# Critical measures on higher genus Riemann surfaces

Marco Bertola 111Department of Mathematics and Statistics, Concordia
University, 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8, Email:
<EMAIL_ADDRESS>and SISSA, International School for Advanced
Studies, via Bonomea 265, Trieste, Italy. Email<EMAIL_ADDRESS>Supported in part by the Natural Sciences and Engineering Research Council of
Canada (NSERC) grant RGPIN-2016-06660.  Alan Groot 222Department of
Mathematics, Katholieke Universiteit Leuven, Leuven, Belgium, Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Supported by long term structural
funding-Methusalem grant of the Flemish Government. Arno B.J. Kuijlaars
333Department of Mathematics, Katholieke Universiteit Leuven, Leuven, Belgium,
Email<EMAIL_ADDRESS>Supported by long term structural funding-
Methusalem grant of the Flemish Government, and by FWO Flanders projects EOS
30889451 and G.0910.20.

###### Abstract

Critical measures in the complex plane are saddle points for the logarithmic
energy with external field. Their local and global structure was described by
Martínez-Finkelshtein and Rakhmanov. In this paper we start the development of
a theory of critical measures on higher genus Riemann surfaces, where the
logarithmic energy is replaced by the energy with respect to a bipolar Green’s
kernel. We study a max-min problem for the bipolar Green’s energy with
external fields $\operatorname{Re}V$ where $dV$ is a meromorphic differential.
Under reasonable assumptions the max-min problem has a solution and we show
that the corresponding equilibrium measure is a critical measure in the
external field. In a special genus one situation we are able to show that the
critical measure is supported on maximal trajectories of a meromorphic
quadratic differential.

We are motivated by applications to random lozenge tilings of a hexagon with
periodic weightings. Correlations in these models are expressible in terms of
matrix valued orthogonal polynomials. The matrix orthogonality is interpreted
as (partial) scalar orthogonality on a Riemann surface. The theory of critical
measures will be useful for the asymptotic analysis of a corresponding
Riemann-Hilbert problem as we outline in the paper.

##  1 Introduction

The notion of critical measures in the complex plane was developed by
Martínez-Finkelshtein and Rakhmanov [41, 42, 44] with the aim of studying
asymptotic zero distributions of Heine-Stieltjes polynomials. It is related to
the asymptotics of orthogonal polynomials with a non-hermitian orthogonality
on contours in the complex plane as initiated by Stahl [49, 50] and Gonchar
and Rakhmanov [31]. The goal of this paper is to generalize the notion of
critical measures to higher genus Riemann surfaces. Our motivation to do so
comes from the analysis of certain tiling problems with periodic weightings as
we will explain in Section 4 of this paper.

###  1.1 Critical measure in the complex plane

We start by recalling the notion of critical measure in the plane following
[41]. It can be defined in a more general situation, but we restrict here to
the case of an external field $\operatorname{Re}V$ where $V^{\prime}$ is a
rational function on $\mathbb{C}$. In fact, $V$ itself can be multi-valued,
but $\operatorname{Re}V$ is assumed to be well-defined and single-valued on
$\mathbb{C}$. The logarithmic energy of a probability measure $\mu$ in the
external field $\varphi=\operatorname{Re}V$ is

$E_{\varphi}\left[\mu\right]=\iint\log\frac{1}{|s-t|}d\mu(s)d\mu(t)+\int\varphi
d\mu.$ (1.1)

A critical measure is a probability measure $\mu$ such that (1.1) is
stationary with respect to certain perturbations of $\mu$, known as Schiffer
variations. For a continuous function $h:\mathbb{C}\to\mathbb{C}$, and a
probability measure $\mu$, the one parameter family
$(\mu_{\varepsilon,h})_{\varepsilon\in\mathbb{R}}$ of probability measures is
defined through their action on continuous functions $f$,

$\int fd\mu_{\varepsilon,h}=\int f\left(s+\varepsilon
h(s)\right)d\mu(s),\qquad\varepsilon\in\mathbb{R}.$ (1.2)

######  Definition 1.1.

The probability measure $\mu$ is a critical measure in the external field
$\varphi$ if

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}\left[\mu_{\varepsilon,h}\right]-E_{\varphi}\left[\mu\right]}{\varepsilon}=0$
(1.3)

for every $C^{1}$ function $h$ with compact support.

The definition is a special case of [41, Definition 3.2]. By [41, Lemma 3.1],
the limit (1.3) exists and is equal to

$-\operatorname{Re}\left[\iint\frac{h(s)-h(t)}{s-t}d\mu(s)d\mu(t)-\int
h(s)V^{\prime}(s)d\mu(s)\right].$

By considering both $h$ and $ih$, it follows that $\mu$ is a critical measure
if and only if

$\iint\frac{h(s)-h(t)}{s-t}d\mu(s)d\mu(t)=\int h(s)V^{\prime}(s)d\mu(s)$ (1.4)

for every $C^{1}$ function $h$ with compact support. With a limiting argument
(1.4) can be extended to $C^{1}$ functions $h$ that are bounded on the support
of $\mu$.

The main result on critical measures is that they are supported on
trajectories of quadratic differentials. Recall that a trajectory of the
quadratic differential $-Qdz^{2}$ is an (open or closed) contour $\Sigma$ such
that $Q(z(s))(z^{\prime}(s))^{2}<0$ where $s\mapsto z(s)$ is any smooth
parametrization of $\Sigma$, see e.g. [51].

######  Theorem 1.2 (Martínez-Finkelshtein and Rakhmanov [41]).

Suppose $\varphi=\operatorname{Re}V$ where $V^{\prime}$ is rational on
$\mathbb{C}$. For a critical measure $\mu$ in the external field $\varphi$, we
let

$Q(z)=\left(\frac{V^{\prime}(z)}{2}\right)^{2}-\int\frac{V^{\prime}(z)-V^{\prime}(s)}{z-s}d\mu(s).$
(1.5)

Then the following hold.

  1. (a)

$Q$ is a rational function with the property that

$\left[\int\frac{d\mu(s)}{z-s}-\frac{V^{\prime}(z)}{2}\right]^{2}=Q(z),\quad
m_{2}\text{-a.e.},$ (1.6)

where $m_{2}$ denotes the planar Lebesgue measure.

  2. (b)

The support $\Sigma=\operatorname{supp}(\mu)$ consists of a finite union of
maximal trajectories of the quadratic differential $-Q(s)ds^{2}$, and on each
trajectory we have (with $ds$ the complex line element and an appropriate
branch of the square root)

$d\mu(s)=\frac{1}{\pi i}Q(s)^{1/2}ds,\qquad s\in\Sigma.$

  3. (c)

The logarithmic potential
$U^{\mu}(z)=\displaystyle\int\log\frac{1}{|z-s|}d\mu(s)$ of $\mu$ satisfies

$2U^{\mu}(z)+\operatorname{Re}V(z)=c_{j},\qquad z\in\Sigma_{j},$ (1.7)

with a constant $c_{j}$ that can be different for each connected component
$\Sigma_{j}$ of $\Sigma$.

  4. (d)

Any point $z\in\Sigma$ that is not a zero of $Q$ has a neighborhood $D$ such
that $D\cap\Sigma$ is an analytic arc, and

$\frac{\partial}{\partial
n_{+}}\left(2U^{\mu}+\operatorname{Re}V\right)(z)=\frac{\partial}{\partial
n_{-}}\left(2U^{\mu}+\operatorname{Re}V\right)(z),\quad z\in D\cap\Sigma,$
(1.8)

where $\frac{\partial}{\partial n_{\pm}}$ denote the two normal derivatives to
$\Sigma$ at $z$.

###### Proof.

See Theorem 5.1, Lemma 5.3 and Lemma 5.4 in [41]. ∎

The identity (1.8) is known as the $S$-property in the external field
$\operatorname{Re}V$ and $\Sigma$ is called an $S$-contour or $S$-curve in the
external field $\operatorname{Re}V$. Together with (1.7), it implies that the
$g$-function

$g(z)=\int\log(z-s)d\mu(s)$

satisfies

$g_{+}(z)+g_{-}(z)-V(z)=\ell_{j}\qquad\text{ on }\Sigma_{j}$ (1.9)

for a complex constant $\ell_{j}$ that can be different for each connected
component $\Sigma_{j}$ of $\Sigma$. Indeed, the real part of $g_{+}+g_{-}-V$
is constant on each component by (1.7), while (1.8) implies that the imaginary
part is constant on each component as well, as can be seen from an application
of the Cauchy-Riemann equations.

