be written as

$R_{1}-R_{0}=\frac{1}{32\pi^{2}}\left(IH^{\prime}(I)-H(I)\right)$ (5.34)

Since $H^{\prime}(I)=2\pi/T(I)>0$ and $T^{\prime}(I)>0$ [8] (p. 315), we
conclude that $H^{\prime\prime}(I)=-2\pi T^{\prime}(I)/T^{2}(I)<0$. This fact
and $H(0)=0$ imply that $R_{1}-R_{0}<0$. In Figure 1 we present a numerical
estimate of the difference $R_{1}-R_{0}$.

Figure 1: LEFT: Difference $R_{1}-R_{0}$ as a function of $a$, where $R_{1}$
($R_{0}$) is the Robin function of the non flat torus
$\\{{\mathbb{R}}^{2}/(a{\mathbb{Z}}\times a^{-1}{\mathbb{Z}}),g_{1}\\}$ (flat
torus $\\{{\mathbb{R}}^{2}/(a{\mathbb{Z}}\times
a^{-1}{\mathbb{Z}}),g_{0}\\}$). RIGHT: Graphs of $R_{1}$ and $R_{0}$ as a
function of $a$. The horizontal line represents the value of the Robin
function $R_{S}$ for a round sphere of area 1. According to [36] (Appendix):
$R_{0}(a)=-\frac{\log(2\pi)}{2\pi}-\frac{\log(|\eta(ia^{2})|^{4}a^{2})}{4\pi}$
and $R_{S}=-\frac{1+\log\pi}{4\pi}$, where $\eta$ is the Dedekind eta
function.

The torus $\big{\\{}{\mathbb{R}}^{2}/(a{\mathbb{Z}}\times
a^{-1}{\mathbb{Z}}),g_{1}\big{\\}}$ can be represented as the quotient of a
cylinder that is infinite along the $x$-axis and periodic with period $a$. We
will show that this cylinder can be isometrically embedded in the Euclidean
three-space. Let $X,Y,Z$ be Cartesian coordinates in ${\mathbb{R}}^{3}$. We
will look for an embedding of the form $X=X(x)$, $Y=F(x)\sin(2\pi ay)$ and
$Z=F(x)\cos(2\pi ay)$, where
$x\in{\mathbb{R}}\,,y\in{\mathbb{R}}/a^{-1}{\mathbb{Z}}$. The pull-back of the
Euclidean metric by the embedding is
$(\dot{X}^{2}+\dot{F}^{2})dx^{2}+4\pi^{2}a^{2}F^{2}dy^{2}$. We impose that the
pull-back coincides with $g_{1}={\rm e}^{f(x)}(dx^{2}+dy^{2})$ and obtain that
$4\pi^{2}a^{2}F^{2}={\rm e}^{f}$ and $\dot{X}^{2}+\dot{F}^{2}={\rm e}^{f}$.
This implies that $F(x)={\rm e}^{f(x)/2}/(2\pi a)$ and

$\dot{X}^{2}={\rm e}^{f}\left(1-\frac{\dot{f}^{2}}{16\pi^{2}a^{2}}\right)$
(5.35)

Since $X:{\mathbb{R}}\to{\mathbb{R}}$ must be a diffeomorphism, the right-hand
side of equation (5.35) must be strictly positive for all $x\in[0,a]$. We will
show this in the following paragraph.

The first integral (5.30) and $({\rm e}^{f}-f-1)\geq 0$ imply that
$\dot{f}^{2}(x)\leq 2E$ for $x\in[0,a]$, where $E$ is the value of $H$ for the
solution with period $T(E)=a$. This and inequality (5.31) imply

$1-\frac{\dot{f}^{2}}{16\pi^{2}a^{2}}\geq 1-\frac{2E}{16\pi^{2}T(E)^{2}}>0\,.$

In Figure 2 we show the curves $x\to\\{X(x),Z(x)\\}$, $x\in[0,a]$ and $y=0$,
that when rotated about the $X-$axis generate the embedded cylinders. These
curves were obtained by the numerical integration of equations
$\ddot{f}=8\pi(1-{\rm e}^{f})$ and (5.35) for: $a=1.255$, $a=1.50$, and
$a=3.0$. Only one fundamental cell of the periodic cylinder is shown. There
are two different tori with $a=3$: one for which the minimal period of $f$ is
$3$ and another for which the minimal period of $f$ is $1.5$, and so $f$
oscillates twice inside a fundamental cell. In Figure 3 we show a
3-dimensional representation of a single cell of each one of the cylinders
whose generators are in Figure 2. It is clear from Figure 3 that for $a\gg 1$
the cylinder becomes a collection of aligned spheres each one touching its
neighbors at a single point. This is in agreement with the interpretation
given in [36]: (the non-flat torus) “is approximately spherical except for a
short wormhole joining the poles”. Note: as shown in the right panel of Figure
1, in the limit as $a\to\infty$ the tori converge to a punctured sphere and
$R_{1}(a)\to R_{S}$ where $R_{S}$ is the Robin function of the round sphere.

Figure 2: Generating functions of four periodic cylinders (each cylinder is
constructed rotating the graph of $X\to Z(X)$ about the $X$-axis). The
quotient of a cylinder by the group of periodic translations gives a torus
that is isometric to a non-flat torus with a steady-vortex metric. The value
of the period $a$ of each torus is shown in the corresponding figure. There
are two different tori with $a=3$: one for which the minimal period of $f$ is
$3$ and another for which the minimal period of $f$ is $1.5$, and so $f$
oscillates twice inside a fundamental cell.

Figure 3: Three-dimensional representation of the tori whose generators are
shown in Figure 2. See the caption of Figure 2 for explanations.

##  Appendix A Proofs of theorems 2.1 and 2.2.

######  Lemma A.1.

Let $g_{0}$ and $g_{1}$ be two different Riemannian metrics on $S$ in the same
conformal class, $g_{1}=\lambda^{2}g_{0}$ . Let $G_{j}$, $R_{j}$, $\mu_{j}$,
$K_{j}$, $\Delta_{j}$, $j=0,1$, be the: Green’s function, Robin function,
volume form, Gaussian curvature, and Laplace operator, of $g_{j}$. Let the
conformal factor $\lambda$ be normalized such that the volumes
$\int_{S}\mu_{0}=\int_{S}\mu_{1}=V$ are the same. Let $\phi$ be the unique
solution of

$d\ast
d\phi=\frac{\mu_{1}-\mu_{0}}{V}\quad\text{with}\quad\int_{S}\phi\mu_{0}=0,$

that is given by

$\phi(p)=-\frac{1}{V}\int_{S}G_{0}(q,p)\lambda^{2}(q)\mu_{0}(q)=-\frac{1}{V}\int_{S}G_{0}(q,p)\mu_{1}(q)$
(A.36)

Then $G_{0}$, $G_{1}$, $R_{0}$ and $R_{1}$ satisfy the following relations:

$\displaystyle G_{1}(q,p)-G_{0}(q,p)=\phi(q)+\phi(p)+c$ (A.37) $\displaystyle
R_{1}(p)=R_{0}(p)+\frac{1}{2\pi}\log\lambda(p)+2\phi(p)+c$ (A.38)

where

$c=-\frac{1}{V}\int_{S}\phi(p)\mu_{1}(p)=\frac{1}{V^{2}}\int_{S}\int_{S}G_{0}(q,p)\mu_{1}(q)\mu_{1}(p)$

is a constant. Equation (A.37) is in [34] (equation (8)) and Equation (A.38)
is in [43] (Theorem 4).

Proof. Let $p$ and $q$ be sufficiently close to be in a domain $U$ of a local
uniformizer $z$. Suppose that $U$ is such that any two points in $U$ are
connected by a single geodesic in $U$. In this coordinates the length elements
of the metrics $g_{0}$ and $g_{1}$ are $\lambda_{0}|dz|$ and
$\lambda_{1}|dz|$, respectively. Notice that $\lambda_{1}=\lambda\lambda_{0}$.
If $\mu=dx\wedge dy$ and $\Delta=\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}$ denote the area form and the usual
Laplacian in the coordinates $z=(x,y)$, respectively, then

$\Delta_{j}=\frac{1}{\lambda_{j}^{2}}\Delta,\qquad-\Delta\log\lambda_{j}=\lambda_{j}^{2}K_{j},\qquad\mu_{j}=\lambda^{2}_{j}\mu,\qquad
j=0,1.$ (A.39)

The Dirac-delta distributions associated to the volume forms $\mu_{0}$ and
$\mu_{1}$ satisfy

$\delta_{j,w}=\frac{1}{\lambda^{2}_{j}}\delta_{w}\,,\quad\text{where}\quad\psi(w)=\int\psi(x,y)\delta_{w}(x,y)dx\wedge
dy\,.$