###  1.2 Equilibrium measures

For a compact set $F\subset\mathbb{C}$ the equilibrium measure in the external
field $\varphi=\operatorname{Re}V$ is the probability measure on $F$ that
minimizes the functional (1.1) among probability measures on $F$, see e.g.
[21, 45, 46]. If $F$ is a continuum (i.e., a compact connected set with more
than one point), or a union of continua, and $\operatorname{Re}V$ is bounded
from below on $F$, then there is a unique equilibrium measure $\mu^{F}$ in the
external field $\operatorname{Re}V$. It satisfies

$\displaystyle 2U^{\mu^{F}}(z)+\operatorname{Re}V(z)$
$\displaystyle=c,\quad\text{ on }\operatorname{supp}(\mu^{F}),$ (1.10)
$\displaystyle 2U^{\mu^{F}}(z)+\operatorname{Re}V(z)$ $\displaystyle\geq
c,\quad\text{ on }F,$ (1.11)

for some constant $c$. We emphasize that $c$ is the same on all connected
components of $\operatorname{supp}(\mu^{F})$, which is in contrast to the
situation in Theorem 1.2 (c). For a critical measure the constant can be
different on each connected component of its support. We are interested in
equilibrium measures that are also critical measures, and, given $V$, this
will depend on the choice of a good compact set $F$.

To determine such $F$, Kuijlaars and Silva [40] introduced the notion of a
critical set, based on the extremal compact considered by Rakhmanov in [44].
To describe it, we use

$E_{\varphi}(F)=E_{\varphi}[\mu^{F}].$ (1.12)

######  Definition 1.3.

Let $F$ be a compact set such that $\varphi=\operatorname{Re}V$ is bounded
from below on $F$. Then $F$ is a critical set in the external field $\varphi$
if

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}(F_{\varepsilon,h})-E_{\varphi}(F)}{\varepsilon}=0$ (1.13)

for every $C^{1}$ function $h$ with compact support, where

$F_{\varepsilon,h}=\\{x+\varepsilon h(x)\mid x\in F\\}.$ (1.14)

Note that (1.14) is the support of the deformed measure $\mu_{\varepsilon,h}$
if $\operatorname{supp}(\mu)=F$.

Rakhmanov [44] essentially proved the following, see also [40].

######  Proposition 1.4.

Let $F$ be a union of continua such that $\varphi=\operatorname{Re}V$ is
bounded from below on $F$. Let $\mu^{F}$ be its equilibrium measure in the
external field $\varphi$.

  1. (a)

Then the limit in (1.13) exists and it is equal to the limit in (1.3), i.e.,

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}(F_{\varepsilon,h})-E_{\varphi}(F)}{\varepsilon}=\lim_{\varepsilon\to
0}\frac{E_{\varphi}[\mu^{F}_{\varepsilon,h}]-E_{\varphi}[\mu^{F}]}{\varepsilon}.$

  2. (b)

$F$ is a critical set if and only if $\mu^{F}$ is a critical measure in the
external field.

  3. (c)

Suppose $F$ belongs to a family $\mathcal{F}$ of union of continua such that
for every $C^{1}$ function $h$ there is $\varepsilon_{0}>0$ such that
$F_{\varepsilon,h}\in\mathcal{F}$ for every
$\varepsilon\in(-\varepsilon_{0},\varepsilon_{0})$, where $F_{\varepsilon,h}$
is given by (1.14). Suppose also that

$E_{\varphi}(F)=\max_{F^{\prime}\in\mathcal{F}}E_{\varphi}(F^{\prime}).$
(1.15)

Then $F$ is a critical set and $\mu^{F}$ is a critical measure in the external
field $\varphi$.

###### Proof.

The proof of part (a) follows from the property that

$E_{\varphi}(F_{\varepsilon,h})-E_{\varphi}[\mu^{F}_{\varepsilon,h}]=o(\varepsilon)\quad\text{
as }\varepsilon\to 0,$ (1.16)

see [40, Section 4] or [44, Section 9.10] for details, and see also
Proposition 3.8 below where the analogous statement is proved in the higher
genus case. Part (b) is immediate from part (a) and the definitions of
critical measure and critical set. If $F$ satisfies the conditions of part
(c), then $E_{\varphi}(F_{\varepsilon,h})$ has a local maximum at
$\varepsilon=0$ for every $C^{1}$ function $h$. Hence (1.13) holds and $F$ is
a critical set. Then also $\mu^{F}$ is a critical measure by part (b). ∎

Part (c) shows that one may find critical sets from solving a max-min energy
problem

$\max_{F\in\mathcal{F}}\min_{\operatorname{supp}(\mu)\subset
F}E_{\varphi}[\mu]$

where the minimum is over probability measures on $F$ and the maximum is over
a suitable family $\mathcal{F}$ satisfying the condition of part (c) of
Proposition 1.4. If a maximizer $F$ exists then clearly (1.15) holds and
$\mu^{F}$ is a critical measure in the external field $\varphi$. Then Theorem
1.2 applies and it follows in particular that the support of $\mu^{F}$ is a
union of maximal trajectories of a quadratic differential.

It can be shown that under suitable conditions, a maximizer indeed exists.
This program was first carried out by Kamvissis and Rakhmanov [37] in the case
of weighted Green’s energy in the upper half plane, and later by Rakhmanov
[44] in the case of weighted logarithmic energy in the complex plane; see also
the work of Kuijlaars and Silva [40] for the case of a polynomial $V$. Our
extension to higher genus (see Theorem 3.11 below and its proof in Section 6)
follows this approach.

There is a substantial literature on determining critical equilibrium measures
in the external field $\operatorname{Re}V$ where $V$ is a polynomial. This is
motivated in part by questions in random matrix theory, see the recent papers
[2, 10, 11] and references cited therein. See also [43] for an extension to
vector equilibrium measures.

##  2 Potential theory on Riemann surfaces

Before we can state the results of this paper we need to introduce certain
notions from potential theory on a higher genus Riemann surface, see also
Skinner [48] and a recent series of papers by Chirka [18, 19, 20]. Our
exposition will focus on equilibrium measures in an external field.

Throughout, we use $X$ to denote a compact Riemann surface with a
distinguished point $p_{\infty}$, which we refer to as the point at infinity.

###  2.1 Bipolar Green’s function

To extend Theorem 1.2 we first of all need the appropriate analogue of the
logarithmic kernel that appears in (1.1) to define the logarithmic energy.
This is provided by the bipolar Green’s function.

######  Proposition 2.1.

Let $X$ be a compact Riemann surface with a distinguished point $p_{\infty}\in
X$. There is a function $(p,q)\mapsto G(p,q)$ defined on $X\times X$ such that

  1. (a)

for a fixed $q\in X\setminus\\{p_{\infty}\\}$ the function $p\mapsto G(p,q)$
is real-valued and harmonic on $X\setminus\\{p_{\infty},q\\}$,

  2. (b)

if $z$ is a local coordinate at $q$, then

$G(p,q)=-\log|z(p)|+O(1)\quad\text{ as }p\to q,$ (2.1)

  3. (c)

if $z_{\infty}$ is a local coordinate at $p_{\infty}$, then

$G(p,q)=\log|z_{\infty}(p)|+O(1)\quad\text{ as }p\to p_{\infty},$ (2.2)

  4. (d)

$G(p,q)=G(q,p)$.

As usual, a local coordinate at a point on a Riemann surface means a
holomorphic chart in which the point corresponds to $0\in\mathbb{C}$.

###### Proof.

The existence of $G$ satisfying parts (a), (b) and (c) of Proposition 2.1 can
be found in Gamelin [30], Simon [47, Section 3.8]. We could not find an
appropriate reference for part (d), although it may be known. The parts (a),
(b) and (c) determine $p\mapsto G(p,q)$ up to an additive constant that may
depend on $q$. It is noted by Skinner [48, p. 25] that it is not immediately
obvious how to choose the constant such that $G$ is symmetric. We give a proof
in the appendix. ∎

######  Definition 2.2.

The function $G$ satisfying the conditions of Proposition 2.1 is called the
bipolar Green’s function with one pole at $p_{\infty}$, or simply bipolar
Green’s function.

The bipolar Green’s function is unique up to an additive constant, but the
constant will not be important for us. Note that $G(p,q)\to-\infty$ as $p\to
p_{\infty}$ and $G(p,q)\to+\infty$ as $p\to q$.

######  Example 2.3.

(a) If $X$ is the Riemann sphere with $p_{\infty}$ the point at infinity, then
$G(p,q)=\log\frac{1}{|p-q|}$.