To simplify the notation we write $z(q)=z$ and $z(p)=w$. In the coordinates
$(z,w)$ equation (2.1) becomes

$-\Delta_{z}G_{j}(z,w)=\delta_{w}(z)-\frac{\lambda_{j}^{2}(z)}{V}\,.$ (A.40)

The Green’s function can be written as

$G_{j}(z,w)=-\frac{1}{2\pi}\log|z-w|+f_{j}(z,w)$ (A.41)

where $f_{j}(z,w)=f_{j}(w,z)$. Since $\Delta_{z}\log|z-w|=2\pi\delta_{w}$, we
obtain

$\Delta_{z}f_{j}(z,w)=\frac{\lambda^{2}_{j}(z)}{V}.$ (A.42)

Let $\ell_{j}(z,w)$ be the length with respect to the metric $g_{j}$ of the
unique geodesic connecting $z$ to $w$. It can be shown that (see for instance
[40] proof of Theorem 5.1):

$\ell_{j}(z,w)=|w-z|\sqrt{\lambda_{j}(z)\lambda_{j}(w)}[1+{\cal O}(|z-w|)]$

Therefore

$G_{j}(z,w)+\frac{1}{2\pi}\log\ell_{j}(z,w)=f_{j}(z,w)+\frac{1}{4\pi}\log[\lambda_{j}(z)\lambda_{j}(w)]+{\cal
O}(|z-w|).$

Taking the limit as $|z-w|\to 0$ we obtain

$R_{j}(z)=f_{j}(z,z)+\frac{1}{2\pi}\log\lambda_{j}(z).$ (A.43)

If we subtract equation (A.40) for $j=0$ from that for $j=1$ we obtain

$\Delta_{z}G_{1}(z,w)-\Delta_{z}G_{0}(z,w)=\frac{\lambda_{1}^{2}(z)-\lambda_{0}^{2}(z)}{V}\,.$
(A.44)

This equation can be written intrinsically in terms of two-forms as

$d_{q}\ast d_{q}G_{1}(q,p)-d_{q}\ast
d_{q}G_{2}(q,p)=\frac{\mu_{1}-\mu_{0}}{V}=d_{q}\ast d_{q}\phi(q)$

where $\phi$ is the function in the statement of the theorem. Thus
$G_{1}(q,p)-G_{0}(q,p)=\phi(q)+\psi(p)$ that, due to the symmetry
$G_{j}(p,q)=G_{j}(q,p)$, implies equation (A.37). Equation $d_{q}\ast
d_{q}\phi(q)=\frac{\mu_{1}-\mu_{0}}{V}$ can be written as
$\Delta_{0}\phi=(\lambda^{2}-1)/V$. The representation formula (2.2) for
$\phi$ plus the relations $\int_{S}\phi\mu_{0}=0$ and
$\int_{S}G_{0}(q,p)\mu_{0}(q)=0$ imply that $\phi$ can be written as in
equation (A.36). Integrating both sides of equation (A.37) with respect to
$\mu_{1}(q)$ over $S$ we obtain the expression for $c$ in the lemma. In the
$z$-coordinates, equation (A.37) implies
$f_{1}(z,w)-f_{0}(z,w)=\phi(z)+\phi(w)+c$. This equation and equation (A.43)
imply equation (A.38).

$\Box$

######  Lemma A.2.

Let $g_{0}$ and $g_{1}$ be two different Riemannian metrics on $S$ in the same
conformal class, as in Lemma A.1. Let $z=x+iy$ be a local uniformizer and to
simplify the notation write $z(q)=w$ and $z(p)=z$. Then

$\left(\Delta_{1}R_{1}+\frac{K_{1}}{2\pi}-\frac{2}{V}\right)\mu_{1}=\left(\Delta_{0}R_{0}+\frac{K_{0}}{2\pi}-\frac{2}{V}\right)\mu_{0}=-\tilde{\sigma}\,,$
(A.45)

where

$-\tilde{\sigma}=8h(z)dx\wedge dy=4ih(z)dz\wedge d\overline{z},$

with

$h(z)=\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}G_{0}(z,w)\Bigr{|}_{w=z}=\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}G_{1}(z,w)\Bigr{|}_{w=z}\,.$

Proof. In this proof we follow the notation of the proof of Lemma A.1. In the
$z$-coordinates, equation (A.38) becomes

$R_{1}(z)=R_{0}(z)+\frac{1}{2\pi}\log\lambda_{1}(z)-\frac{1}{2\pi}\log\lambda_{0}(z)+2\phi(z)+c.$

Taking the Laplacian $\Delta_{z}$ of both sides of this equation, using
$\Delta_{z}\phi_{z}=(\lambda^{2}_{1}-\lambda_{0}^{2})/V$, and the relations
(A.39) for conformal metrics we obtain the first equality in equation (A.45).
We recall that $\frac{\partial}{\partial
z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial
y}\right)$,
$\frac{\partial}{\partial\overline{z}}=\frac{1}{2}\left(\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y}\right)$, $\Delta_{z}=4\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{z}}$, and $dx\wedge dy=\frac{i}{2}dz\wedge
d\overline{z}$. From equation (A.43) we obtain for $j=0,1$

$\displaystyle\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
z}R_{j}(z)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
z}f_{j}(z,w)\Bigr{|}_{w=z}+\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
w}f_{j}(z,w)\Bigr{|}_{w=z}$
$\displaystyle+\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
w}f_{j}(z,w)\Bigr{|}_{w=z}+\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}f_{j}(z,w)\Bigr{|}_{w=z}+\frac{1}{2\pi}\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
z}\log\lambda_{j}(z).$

From equation (A.42) and $f_{j}(z,w)=f_{j}(w,z)$ we get

$\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
z}f_{j}(z,w)\Bigr{|}_{w=z}=\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
w}f_{j}(z,w)\Bigr{|}_{w=z}=\frac{1}{4}\Delta_{z}f_{j}(z,w)\Bigr{|}_{w=z}=\frac{1}{4}\frac{\lambda^{2}_{j}(z)}{V}$

From equation (A.41) and from the symmetry
$\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
w}f_{j}(z,w)=\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
w}f_{j}(w,z)$ we get

$\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}G_{j}(z,w)\Bigr{|}_{w=z}=\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}f_{j}(z,w)\Bigr{|}_{w=z}=\frac{\partial}{\partial\overline{z}}\frac{\partial}{\partial
w}f_{j}(z,w)\Bigr{|}_{w=z}.$ (A.46)

Finally, from the above equations and from equation (A.39) we obtain

$\Delta_{z}R_{j}(z)+\frac{\lambda_{j}^{2}(z)}{2\pi}K_{j}(z)-\frac{2\lambda^{2}_{j}(z)}{V}=8\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}G_{j}(z,w)\Bigr{|}_{w=z}$

If we multiply both sides of this equation by $dx\wedge dy$ we obtain

$\left(\Delta_{j}R_{j}+\frac{K_{j}}{2\pi}-\frac{2}{V}\right)\mu_{j}=8\frac{\partial}{\partial\overline{w}}\frac{\partial}{\partial
z}G_{j}(z,w)\Bigr{|}_{w=z}dx\wedge dy$

for $j=0,1$. Since we have already shown that the left hand side of this
equation gives the same 2-form for $j=0$ and $j=1$, then the right hand side
has the same property. $\Box$

The expression $\frac{\partial}{\partial\bar{w}}\frac{\partial}{\partial
z}G_{j}(z,w)$ is formally analogous to the traditional Bergman kernel for
bounded domains in the complex plane. Indeed, equation (A.41), which
represents the decomposition of the Green’s function into its singular and
regular parts, applies as well to the Green’s function for bounded domains in
the plane. The distinction between the two situations lies in the regular part
$f$, which is harmonic in bounded domains, whereas, in this paper, the non-
harmonicity of $f$ stems from the additional term of constant “background
vorticity.”

Following [42], let $\partial$ be an operator defined on complex valued
functions by $\partial=\frac{1}{2}(d+i\ast d)$ and
$\overline{\partial}=\frac{1}{2}(d-i\ast d)$. In terms of a local uniformizer
$z$ we have $\partial f=\frac{\partial f}{\partial z}dz$ and
$\overline{\partial}f=\frac{\partial f}{\partial\overline{z}}d\overline{z}$.