(b) On a complex torus $X=\mathbb{C}/\Lambda$ with lattice
$\Lambda=\mathbb{Z}+\tau\mathbb{Z}$ and $\operatorname{Im}\tau>0$, the bipolar
Green’s function with pole at $p_{\infty}=0$ (modulo $\Lambda$) is explicitly
given by, see e.g. [18], [38, Section 2], [48],

$G(p,q)=\log\left|\frac{\theta_{1}(p)\theta_{1}(q)}{\theta_{1}(p-q)}\right|-\frac{2\pi}{\operatorname{Im}\tau}\left(\operatorname{Im}p\right)\left(\operatorname{Im}q\right)$
(2.3)

in terms of the Jacobi elliptic function $\theta_{1}$ that has a zero at $0$
(see [24, Chapter 20] which however uses a different scaling of elliptic
functions with periods $\pi$ and $\pi\tau$ instead of $1$ and $\tau$), i.e.,

$\theta_{1}(z)=\theta_{1}(z;\tau)=-i\sum_{k=-\infty}^{\infty}(-1)^{k}e^{\pi
i\tau(k+\frac{1}{2})^{2}+(2k+1)\pi iz}.$ (2.4)

The following properties of $\theta_{1}$ can be found in [24, Chapter 20]. The
Jacobi elliptic function is an odd entire function with simple zeros at every
lattice point, and no other zeros. Moreover, it has the quasi-periodicity
properties

$\theta_{1}(z+1)=-\theta_{1}(z),\qquad\theta_{1}(z+\tau)=-e^{-\pi i\tau-2\pi
iz}\theta_{1}(z).$ (2.5)

We can use (2.3) to construct the bipolar Green’s function on an arbitrary
genus one Riemann surface $X$, by composing it with the Abel map from $X$ to
$\mathbb{C}/\Lambda$ that maps $p_{\infty}$ to $0$.

The local behavior of $G(p,q)$ is determined by the logarithmic kernel
$\log\frac{1}{|p-q|}$ in the following sense.

######  Lemma 2.4.

We have

  1. (a)

If $z_{\infty}$ is a local coordinate at $p_{\infty}$ (that maps $p_{\infty}$
to $0$) then we have

$G(p,q)-\log\frac{1}{|z_{\infty}(p)^{-1}-z_{\infty}(q)^{-1}|}=O(1)$ (2.6)

uniformly for $p,q$ in a neighborhood of $p_{\infty}$.

  2. (b)

If $z_{0}$ is a local coordinate at $p_{0}\neq p_{\infty}$ (that maps $p_{0}$
to $0$) then we have

$G(p,q)-\log\frac{1}{|z_{0}(p)-z_{0}(q)|}=O(1)$ (2.7)

uniformly for $p,q$ in a neighborhood of $p_{0}$ that does not contain
$p_{\infty}$.

###### Proof.

The two terms in the left hand sides of (2.6) and (2.7) have the same
logarithmic singularities when $p=q$. The two terms in (2.6) also have the
same logarithmic singularity when $p=p_{\infty}$ and by symmetry of $G$ also
when $q=p_{\infty}$. Therefore the differences in both (2.6) and (2.7) are
harmonic in both variables near $p_{\infty}$ and $p_{0}$, respectively, and
therefore they remain bounded. ∎

###  2.2 Equilibrium measure in external field

The bipolar Green’s function allows us to extend concepts of logarithmic
potential theory in the complex plane [21, 45, 46] to compact Riemann
surfaces.

######  Definition 2.5.

Let $\mu$ be a measure on $X$ with compact support in
$X\setminus\\{p_{\infty}\\}$. Then we define its bipolar Green’s energy

$E\left[\mu\right]=\iint G(p,q)d\mu(p)d\mu(q).$ (2.8)

For a lower semi-continuous
$\varphi:\operatorname{supp}(\mu)\to\mathbb{R}\cup\\{+\infty\\}$ we define the
bipolar Green’s energy of $\mu$ in the external field $\varphi$ by

$E_{\varphi}\left[\mu\right]=\iint G(p,q)d\mu(p)d\mu(q)+\int\varphi d\mu.$
(2.9)

We extend the definition (2.8) to signed measures $\nu=\mu_{1}-\mu_{2}$,
provided $E[\mu_{1}]$ and $E[\mu_{2}]$ are finite. We need the crucial
property

######  Proposition 2.6.

If $\mu_{1},\mu_{2}$ are two compactly supported probability measures on
$X\setminus\\{p_{\infty}\\}$ with finite bipolar Green’s energy and
$\nu=\mu_{1}-\mu_{2}$, then

$E[\nu]=\iint G(p,q)d\nu(p)d\nu(q)\geq 0,$ (2.10)

and $E[\nu]=0$ if and only if $\mu_{1}=\mu_{2}$.

###### Proof.

A number of proofs are known in the literature. One proof involves the
eigenfunctions of the Laplace operator as in [19] and [48]. Another proof
relates (2.10) to the norm of certain functions in a Sobolev space; see [19]
and [38]. ∎

Let $\mathcal{E}^{1}(K)$ denote the set of all probability measures $\mu$ on
$K$ with $E[\mu]<+\infty$. If $\mu\in\mathcal{E}^{1}(K)$ and $(\mu_{n})_{n}$
is a sequence in $\mathcal{E}^{1}(K)$ such that

$\lim_{n\to\infty}E[\mu_{n}-\mu]=0,$ (2.11)

then we say that $(\mu_{n})_{n}$ converges to $\mu$ in energy norm.
Convergence in energy norm implies weak∗ convergence (see the second bullet
point on [19, p. 306], or [35, Theorem 7.3.10] for the case of potential
theory in $\mathbb{R}^{n}$), i.e.,

$\lim_{n\to\infty}\int fd\mu_{n}=\int fd\mu$

for every continuous $f$ on $K$.

For a compact set $F\subset X\setminus\\{p_{\infty}\\}$ and a lower semi-
continuous $\varphi:F\to\mathbb{R}\cup\\{+\infty\\}$, we denote

$E_{\varphi}(F)=\inf_{\mu\in\mathcal{E}^{1}(F)}E_{\varphi}\left[\mu\right].$
(2.12)

######  Definition 2.7.

Let $F\subset X\setminus\\{p_{\infty}\\}$ be compact and let
$\varphi:F\to\mathbb{R}\cup\\{+\infty\\}$ $E_{\varphi}(F)<\infty$ be a lower
semi-continuous function. Suppose $E_{\varphi}(F)<+\infty$. Then a probability
measure $\mu$ on $F$ that satisfies
$E_{\varphi}(F)=E_{\varphi}\left[\mu\right]$ is called an equilibrium measure
of $F$ in the external field $\varphi$, or simply an equilibrium measure.

There is only one equilibrium measure, so that we speak of the equilibrium
measure of $F$ in the external field $\varphi$, as is part of the following
proposition.

######  Proposition 2.8.

Suppose $F\subset X\setminus\\{p_{\infty}\\}$ is a compact set, and
$\varphi:F\to\mathbb{R}\cup\\{+\infty\\}$ is lower semi-continuous with
$E_{\varphi}(F)<+\infty$. Then there is a unique equilibrium measure $\mu^{F}$
on $F$ in the external field $\varphi$. The equilibrium measure is the unique
probability measure $\mu$ on $F$ such that for some constant $c$,

$\displaystyle 2\int G(p,q)d\mu(q)+\varphi(p)$
$\displaystyle=c,\quad\text{q.e.\ on }\operatorname{supp}(\mu),$ (2.13)
$\displaystyle 2\int G(p,q)d\mu(q)+\varphi(p)$ $\displaystyle\geq
c,\quad\text{q.e.\ on }F.$ (2.14)

Here q.e. means quasi-everywhere, i.e., except for a set of zero capacity.

###### Proof.

With the aid of Proposition 2.6, the proof is similar to the proof for
equilibrium measures in logarithmic potential theory given in [46, Theorem
I.1.3], see also [19, Section 2.3] or [48, Lemma 3.4.5].

For the variational conditions (2.13) and (2.14) it is important that $G$ is
symmetric, see Proposition 2.1 (d), since otherwise $2\int G(p,q)d\mu(q)$ in
(2.13) and (2.14) should be replaced by
$\int\left(G(p,q)+G(q,p)\right)d\mu(q)$. ∎

In the present work we will have that $F$ is a connected compact set with more
than one point (i.e., a continuum), or a finite disjoint union of such sets,
and in such a situation there are no exceptional sets of zero capacity, and
(2.13) and (2.14) hold everywhere on their respective sets.

###  2.3 $S$-property

For a compactly supported measure $\mu$ with
$\operatorname{supp}(\mu)\setminus X\setminus\\{p_{\infty}\\}$, note that
$p\mapsto\int G(p,q)\mu(q)$ is a harmonic function on
$X\setminus(\\{p_{\infty}\\}\cup\operatorname{supp}(\mu))$. It is real-valued,
but being harmonic it is locally the real part of a holomorphic function that
we denote by $g$ and we call it the $g$-function. Depending on the situation,
we may need to define certain branch cuts in order to make $g$ single-valued.
Its real part

$\operatorname{Re}g(p)=-\int G(p,q)\mu(q)$

is always single-valued, however.

In case $F=\gamma$ is a contour and $\varphi=\operatorname{Re}V$ for a multi-
valued, locally meromorphic function on $X$ with single-valued real part, the
variational conditions (2.13)–(2.14) can also be formulated as

$\displaystyle\operatorname{Re}\left(g_{+}(p)+g_{-}(p)-V(p)\right)$
$\displaystyle=c,\qquad p\in\operatorname{supp}(\mu),$ (2.15)
$\displaystyle\operatorname{Re}\left(g_{+}(p)+g_{-}(p)-V(p)\right)$
$\displaystyle\leq c,\qquad p\in\gamma.$

The contour will be oriented, which defines the $+$ and $-$ sides; the
notation $g_{+}$ and $g_{-}$ refers to limiting values of $g$ on the $+$ and
$-$ sides of $\gamma$.