######  Lemma A.3.

If $G(q,p)$ is the Green’s function associated to a given metric and
$\\{\theta_{1},\ldots,\theta_{2{\cal G}}\\}$ is an orthonormal basis of
harmonic forms then

$-2(\partial_{p}\overline{\partial}_{q}G+\overline{\partial}_{p}\partial_{q}G)=-(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)=\sum_{k=1}^{2{\cal
G}}\theta_{k}(q)\theta_{k}(p)$ (A.47)

is the Bergman reproducing kernel for harmonic forms in $S$. Moreover, if $q$
and $p$ are in the domain of a local uniformizer with $z(q)=w$ and $z(p)=z$
then

$2(\partial_{p}\overline{\partial}_{q}G+\overline{\partial}_{p}\partial_{q}G)=4{\rm
Re}\\{\partial_{p}\overline{\partial}_{q}G\\}=4{\rm
Re}\left\\{\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{w}}G(w,z)d\overline{w}dz\right\\}$ (A.48)

Proof. The equality
$2(\partial_{p}\overline{\partial}_{q}G+\overline{\partial}_{p}\partial_{q}G)=d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G$
and equation (A.48) are direct consequences of the definition of the operators
$\partial$ and $\overline{\partial}$. Due to equation (A.46) the function
$\frac{\partial}{\partial z}\frac{\partial}{\partial\overline{w}}G(w,z)$ is
$\hbox{\rm C}^{\infty}$ for all values of $z$ and $w$ including $z=w$. So the
double one-form $d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G$ is $\hbox{\rm
C}^{\infty}$ on $S\times S$.

The Bergman reproducing kernel for harmonic forms $H(q,p)=\sum_{k=1}^{2{\cal
G}}\theta_{k}(q)\theta_{k}(p)$ is characterized by the following properties:

  * For an arbitrary function $\psi$ on $S$:

$\displaystyle\quad\int_{S}d\psi(p)\wedge H(q,p)=0$
$\displaystyle\quad\int_{S}\ast_{p}d\psi(p)\wedge H(q,p)=0\,,$

where the integrations are with respect to the variable $p$; and for any
harmonic one-form $\nu$ on $S$

$\nu(q)=\int_{S}\nu(p)\wedge\ast_{p}H(q,p)=\sum_{k=1}^{2{\cal
G}}\theta_{k}(q)\int_{S(p)}\nu(p)\wedge\ast_{p}\theta_{k}(p).$

In order to prove the equality
$d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G=-H(q,p)$ we use the regularity of
$d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G$ on $S\times S$. So, for any function
$\psi$ on $S$

$\begin{split}&\int_{S}d_{p}\psi(p)\wedge(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)=-\int_{S}\psi(p)\wedge
d_{p}(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)\\\
&\qquad\qquad=-\lim_{\epsilon\to 0}\int_{S-B_{\epsilon}(q)}\psi(p)\wedge
d_{p}(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)\end{split}$

where $B_{\epsilon}(q)$ is a small ball (with respect to any local
uniformizer) of radius $\epsilon$ with center at $q$. For $p$ outside
$B_{\epsilon}(q)$,

$d_{p}(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)=\ast_{q}d_{q}(d_{p}\ast_{p}d_{p}G)=\ast_{q}d_{q}\left(\frac{\mu(p)}{V}\right)=0,$

so $\int_{S}d_{p}\psi(p)\wedge(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)=0$. In
the same way it is possible to prove that
$\int_{S}\ast_{p}d\psi(p)\wedge(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)=0$.

It remains to show that
$\nu(q)=-\int_{S}\nu(p)\wedge\ast_{p}(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)$
for any harmonic one-form $\nu$ on $S$. This is a consequence of

$\begin{split}&\int_{S}\nu(p)\wedge\ast_{p}(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)=\lim_{\epsilon\to
0}\int_{S-B_{\epsilon}(q)}\nu(p)\wedge\ast_{p}(d_{p}d_{q}G+\ast_{p}\ast_{q}d_{p}d_{q}G)\\\
&\qquad=\lim_{\epsilon\to 0}\int_{-\partial
B_{\epsilon}(q)}\ast_{p}\nu(p)d_{q}G+\int_{-\partial
B_{\epsilon}(q)}\nu(p)\ast_{q}d_{q}G\end{split}$

An explicit computation using a local uniformizer gives that this last
integral is equal to $-\nu(q)$. $\Box$

Theorem 2.2 is a consequence of lemmas A.2 and A.3 and the following
reasoning. Let $z(p)=z=x+iy$ and $z(q)=w=\xi+i\eta$ be the components of the
local uniformizer used in lemma A.3 and
$\theta_{k}(p)=\theta_{k1}(z)dx+\theta_{k2}(z)dy$ and
$\theta_{k}(q)=\theta_{k1}(w)d\xi+\theta_{k2}(w)d\eta$ be the components of
$\theta_{k}$. Lemma A.3 implies that

$\begin{split}\sum_{k=1}^{2{\cal
G}}\theta_{k}(q)\theta_{k}(p)=&\left(\sum_{k=1}^{2{\cal
G}}\theta_{k1}(w)\theta_{k1}(z)\right)dxd\xi+\left(\sum_{k=1}^{2{\cal
G}}\theta_{k2}(w)\theta_{k2}(z)\right)dyd\eta\\\ &+\left(\sum_{k=1}^{2{\cal
G}}\theta_{k2}(w)\theta_{k1}(z)\right)dxd\eta+\left(\sum_{k=1}^{2{\cal
G}}\theta_{k1}(w)\theta_{k2}(z)\right)dyd\xi\\\ &=-4{\rm
Re}\left\\{\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{w}}G(w,z)d\overline{w}dz\right\\}\end{split}$

For $q=p$ and $dz=dw$, the right hand side of this equation becomes

$-4\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{w}}G(w,z)\Bigr{|}_{w=z}(dx^{2}+dy^{2})$

that implies

$\sum_{k=1}^{2{\cal G}}\theta^{2}_{k1}(z)=\sum_{k=1}^{2{\cal
G}}\theta^{2}_{k2}(z)=-4\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{w}}G(w,z)\Bigr{|}_{w=z},\quad\text{and}\quad\sum_{k=1}^{2{\cal
G}}\theta_{k1}(z)\theta_{k2}(z)=0$

So, the form $\sigma$ in theorem 2.2 can be written as

$\begin{split}\sigma(z)&=\sum_{k=1}^{2{\cal
G}}\theta_{k}(z)\wedge*\theta_{k}(z)=\sum_{k=1}^{2{\cal
G}}[\theta_{k1}(z)dx+\theta_{k2}(z)dy]\wedge[\theta_{k1}(z)dy-\theta_{k2}(z)dx]\\\
&=\sum_{k=1}^{2{\cal G}}[\theta^{2}_{k1}(z)+\theta^{2}_{k2}(z)]dx\wedge
dy=-8\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{w}}G(w,z)\Bigr{|}_{w=z}dx\wedge
dy=\tilde{\sigma},\end{split}$

where $\tilde{\sigma}$ is the form in equation (A.45). This proves that
equation (2.5) holds and finishes the proof of theorem 2.2. $\Box$

Now we prove theorem 2.1. The Robin function on a Riemannian manifold $(S,g)$
is constant whenever $(S,g)$ admits a transitive Lie group action of
isometries. So, the Robin function is constant for the round sphere and for
all flat tori. Let $S$ be a sphere (torus) endowed with a Riemannian metric
$g_{0}$. The uniformization theorem implies the existence of a diffeomorphism
from $(S,g_{0})$ to the round sphere (a flat torus) $({\mathbb{S}}^{2},g_{1})$
such that the pull-back of $g_{1}$ is conformal to $g_{0}$. So, the existence
of a steady vortex metric on the sphere (torus) is proved.

The proof is more complicated when $S$ is compact and has a genus larger than
one. Equation (A.38) implies:

$\Delta_{0}R_{1}(p)=\Delta_{0}R_{0}(p)+\frac{1}{2\pi}\Delta_{0}\log\lambda(p)+2\frac{\lambda^{2}-1}{V}$
(A.49)

Imposing that $R_{1}$ is constant, normalizing the volume $V$ of $S$ to be
equal to one, and defining

$u=4\pi R_{0}+\log\lambda^{2}$

we get the following equation for $u$

$\Delta_{0}u=8\pi-8\pi h{\rm e}^{u}$ (A.50)

where $h={\rm e}^{-4\pi R_{0}}$. To each solution of this equation corresponds
a Riemannian metric $g_{1}$ conformal to $g_{0}$ such that $\Delta R_{1}=0$
and therefore $R_{1}$ is constant. Equation (A.50) was very much studied for
several reasons. It appears in the problem of finding a Riemannian metric on
the sphere with a prescribed curvature $h$ that is conformal to the standard
metric with curvature $4\pi$ (the conformal factor is ${\rm e}^{u}$). It also
appears in the so-called Chern-Simons-Higgs theory (see [10] for references).
The following theorem was taken from [10] (it is a combination of their
theorem 1.2 plus their remark 1.3).