######  Definition 2.9.

A contour $\gamma$ has the $S$-property in external field $\operatorname{Re}V$
if the $g$-function of its equilibrium measure $\mu$ in the external field
$\operatorname{Re}V$ has the following property: not only the real part of
$g_{+}+g_{-}-V$ is constant on $\operatorname{supp}(\mu)$, see (2.15), but
also the imaginary part of $g_{+}+g_{-}-V$ is piecewise constant on each
connected component of $\operatorname{supp}(\mu)$.

In case $\gamma$ has more than one connected component, then the imaginary
part of the constant can be different on the different components. The
imaginary part can also be different due to a different choice of branches of
$g$ in case a branch cut intersects $\gamma$.

##  3 Statement of results

###  3.1 Cauchy kernel

We would like to have the analogue of Theorem 1.2 (a) which in particular
states that for a critical measure

$\left[\int C(p,q)d\mu(q)-\frac{dV(p)}{2}\right]^{2}$ (3.1)

is a meromorphic quadratic differential on $X$ where $C(p,q)$ is an
appropriate Cauchy kernel.

######  Definition 3.1.

The Cauchy kernel $C(p,q)$ is given in terms of the bipolar Green’s function
with pole at $p_{\infty}$, see Definition 2.2, by

$C(p,q)=-2\partial_{p}G(p,q)dp.$ (3.2)

Here $2\partial_{p}=\partial_{x}-i\partial_{y}$ if $p=x+iy$ in a local
coordinate. Since $p\mapsto G(p,q)$ is harmonic for $p\in
X\setminus\\{q,p_{\infty}\\}$, we find from (3.2) that $C(p,q)$ is a
meromorphic differential in the $p$-variable (for any fixed $q\in
X\setminus\\{p_{\infty}\\}$) and a harmonic function in the $q$-variable. More
precisely,

  * •

$C(\cdot,q)$ is a meromorphic differential on $X$, with simple poles at $q$
and at $p_{\infty}$, and holomorphic otherwise, the pole at $q$ has residue
$1$ and the pole at $p_{\infty}$ has residue $-1$ (these residues come from
the behavior (2.1), (2.2) of the bipolar Green’s function)

  * •

$C(p,\cdot)$ is a harmonic function on $X\setminus\\{p,p_{\infty}\\}$.

We also note that, since $G(p,q)$ is harmonic and single-valued,

  * •

$C(\cdot,q)$ has purely imaginary periods.

since $G(p,q)$ is harmonic and single-valued. The above three properties
actually characterize the Cauchy kernel.

For any measure $\mu$ with compact support, the expression (3.1) is a
quadratic differential that is meromorphic on
$X\setminus\operatorname{supp}(\mu)$ with poles at $p_{\infty}$ and at the
poles of $dV$.

######  Example 3.2.

(a) On the Riemann sphere we have

$C(p,q)=-2\partial_{p}\left(\log\frac{1}{|p-q|}\right)dp=\frac{dp}{p-q},$

which is the usual Cauchy kernel from complex analysis.

(b) On a complex torus $X=\mathbb{C}/\Lambda$ as in Example 2.3 (b) we have

$\displaystyle C(p,q)$
$\displaystyle=-2\partial_{p}\left(\log\left|\frac{\theta_{1}(p)\theta_{1}(q)}{\theta_{1}(p-q)}\right|-\frac{2\pi}{\operatorname{Im}\tau}\left(\operatorname{Im}p\right)\left(\operatorname{Im}q\right)\right)dp$
$\displaystyle=-\left(\frac{\theta_{1}^{\prime}(p)}{\theta_{1}(p)}-\frac{\theta_{1}^{\prime}(p-q)}{\theta_{1}(p-q)}+\frac{2\pi
i}{\operatorname{Im}\tau}\operatorname{Im}q\right)dp.$ (3.3)

The formula shows that $q\mapsto C(p,q)$ is not meromorphic in the genus one
case, but it is harmonic.

The notion of Cauchy kernel we use here could be further qualified as
imaginary normalized and it is only harmonic with respect to $q\neq p$. More
commonly the term is used to refer to a kernel which is meromorphic in $q$,
with an additional pole at a prescribed collection of $g$ points (where $g$ is
the genus of the Riemann surface). Possibly the first occurrence is in [3] but
the notion appears ubiquitously in the literature on Riemann surfaces, notably
in [28, 34, 52]. A related notion is used also later in this paper, see
Section 3.5.

###  3.2 Critical measures

In order to define the notion of a critical measure, we need an analogue of
the Schiffer variation formula (1.2). Instead of considering a function $h$,
as we did in the genus zero case, we identify $h$ as a vector field on $X$,
i.e., a section of the tangent bundle in the higher genus case. A vector field
$h$ induces a flow $\Phi(t,p)$ on $X$ (for $t\in\mathbb{R}$ and $p\in X$, we
have $\Phi(t,p)\in X$) satisfying $\frac{d\Phi}{dt}=h$ and $\Phi(0,p)=p$.
Then, given a measure $\mu$ on $X$ we define $\mu_{\varepsilon,h}$ by its
action on continuous functions $f$

$\int fd\mu_{\varepsilon,h}=\int
f(\Phi(\varepsilon,p))d\mu(p),\qquad\varepsilon\in\mathbb{R},$ (3.4)

which is the analogue of (1.2). Thus $\mu_{\varepsilon,h}$ is the image of
$\mu$ along the flow induced by $h$, and we may alternatively write

$\mu_{\varepsilon,h}=\Phi(\varepsilon,\mu).$

######  Definition 3.3.

Let $\mu$ be a measure on $X$ with support in $X\setminus\\{p_{\infty}\\}$ and
suppose that the external field $\varphi$ is a real-valued $C^{1}$ function on
a neighborhood of $\operatorname{supp}(\mu)$. Then $\mu$ is a critical measure
in the external field $\varphi$ if

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}[\mu_{\varepsilon,h}]-E_{\varphi}[\mu]}{\varepsilon}=0$
(3.5)

for every $C^{1}$ vector field $h$.

Similar to [41, Lemma 3.1], there is a convenient identity for the limit in
(3.5) in case $\varphi$ is given by the real part of $V$.

######  Proposition 3.4.

Suppose that the external field $\varphi$ is given by
$\varphi=\operatorname{Re}V$ and suppose that $\mu$ is a measure with support
in $X\setminus\\{p_{\infty}\\}$ such that $E[\mu]<\infty$ and $\varphi$ is
bounded on $\operatorname{supp}(\mu)$. Then the following holds.

  1. (a)

For every $C^{1}$ vector field $h$, the limit in (3.5) exists with

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}[\mu_{\varepsilon,h}]-E_{\varphi}[\mu]}{\varepsilon}=\operatorname{Re}D_{V,h}(\mu),$
(3.6)

where $D_{V,h}(\mu)$ is given by

$D_{V,h}(\mu)=-\iint\left(h(p)C(p,q)+h(q)C(q,p)\right)d\mu(p)d\mu(q)+\int
hdVd\mu$ (3.7)

and $C(p,q)$ is the Cauchy kernel from Definition 3.1, see Remark 3.5.

  2. (b)

A critical measure $\mu$ satisfies $D_{V,h}(\mu)=0$ for every $C^{1}$ vector
field $h$.

The proof is given in Section 5.

######  Remark 3.5.

Let us comment on the expression (3.7). The vector field $h$ acts on the
meromorphic differential $dV$ in a natural way: the expression $hdV$ defines a
$C^{1}$ function on $X$ away from the poles of $dV$. The function can be
integrated against a measure and this is how the term $\int hdVd\mu$ should be
interpreted in (3.7). It is the analogue of the right-hand side of (1.4).
Also, the fact that the external field is harmonic means that we only need to
consider vector fields in the holomorphic tangent bundle. See Remark 3.6 for
more details.

The analogue of the left-hand side of (1.4) is the double integral with the
Cauchy kernel $C(p,q)$. Since $C(p,q)$ is a differential in $p$, the product
$h(p)C(p,q)$ is a function in both variables $p$ and $q$ that becomes infinite
when $p=q$. The same holds true for $h(q)C(q,p)$, but the infinities for $p=q$
disappear in the sum $h(p)C(p,q)+h(q)C(q,p)$ if $h$ is a $C^{1}$ vector field.
The sum is a well-defined bounded and continuous function on
$(X\setminus\\{p_{\infty}\\})\times(X\setminus\\{p_{\infty}\\})$ that plays
the role of the divided difference $\frac{h(s)-h(t)}{s-t}$ in (1.4).

######  Remark 3.6.

The energy functionals we are considering involve harmonic functions
$\varphi=\operatorname{Re}V$ for their external field, and the bipolar Green’s
function is also a harmonic function. This has the following effect: when we
consider the Schiffer variations, it is sufficient to consider vector fields
in the holomorphic tangent bundle of the Riemann surface $X$, usually denoted
by $T^{1,0}=T^{1,0}X$. In concrete terms, this means that when we consider $X$
as a two-dimensional real manifold with local coordinates $x,y$ (forming the
real and imaginary part of a local coordinate $z=x+iy$), a vector field is
simply an expression of the form $\mathbb{V}=a(x,y)\frac{\partial}{\partial
x}+b(x,y)\frac{\partial}{\partial y}$, with $a,b$ smooth real-valued
functions. The vector field can then be written in terms of the Cauchy-Riemann
operators $\frac{\partial}{\partial
z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial
y}\right),\frac{\partial}{\partial\overline{z}}=\frac{1}{2}\left(\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y}\right)$ as

$\mathbb{V}=(a+ib)\frac{\partial}{\partial
z}+(a-ib)\frac{\partial}{\partial\overline{z}}=\mathbb{V}^{1,0}+\mathbb{V}^{0,1}.$
(3.8)

A real-valued harmonic function $H(x,y)$ can be written locally as the real
part of a holomorphic (or anti-holomorphic) function $f$ as
$H(x,y)=\operatorname{Re}f(z)$. Acting with $\mathbb{V}$ on such a function
yields

$\mathbb{V}H(x,y)=\frac{1}{2}\left((a+ib)\frac{\partial f(z)}{\partial
z}+(a-ib)\frac{\partial\overline{f(z)}}{\partial\overline{z}}\right)=\operatorname{Re}\left(\mathbb{V}^{1,0}f\right).$
(3.9)

Having thus established that we need only to consider vectors in $T^{1,0}$, we
also now point out that given any (meromorphic) differential, written in local
coordinate as $\omega=f(z)dz$, the expression $\frac{1}{\omega}$ can be
interpreted as a (meromorphic) vector field in $T^{1,0}$. This is easily
verified by observing that it transforms as a vector in $T^{1,0}$ does under
holomorphic change of coordinates. In fact this also holds for smooth sections
of the canonical bundle, namely, expressions $\omega=f(z,\overline{z})dz$,
with $f$ a smooth (i.e. not necessarily satisfying the Cauchy-Riemann
equations) complex-valued function. Even in this case, $\frac{1}{\omega}$
again defines a smooth vector field wherever $\omega$ is not zero.

###  3.3 Critical sets

For a vector field $h$ with an associated flow $\Phi$, we also consider the
flow

$F_{\varepsilon,h}=\Phi(\varepsilon,F)$ (3.10)

of a compact set $F$.

######  Definition 3.7.

$F$ is a critical set in the external field $\varphi=\operatorname{Re}V$ if

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}(F_{\varepsilon,h})-E_{\varphi}(F)}{\varepsilon}=0$ (3.11)

for every $C^{1}$ vector field $h$.

The following analogue of Proposition 1.4 holds.

######  Proposition 3.8.

Let $F\subset X\setminus\\{p_{\infty}\\}$ be a union of continua such that
$\operatorname{Re}V$ is bounded from below on $F$. Let $\mu^{F}$ be the
equilibrium measure of $F$ in the external field $\varphi=\operatorname{Re}V$.

  1. (a)

Then for every $C^{1}$ vector field $h$,

$\lim_{\varepsilon\to
0}\frac{E_{\varphi}(F_{\varepsilon,h})-E_{\varphi}(F)}{\varepsilon}=\lim_{\varepsilon\to
0}\frac{E_{\varphi}[\mu^{F}_{\varepsilon,h}]-E_{\varphi}[\mu^{F}]}{\varepsilon}$
(3.12)

  2. (b)

$F$ is a critical set in the external field $\varphi$ if and only its
equilibrium measure $\mu^{F}$ is a critical measure in the external field
$\varphi$.

The proof is given in Section 5, and it is similar to the proof of Lemma 3.5
in [44] and the proof of Proposition 4.2 in [40]. Proposition 3.8 is the
higher genus analogue of parts (a) and (b) of Proposition 1.4. The analogue of
part (c) is now immediate as well.

######  Corollary 3.9.

Suppose $F$ belongs to a family $\mathcal{F}$ of union of continua such that
for every $C^{1}$ vector field $h$ there is $\varepsilon_{0}>0$ such that
$F_{\varepsilon,h}\in\mathcal{F}$ for every
$\varepsilon\in(-\varepsilon_{0},\varepsilon_{0})$, where $F_{\varepsilon,h}$
is given by (3.10). Suppose also that

$E_{\varphi}(F)=\max_{F^{\prime}\in\mathcal{F}}E_{\varphi}(F^{\prime}).$
(3.13)

Then $F$ is a critical set and $\mu^{F}$ is a critical measure in the external
field $\varphi$.

###  3.4 Max-min energy problem

Next we describe a situation to which Corollary 3.9 applies and where we can
prove the existence of a critical set $F$ and a critical measure $\mu^{F}$ in
external field. We are not aiming at the most general set-up here, but instead
we confine ourselves to a case that will be useful for the analysis of
periodic tiling models, see Section 4.

We assume $\varphi=\operatorname{Re}V$, where $V$ is locally meromorphic on
$X$ with logarithmic singularities at a finite number of points. The
differential $dV$ is meromorphic and single-valued on $X$ with at most simple
poles (i.e., an Abelian differential of the third kind). The logarithmic
singularities of $V$ correspond to simple poles of $dV$. We assume the
residues are real and negative, except for a pole $p_{0}\in
X\setminus\\{p_{\infty}\\}$ with positive residue $r_{0}>0$ and a possible
pole at $p_{\infty}$. This implies that in a local coordinate $z_{0}$ at
$p_{0}$ one has

$\operatorname{Re}V(p)=r_{0}\log|z_{0}(p)|+O(1),\quad\text{ as }p\to p_{0}.$
(3.14)

We fix a local coordinate at $p_{0}$, that identifies a certain neighborhood
of $p_{0}$ in $X$ with a disk $D(0,\delta_{0})=\\{|z_{0}|<\delta_{0}\\}$ for
some $\delta_{0}>0$. For $\delta<\delta_{0}$ we use $U_{\delta}(p_{0})$ to
denote the neighborhood of $p_{0}$ that corresponds to $D(0,\delta)$ in this
local coordinate.

The differential $dV$ could have simple pole at $p_{\infty}$ as well, and then
we use $r_{\infty}=\operatorname{Res}(dV,p_{\infty})$ to denote the residue
which could be positive or negative. If $dV$ is holomorphic at $p_{\infty}$,
then we write $r_{\infty}=0$. In either case we have in a fixed local
coordinate $z_{\infty}$ at $p_{\infty}$

$\operatorname{Re}V(p)=r_{\infty}\log|z_{\infty}(p)|+O(1),\quad\text{ as }p\to
p_{\infty}.$ (3.15)

A neighborhood of $p_{\infty}$ is identified with the disk
$D(0,\delta_{\infty})$ for some $\delta_{\infty}>0$. For
$\delta<\delta_{\infty}$ we use $U_{\delta}(p_{\infty})$ for the neighborhood
of $p_{\infty}$ that corresponds to the smaller disk $D(0,\delta)$.

At all other poles (if any) there is a negative residue, which means that
$\operatorname{Re}V$ tends to $+\infty$ at those poles. Thus the external
field $\operatorname{Re}V$ is bounded from below on
$X\setminus(U_{\delta}(p_{0})\cup U_{\delta}(p_{\infty}))$ for any small
enough $\delta>0$. See Figure 1 for a sketch of the neighborhoods
$U_{\delta}(p_{0})$ and $U_{\delta}(p_{\infty})$.

$\gamma_{1}$$\gamma_{2}$$U_{\delta}(p_{0})$$U_{\delta}(p_{\infty})$$\gamma_{3}$

Figure 1:  Neighborhoods $U_{\delta}(p_{0})$ and $U_{\delta}(p_{\infty})$, and
an example of a multi-contour $\gamma=\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}$
in $\mathcal{T}_{\delta}$ as defined in Definition 3.10. The three components
belong to $\mathcal{S}_{\delta}$.

######  Definition 3.10.

Suppose $\delta_{0}$ and $\delta_{\infty}$ are as above with
$\delta_{\infty}\geq\delta_{0}$ and such that the neighborhoods
$U_{\delta}(p_{0})$ and $U_{\delta}(p_{\infty})$ are disjoint for every
$\delta\in(0,\delta_{0}]$. Then for $\delta\in(0,\delta_{0}]$, we define the
following

  1. (a)

We use $\mathcal{S}_{\delta}$ to denote the set of simple closed oriented
contours in $X\setminus(U_{\delta}(p_{0})\cup U_{\delta}(p_{\infty}))$ that
are not homotopic to a point in $X\setminus\\{p_{0},p_{\infty}\\}$.