######  Theorem A.1 (Ding, Jost, Li,and Wang).

Let $(S,g_{0})$ be a compact Riemann surface and let $K_{0}$ be its Gauss
curvature. Let $h$ be a positive smooth function on $S$. Suppose that the
function $8\pi R_{0}+2\log h$ achieves its maximum at $p$. If $\Delta_{0}\log
h(p)>-(8\pi-2K_{0}(p))$ then equation (A.50) has a smooth solution.

It is remarkable that in the case we are interested in $h={\rm e}^{-4\pi
R_{0}}$ and $8\pi R_{0}+2\log h=0$. So, any point in $S$ is a point of maximum
and therefore to finish the proof it is sufficient to show the existence of a
point $p$ in $S$ where the inequality $0>-\Delta_{0}\log
h(p)-(8\pi-2K_{0}(p))$ holds. The Gauss-Bonet theorem implies
$\int_{S}K_{0}\mu_{0}=2\pi(2-2{\cal G})$, where ${\cal G}$ is the genus of
$S$. Since $\int\mu_{0}=1$, the integral of the right hand side of the
inequality above is $-8\pi{\cal G}<0$. This finishes the proof of existence of
a natural vortex metric if ${\cal G}>1$. $\Box$

##  Appendix B The Robin function and the Minakshisundaram–Pleijel zeta
function.

The Minakshisundaram–Pleijel zeta function, which will be referred as the zeta
function, is defined as

$\zeta(q,p,s)=\sum_{k=1}^{\infty}\frac{\phi_{k}(q)\phi_{k}(p)}{\lambda_{k}^{s}}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\left(K(q,p,t)-\frac{1}{V}\right)t^{s-1}dt\,,$
(B.51)

where $s\in{\mathbb{C}}$ and Re $s>n/2$ (the convergence is a consequence of
inequality (3.16)).

According to the theorem in Section 5 of [32], the function

$\zeta(p,s):=\zeta(p,p,s)$

can be extended as a meromorphic function to the whole complex plane. If
dimension $n\geq 3$ is odd, then the only possible poles of $\zeta(p,s)$ are
located at $s=n/2,n/2-1,\ldots,3/2,1/2,-1/2,\ldots$. If the dimension $n$ is
even, then $\zeta(p,s)$ has at most a finite number of poles that are possibly
located at $s=n/2,n/2-1,\ldots,2,1$ and the residue at the poles can be
computed [32]. In particular, if $n$ is even and $s$ is close to $s=1$, then

$\zeta(p,s)=\frac{1}{(4\pi)^{n/2}}\frac{a_{n/2-1}(p)}{s-1}+\text{convergent
power series in }(s-1)\,,$ (B.52)

where $a_{n/2-1}(p)$ is the function that appears in equation (3.17).

If $s$ is made equal to one in equation (B.51), then we obtain a formal
expression

$G(q,p)=\int_{0}^{\infty}\left(K(q,p,t)-\frac{1}{V}\right)dt=\sum_{k=1}^{\infty}\frac{\phi_{k}(q)\phi_{k}(p)}{\lambda_{k}}\
{}^{\prime}\\!\\!=^{\prime}\zeta(q,p,1)\,$ (B.53)

that indicates a possible relation between the regularization of $G(q,p)$ and
$\zeta(q,p,s)$ as $q\to p$ and $s\to 1$. Indeed, for $n=2$ the following
result holds (see, e.g. [43], Proposition 2 and the Appendix):

$\begin{split}R(p)=&\lim_{\ell(q,p)\to
0}\left[G(q,p)+\frac{1}{2\pi}\log\ell(q,p)\right]\\\ =&\lim_{s\to
1}\left[\zeta(p,s)-\frac{1}{(4\pi)}\frac{1}{s-1}\right]+\frac{\log
4-2\gamma}{4\pi}\end{split}$ (B.54)

where $\gamma$ is the Euler’s constant. In the following theorem we show that
this result can be generalized to higher dimensions. The same result, for an
elliptic operator that appears in the context of quantum field theory in
curved spacetime, was obtained by Bilal and Ferrari in [4] (Section 3). If the
parameters $m$ and $\psi$ that appear in their elliptic operator are set equal
to zero, then the formulas in equations (3.45) and (3.46) of [4] are exactly
ours in theorem (B.1).