  2. (b)

We use $\mathcal{T}_{\delta}$ to denote the set

$\mathcal{T}_{\delta}=\left\\{\gamma=\gamma_{1}\cup\cdots\cup\gamma_{n}\mid\begin{array}[]{ll}n\in\mathbb{N},\gamma_{j}\in\mathcal{S}_{\delta}\text{
for every $j$, and }\gamma\text{ is homologous}\\\ \text{in
}X\setminus\\{p_{0},p_{\infty}\\}\text{ to a circle around $p_{0}$ in
}U_{\delta}(p_{0})\end{array}\right\\}.$ (3.16)

  3. (c)

We define

$\mathcal{F}_{\delta}=\overline{\mathcal{T}}_{\delta},\qquad\mathcal{F}=\bigcup_{0<\delta<\delta_{0}}\mathcal{F}_{\delta},$
(3.17)

where the closure is with respect to the topology corresponding to Hausdorff
distance on compact subsets of $X$.

An illustration of a possible multi-contour in $\mathcal{T}_{\delta}$ is
provided in Figure 1.

The motivation for the definition (3.16) comes from the study of orthogonality
on the Riemann surface, see the discussion in Section 4.5 below. The
orthogonality is expressed through the vanishing of certain integrals of
meromorphic differentials over a small contour around $p_{0}$. The meromorphic
differentials have poles at $p_{0}$ and $p_{\infty}$, and in the absence of
further poles, the integral can be deformed to a system of contours $\gamma$
as in (3.16) by Cauchy’s theorem. Within the class $\mathcal{T}_{\delta}$
(with $\delta$ sufficiently small) we want to find an ideal $\gamma$.

The Hausdorff distance on compact subsets of $X$ that is used in (3.17) comes
from a distance function $d$ on $X$ that is compatible with the complex
structure; see (A.1). Then we write

$d(p,F)=\min_{q\in F}d(p,q)$

for the distance of $p\in X$ to a compact set $F\subset X$, and

$d_{H}(F_{1},F_{2})=\max\left(\max_{p\in F_{1}}d(p,F_{2}),\max_{p\in
F_{2}}d(p,F_{1})\right)$

for the Hausdorff distance between compact sets $F_{1}$ and $F_{2}$.

The Hausdorff distance restricted to compact subsets of a fixed compact
defines a compact metric space. Thus $\mathcal{F}_{\delta}$ is compact for
every $\delta\in(0,\delta_{0})$. The set $\mathcal{F}$ from (3.17) does not
depend on the precise choice of distance function on $X$. Recall that
$E_{\varphi}(F)$ is given by (2.12).

######  Theorem 3.11.

In the above setting, suppose that $r_{0}=\operatorname{Res}(dV,p_{0})>1$ and
$r_{\infty}=\operatorname{Res}(dV,p_{\infty})>-1$. Then there exists
$F\in\mathcal{F}$ (as defined in Definition 3.10) such that

$E_{\varphi}(F)=\max_{F^{\prime}\in\mathcal{F}}E_{\varphi}(F^{\prime}).$
(3.18)

The set $F$ is critical and its equilibrium measure $\mu^{F}$ is a critical
measure in the external field $\varphi$.

The condition $r_{0}>1$ in Theorem 3.11 implies that any $F\in\mathcal{F}$
that comes very close to $p_{0}$ has a small bipolar Green’s energy
$E_{\varphi}(F)$ in a sense that will be made precise in Proposition 6.2. The
condition $r_{\infty}>-1$ gives the same result for $F$ that come close to
$p_{\infty}$.

The proof of Theorem 3.11 is in Section 6.

###  3.5 Final result

The final result of the paper is a generalization of Theorem 1.2 to the case
of a higher genus Riemann surface, albeit only for a special case of genus one
that is given by a cubic equation

$X:\quad w^{2}=z(z-x_{1})(z-x_{2})$ (3.19)

with $x_{1},x_{2}\in\mathbb{R}$ and $x_{1}<x_{2}<0$. We use $p=(w,z)$ to
denote a generic point on $X$. We use $p_{\infty}$ for the point at infinity
and we use $p_{0}=(0,0)$, $p_{1}=(0,x_{1})$, $p_{2}=(0,x_{2})$ to denote the
branch points of (3.19). Then $X$ has two sheets that are copies of
$\mathbb{C}\setminus((-\infty,x_{1}]\cup[x_{2},0])$. The first sheet is such
that $w>0$ for $z>0$ on the first sheet, while $w<0$ for $z>0$ on the second
sheet. $X$ has an antiholomorphic involution

$\sigma:X\to X:(w,z)\mapsto(\overline{w},\overline{z}).$ (3.20)

The real locus of $X$ is invariant under $\sigma$, and it consists of two
parts. The bounded part is the real oval $C_{1}$ consisting of the points
$p=(w,z)$ with $z\in[x_{1},x_{2}]$, and the unbounded part $C_{2}$ consists of
the points $p=(w,z)$ with $z\in[0,\infty)$ together with the point
$p_{\infty}$ at infinity, see Figure 2.

By means of the Abel map, we identify $X$ with a complex torus
$\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ with $\tau\in i\mathbb{R}^{+}$, with
$p_{\infty}$ identified with $0$. The complex torus has the following
$(2,-1)$-Cauchy kernel, which depends on a parameter $a$. Here $(2,-1)$ refers
to the fact that this kernel is a quadratic differential with respect to one
variable and a meromorphic vector field with respect to the other variable.

######  Proposition 3.12.

For $u,v,a\in\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ we have that

$C^{(2,-1)}(u,v;a)=\frac{\theta_{1}^{\prime}(0)\theta_{1}(u-a)^{2}\theta_{1}(v)^{2}\theta_{1}(u-v+2a)}{\theta_{1}(u-v)\theta_{1}(v-a)^{2}\theta_{1}(u)^{2}\theta_{1}(2a)}\frac{du^{2}}{dv}$
(3.21)

satisfies the following.

  1. (a)

For fixed $u,a$, we have that (3.21) is a meromorphic vector field (i.e. a
$-1$-differential by Remark 3.6) in $v$ with a simple pole at $v=u$, a double
pole at $v=a$, a double zero at $v=0$ and a simple zero at $v=u+2a$.

  2. (b)

For fixed $v,a$, we have that (3.21) is a meromorphic quadratic differential
in $u$, with a simple pole at $u=v$, a double pole at $u=0$, a double zero at
$u=a$, and a simple zero at $u=v-2a$.

  3. (c)

The kernel (3.21) is normalized such that

$C^{(2,-1)}(u,v;a)\frac{dv}{du^{2}}=\frac{1}{u-v}+O(1)$ (3.22)

as $u\to v$ on the complex torus.

###### Proof.

The periodicity properties (2.5) show that (3.21) is doubly periodic in both
variables $u$ and $v$. The statements (a), (b), (c) can be directly verified
from the formula (3.21) with the fact that $\theta_{1}$ has a simple zero at
$0$ and no other zeros (modulo the lattice). ∎

$z$$w$$C_{1}$$C_{2}$

$\frac{\tau}{2}$$C_{1}$$C_{2}$$0$$\tau$$\tau+1$$1$

Figure 2:  Left: depiction of the real locus of an elliptic curve of the form
(3.19) where $C_{1}$ is the bounded real oval and $C_{2}$ is unbounded. Right:
the two real ovals in the complex torus. For a curve of the type (3.19) the
parameter $\tau$ is purely imaginary, $\tau\in i\mathbb{R}^{+}$.

Because of (3.22) we will say that $C^{(2,-1)}(u,v;a)$ has residue $1$ at
$u=v$. This notion of residue is conformally invariant if we transform $u$ and
$v$ simultaneously by the same conformal map.

As a consistency check, we note that the poles in part (a) of Proposition 3.12
add up to $u+2a$, and this is where the simple zero is. The zeros and poles in
part (b) both add up to $v$, as it should be in view of Abel’s theorem. It is
in accordance with the fact that $C^{(2,-1)}(u,v;a)\frac{dv}{du^{2}}$ is
doubly periodic with periods $1$ and $\tau$ in both $u$ and $v$.

With the Abel map we transform (3.21) to the Riemann surface (3.19) where
${p_{\infty}\in X}$ corresponds to $0$ on the complex torus. We still use
$C^{(2,-1)}$ to denote the $(2,-1)$-Cauchy kernel on $X$. There is actually an
explicit expression in terms of the representation (3.19), thus side-stepping
the necessity of using $\theta$–functions. Taking $p=(w,z)$,
$q=(\widetilde{w},\widetilde{z})$ and $a=(w_{0},z_{0})$, one may verify that

$C^{(2,-1)}(p,q;a)\\\
=\left(\frac{w+\widetilde{w}}{z-\widetilde{z}}-\frac{w_{0}+\widetilde{w}}{z_{0}-\widetilde{z}}+\left(\frac{w_{0}^{\prime}}{z_{0}-\widetilde{z}}-\frac{w_{0}+\widetilde{w}}{(z_{0}-\widetilde{z})^{2}}\right)(z-z_{0})\right)\frac{dz^{2}}{2w^{2}}\frac{\widetilde{w}}{d\widetilde{z}},$
(3.23)

where $w_{0}^{\prime}$ is the derivative
$\frac{d}{dz}\sqrt{z(z-x_{1})(z-x_{2})}\bigg{|}_{z=z_{0}}$.

By Proposition 3.4 a critical measure $\mu$ satisfies

$\iint\left(h(p)C(p,q)+h(q)C(q,p)\right)d\mu(p)d\mu(q)=\int hdV\,d\mu$ (3.24)

for every $C^{1}$ vector field $h$. Choosing $h=C^{(2,-1)}(p,\cdot)$ allows us
to extract relevant information from (3.24).

Now we arrive at a generalization of Theorem 1.2 to a symmetric genus one
case.

######  Theorem 3.13.

Let $X$ be given by (3.19). Suppose that the external field $\varphi$ is given
by $\varphi=\operatorname{Re}V$ for a multi-valued, locally meromorphic
function $V$ on $X$ with a single-valued real part and that $dV$ is a
meromorphic differential on $X$. Suppose $\mu$ is a critical measure in the
external field $\varphi=\operatorname{Re}V$ that is invariant under the
involution (3.20). Suppose $\operatorname{supp}(\mu)\subset
X\setminus(\\{p_{\infty}\\}\cup C_{1})$ where $C_{1}$ is the real bounded oval
of $X$. Then there exists an $a\in C_{1}$ such that the following holds.

  1. (a)

Then

$\left[\int C(u,q)d\mu(q)-\frac{dV(u)}{2}\right]^{2}=\upomega(u),\qquad a.e.\
\text{ on }X,$ (3.25)

where $\upomega$ is the meromorphic quadratic differential

$\upomega(u)=\left(\frac{dV(u)}{2}\right)^{2}-\int(C(u,q)dV(u)-C^{(2,-1)}(u,q;a)dV(q))d\mu(q).$
(3.26)

  2. (b)

The support of $\mu$ is a union of analytic arcs or loops that are maximal
trajectories of $-\upomega$. The measure $\mu$ is absolutely continuous with
respect to arclength, and it is given by the differential

$d\mu(u)=\frac{1}{\pi i}\upomega(u)^{1/2},\qquad
u\in\Sigma:=\operatorname{supp}(\mu),$

with an appropriate branch of the square root on each open arc or closed loop.

  3. (c)

The bipolar Green’s potential

$G^{\mu}(p):=\int G(p,q)d\mu(q)$

of $\mu$ satisfies

$2G^{\mu}(p)+\operatorname{Re}V(p)=c_{j},\qquad p\in\Sigma_{j}$

with a constant $c_{j}$ that can be different on each component $\Sigma_{j}$
of $\Sigma$.

  4. (d)

Any point $p\in\Sigma$ that is not a zero of $\upomega$ has a neighborhood $D$
such $D\cap\Sigma$ is an analytic arc, and

$\frac{\partial}{\partial n_{+}}\left(2G^{\mu}+\operatorname{Re}V\right)=\
\frac{\partial}{\partial n_{+}}\left(2G^{\mu}+\operatorname{Re}V\right)\text{
on }D\cap\Sigma,$ (3.27)

where $\frac{\partial}{\partial n_{\pm}}$ denote the two normal derivatives to
$\Sigma$ inside $D\cap\Sigma$.

The proof of Theorem 3.13 can be found in Section 7.2. The equality (3.27) is
an analogue of the $S$-property (1.8) in the planar case, see also [41,
Equation (5.30)] and [40, Definition 2.1]. Note that the equality of the
normal derivatives is independent of the coordinate chart. The identity in
(3.27) is equivalent to the $S$-property defined in Definition 2.9 by the
Cauchy-Riemann equations.

###  3.6 Overview of the rest of the paper

In the next section we motivate the theory developed in the paper by relating
it to certain periodic tiling models. We focus our discussion on lozenge
tilings with periodic weightings.

In Section 5, we prove Propositions 3.4 and 3.8, which generalize the results
[41, Lemma 3.1] on critical measures and [40, Proposition 4.2] (see also [44,
Section 9.10]) on critical sets in the plane to compact Riemann surfaces of
higher genus. In Section 6, we discuss the max-min energy problem and prove
Theorem 3.11. We rely heavily on techniques developed in [37, 40] and
especially [44]. In Section 7, we prove Theorem 3.13; see in particular
Section 7.2. The appendix contains a proof of Proposition 2.1 using the
Green’s function for the Laplacian from Riemannian geometry.

##  4 Motivation: periodic tiling models

###  4.1 Lozenge tilings of a hexagon

Our motivation comes from two-dimensional random tiling models with periodic
weightings [39]. The main examples are domino tilings of the Aztec diamond [5,
17, 25] and lozenge tilings of a hexagon [14, 16]. We focus in this discussion
on the latter one. See the figure for the $ABC$ hexagon and a possible tiling
with lozenges of three types. The hexagon has vertices at $(0,0)$, $(B,0)$,
$(B+C,C)$, $(B+C,A+C)$, $(C,A+C)$ and $(0,A)$, where $A,B,C$ are positive
integers. The vertices of each lozenge in a lozenge tiling have integer
coordinates, see Figure 3.

$A$$A$$B$$B$$C$$C$

Figure 3: The $ABC$ hexagon (left) with a random tiling (right).

We assign a weight to each lozenge depending on its type and on the position
it has in the hexagon. We denote the weights by $w_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)$,
$w_{\leavevmode\hbox to6.09pt{\vbox
to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.5}\pgfsys@color@cmyk@fill{0}{0}{0.5}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{11.38092pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)$, and
$w_{\leavevmode\hbox to11.78pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8,0.8}\pgfsys@color@rgb@fill{1}{0.8}{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@lineto{11.38092pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)$, where
$(x,y)\in\mathbb{Z}^{2}$ is the position of the lower left vertex in case of
$w_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)$,
$w_{\leavevmode\hbox to6.09pt{\vbox
to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.5}\pgfsys@color@cmyk@fill{0}{0}{0.5}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{11.38092pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)$, and the
position of the lower right vertex in case of $w_{\leavevmode\hbox
to11.78pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8,0.8}\pgfsys@color@rgb@fill{1}{0.8}{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@lineto{11.38092pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)$.

The weight $w(\mathcal{T})$ of a tiling $\mathcal{T}$ is the the product of
the weights of the lozenges in the tiling, and the probability of
$\mathcal{T}$ is proportional to its weight

$\mathbb{P}(\mathcal{T})=\frac{1}{Z}w(\mathcal{T}),\qquad
Z=\sum_{\mathcal{T}}w(\mathcal{T}).$ (4.1)

Two weightings are equivalent if they lead to the same probabilities (4.1). It
is known that this model is determinantal, for any choice of positive weights.

###  4.2 Periodic weightings

The paper [25] gives a formula for the correlation kernel as a double contour
integral in case of periodic weighting, where a weighting is periodic if there
exist positive integers $p$ and $q$ such that

$w_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x+mp,y+nq)=w_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y),\qquad(x,y,m,n\in\mathbb{Z}^{2})$
(4.2)

and similarly for $w_{\leavevmode\hbox to6.09pt{\vbox
to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.5}\pgfsys@color@cmyk@fill{0}{0}{0.5}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{11.38092pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and
$w_{\leavevmode\hbox to11.78pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8,0.8}\pgfsys@color@rgb@fill{1}{0.8}{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@lineto{11.38092pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$. To describe the
formula, we need that $w_{\leavevmode\hbox to11.78pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.8,0.8}\pgfsys@color@rgb@fill{1}{0.8}{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@lineto{11.38092pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y)=1$ for every
$(x,y)\in\mathbb{Z}^{2}$, which is an assumption one can make without loss of
generality, as for any weighting there is an equivalent weighting with this
property. Then for each $x\in\mathbb{Z}$ we consider a matrix $T_{x}$ with
entries

$T_{x}(y,y^{\prime})=\begin{cases}w_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y),&\text{ if
}y^{\prime}=y,\\\ w_{\leavevmode\hbox to6.09pt{\vbox
to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{} {}{} {}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.5}\pgfsys@color@cmyk@fill{0}{0}{0.5}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{5.69046pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{11.38092pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(x,y),&\text{ if
}y^{\prime}=y+1,\\\ 0,&\text{ otherwise,}\end{cases}\quad\text{ for
}(y,y^{\prime})\in\mathbb{Z}^{2},$ (4.3)

that describes a transition from the horizontal level $x$ to $x+1$. Each
$T_{x}$ is a two sided infinite matrix with two non-zero diagonals.