######  Theorem B.1.

The Robin function can be written in terms of the analytic extension of the
Minakshisundaram–Pleijel zeta function as

$\begin{array}[]{rcll}R(p)&=&\lim_{s\to
1}\left[\zeta(p,s)-\frac{1}{(4\pi)}\frac{1}{s-1}\right]+\frac{\log
4-2\gamma}{(4\pi)^{n/2}}&\text{if $n$ is even,}\\\ &&&\\\
R(p)&=&\zeta(p,1)&\text{if $n$ is odd}\,.\end{array}$ (B.55)

Proof. We will prove the theorem only for $n$ even, since the proof for $n$
odd is similar. For $s>n/2$ both sides of equation (B.51) converge. The idea
is to add terms to both sides of that equation such that the integral in the
right-hand side of equation (B.51) converges when $s=1$. In analogy to what we
did to define the Robin function we rewrite equation (B.51) for $s>n/2$ as

$\begin{split}&\zeta(p,s)-\sum_{k=0}^{n/2-1}\frac{a_{k}(p)}{(4\pi)^{n/2}\Gamma(s)}\int_{0}^{1}t^{k-n/2}t^{s-1}dt=\frac{1}{\Gamma(s)}\int_{1}^{\infty}\left(K(p,p,t)-\frac{1}{V}\right)t^{s-1}dt\\\
&\quad+\lim_{\epsilon\to
0_{+}}\frac{1}{\Gamma(s)}\int_{\epsilon}^{1}\left(K(p,p,t)-\frac{1}{V}-\sum_{k=0}^{n/2-1}\frac{a_{k}(p)}{(4\pi)^{n/2}}t^{k-n/2}\right)t^{s-1}dt\,.\end{split}$
(B.56)

For $s>n/2$, the left-hand side of this equation can be written as

$\zeta(p,s)-\frac{a_{n/2-1}(p)}{(4\pi)^{n/2}\Gamma(s)}\frac{1}{s-1}-\sum_{k=0}^{n/2-2}\frac{a_{k}(p)}{(4\pi)^{n/2}\Gamma(s)}\frac{1}{s-n/2+k}\,.$
(B.57)

Due to equations (3.16) and (3.17), the integrand in the last line of equation
(B.56) is bounded by a constant times $t^{s-1}$, and therefore the right-hand
side of equation (B.56) is an analytic function of $s$ for Re$\,s>0$. This
implies that the analytic continuation of $\zeta(p,s)$ to Re$\,s>0$ is given
by the regular function at the right-hand side of equation (B.56) plus the
poles at $s=1,2,\ldots,n/2$ explicitly given in the left-hand side of the same
equation. With this understanding, we can compute the regularized value of
$\zeta(p,s)$ at $s=1$ as

$\begin{split}&\lim_{s\to
1}\left[\zeta(p,s)-\frac{a_{n/2-1}(p)}{(4\pi)^{n/2}\Gamma(s)}\frac{1}{s-1}\right]\\\
&=\lim_{s\to
1}\left[\zeta(p,s)-\frac{a_{n/2-1}(p)}{(4\pi)^{n/2}}\frac{1}{s-1}\right]+\frac{a_{n/2-1}(p)}{(4\pi)^{n/2}}\Gamma^{\prime}(1)\\\
&=\sum_{k=0}^{n/2-2}\frac{a_{k}(p)}{(4\pi)^{n/2}}\frac{1}{1-n/2+k}+\int_{1}^{\infty}\left(K(p,p,t)-\frac{1}{V}\right)dt\\\
&\quad+\lim_{\epsilon\to
0_{+}}\int_{\epsilon}^{1}\left(K(p,p,t)-\frac{1}{V}-\sum_{k=0}^{n/2-1}\frac{a_{k}(p)}{(4\pi)^{n/2}}t^{k-n/2}\right)dt\,,\end{split}$
(B.58)

where we used that the integrand in the last line of equation (B.56) is
bounded by a constant times $t^{s-1}$ to exchange the order of the limits.
Performing the integrals of the terms that are polynomials in $t$ in the
right-hand side of equation (B.58), using the definition of the Robin function
given in equation (3.18), and that $\Gamma^{\prime}(1)=-\gamma$ we obtain the
result in the statement of the theorem. $\Box$

Acknowledgments. This paper is dedicated to Jair Koiller who introduced me to
the subject of vortices on surfaces and presented to me the work of Okikiolu
and Steiner. Jair has been a constant source of inspiration.

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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