Consider a periodic weighting with periods $p$ and $q$. Then $T_{x}$ is a
block Toeplitz matrix with blocks of size $q\times q$ because of the
periodicity in the vertical direction. The symbol of $T_{x}$ is the matrix
valued function $z\mapsto A_{x}(z)$ with

$A_{x}(z)=\begin{pmatrix}T_{x}(y,y^{\prime})\end{pmatrix}_{y,y^{\prime}=0}^{q-1}+z\begin{pmatrix}T_{x}(y,y^{\prime}+q)\end{pmatrix}_{y,y^{\prime}=0}^{q-1},\qquad
z\in\mathbb{C}.$ (4.4)

Due to the periodicity in the horizontal direction we have $T_{x+p}=T_{x}$,
and $A_{x+p}=A_{x}$ for every $x$. Write

$A(z)=A_{0}(z)A_{1}(z)\cdots A_{p-1}(z).$ (4.5)

The positions of the  and  lozenges in a random tiling give a random point
process inside the hexagon. This random point process is determinantal and a
formula for the correlation kernel $K$ is given (in a more general setting) in
[25, Theorem 4.7]. Let us assume for simplicity that $A=qN$, $C=qM$, $B+C=pL$,
for certain integers $N,M,L$ where $A,B,C$ are the dimensions of the hexagon.
Then $K((px,qy),(px^{\prime},qy^{\prime}))$ is equal to the $(0,0)$ entry of
the matrix

$=-\frac{\chi_{x>x^{\prime}}}{2\pi
i}\oint_{\gamma}A^{x-x^{\prime}}(z)z^{y^{\prime}-y}\frac{dz}{z}\\\
+\frac{1}{(2\pi
i)^{2}}\oint_{\gamma}\oint_{\gamma}A^{L-x^{\prime}}(w)R_{N}(w,z)A^{x}(z)\frac{w^{y^{\prime}}}{z^{y+1}w^{M+N}}dzdw$
(4.6)

where the integrals are taken entrywise, and $R_{N}$ is a bivariate polynomial
of degrees $\leq N-1$ in both variables that has the reproducing kernel
property

$\frac{1}{2\pi i}\oint_{\gamma}P(w)\frac{A^{L}(w)}{w^{M+N}}R_{N}(w,z)dw=P(z)$
(4.7)

for any matrix valued polynomial $P$ of degree $\leq N-1$. The contour
$\gamma$ in (4.6) and (4.7) is a simple closed contour going around the origin
once in the positive direction. Other entries of the matrix (4.6) give the
correlation kernel at positions
$(px,qy+j),(px^{\prime},qy^{\prime}+j^{\prime})$ when $0\leq j,j^{\prime}\leq
q-1$ and there is a similar, but somewhat more complicated formula for the
correlation kernel at arbitrary points when the first coordinates of the
points are not multiples of the period $p$.

The reproducing kernel $R_{N}$ is related to matrix valued orthogonal
polynomials (MVOPs). The monic MVOP $P_{n}$ of degree $n$ (if it exists)
satisfies

$\frac{1}{2\pi
i}\oint_{\gamma}P_{n}(z)\frac{A(z)^{L}}{z^{M+N}}z^{k}dz=H_{n}\delta_{n,k}\quad\text{
for }0\leq k\leq n,$ (4.8)

with $\det H_{n}\neq 0$. If all MVOP up to degree $N-1$ exist then

$R_{N}(w,z)=\sum_{n=0}^{N-1}P_{n}^{T}(w)H_{n}^{-1}P_{n}(z).$ (4.9)

In our setting it can be shown that the MVOP of degree $N$ exists, as well as
certain other degrees, see also [32, Proposition 1.1], but not for all degrees
up to $N-1$, and the sum formula (4.9) is not valid. There is another formula
for $R_{N}$ in terms of the solution of a Riemann-Hilbert problem which we
describe next.

###  4.3 Riemann-Hilbert problem

Consider

$Y(z)=\begin{pmatrix}P_{N}(z)&\displaystyle\frac{1}{2\pi
i}\oint_{\gamma}\frac{P_{N}(s)A(s)^{L}}{s^{M+N}}\frac{ds}{s-z}\\\
-H_{N-1}^{-1}P_{N-1}(z)&\displaystyle-\frac{H_{N-1}^{-1}}{2\pi
i}\oint_{\gamma}\frac{P_{N-1}(s)A(s)^{L}}{s^{M+N}}\frac{ds}{s-z}\end{pmatrix},\quad
z\in\mathbb{C}\setminus\gamma,$ (4.10)

where $P_{N}$ and $P_{N-1}$ are the monic MVOP of degrees $N$ and $N-1$. This
matrix valued function of size $2q\times 2q$ is defined and analytic for
$z\in\mathbb{C}\setminus\gamma$ and satisfies

$Y_{+}(z)=Y_{-}(z)\begin{pmatrix}I_{q}&\displaystyle\frac{A(z)^{L}}{z^{M+N}}\\\
0&I_{q}\end{pmatrix},\qquad z\in\gamma,$ (4.11)

where $Y_{+}(z)$ and $Y_{-}(z)$ denote the limiting values of $Y(z^{\prime})$
as $z^{\prime}\to z\in\gamma$ from the interior and exterior region,
respectively. In addition, $Y$ has the asymptotic behavior

$Y(z)=(I_{2q}+O(z^{-1}))\begin{pmatrix}z^{N}I_{q}&0\\\
0&z^{-N}I_{q}\end{pmatrix}\quad\text{ as }z\to\infty.$ (4.12)

The Riemann-Hilbert (RH) problem (4.11)–(4.12) for MVOP was formulated in [13,
33] as a generalization of the well-known RH problem for orthogonal
polynomials due to Fokas, Its and Kitaev [29]. The reproducing kernel is
expressed in terms of $Y$ via the formula [23, 25]

$R_{N}(w,z)=\frac{1}{w-z}\begin{pmatrix}0&I_{q}\end{pmatrix}Y^{-1}(w)Y(z)\begin{pmatrix}I_{q}\\\
0\end{pmatrix},$ (4.13)

which can be seen as a matrix valued Christoffel-Darboux formula for the sum
(4.9).

Of interest is the regime where $N,M,L\to\infty$ in such a way that their
ratios approach certain finite, non-zero limits. A steepest descent analysis
of the RH problem would give asymptotic information on the reproducing kernel
(4.13) which could then in turn be used in the double integral (4.6) to find
limiting behavior of the correlation kernel. This has been done successfully
for the case of domino tilings of the Aztec diamond with $2\times 2$ periodic
weightings [25]. For periodic hexagon tilings with periodicities at most two,
the matrix valued orthogonality can be reduced to scalar orthogonality on a
contour in the complex plane [14, 15, 16, 32] where notions from logarithmic
potential theory (such as Theorem 1.2) apply. The asymptotic analysis of
periodic hexagon tilings with higher periodicities remains an open problem,
since one of the difficulties is to normalize the RH problem at infinity. For
the RH problem for orthogonal polynomials on the real line [21, 22], this is
done with an equilibrium measure in an external field [46], and we are faced
with the question of what appropriate equilibrium measures in a matrix-valued
setting are when the matrix valued orthogonality cannot be reduced to scalar
orthogonality in the plane.

###  4.4 Transformation $Y\mapsto Z$

To proceed we follow [25, Section 5.3] to perform a transformation of the RH
problem by using the spectral decomposition $A(z)=E(z)\Lambda(z)E(z)^{-1}$ of
(4.5) where $\Lambda(z)$ is a diagonal matrix with the eigenvalues
$\lambda_{1}(z),\ldots,\lambda_{q}(z)$ of $A(z)$ on the diagonal and the
columns of $E(z)$ are eigenvectors of $A(z)$. The transformation

$Z=Y\begin{pmatrix}E&0\\\ 0&E\end{pmatrix}$ (4.14)

then gives a matrix valued function $Z$ with the jump condition

$Z_{+}(z)=Z_{-}(z)\begin{pmatrix}I_{q}&\displaystyle\frac{\Lambda(z)^{L}}{z^{M+N}}\\\
0&I_{q}\end{pmatrix}\quad z\in\gamma.$ (4.15)

The eigenvalues $\lambda_{j}(z)$ and eigenvectors in $E(z)$ will not be entire
functions of the $z$ variable and will be single-valued only if we apply
certain branch cuts. The definition (4.14) will also create a jump of $Z$ on
these cuts that takes the form (with an appropriate choice of the eigenvectors
in $E(z)$)

$Z_{+}=Z_{-}\begin{pmatrix}\sigma&0\\\ 0&\sigma\end{pmatrix}\quad\text{ on
branch cuts},$ (4.16)

for a permutation matrix $\sigma\in S_{q}$ that could depend on the branch
cut.

###  4.5 Scalar orthogonality on the Riemann surface

The characteristic equation

$X:\quad\det(\lambda I-A(z))=0$ (4.17)

is a polynomial equation in $z$ and $\lambda$. It is the spectral curve in the
sense of [39] (up to a sign change in $\lambda$ in case $n$ is odd), see in
particular [39, p. 1048]. The algebraic curve $X$ is a Harnack curve which
implies that it has the maximal number of real ovals [39, Theorem 5.1]. In
particular it has only real branch points when viewed as a $q$-fold cover of
the $z$-plane, and the branch cuts in (4.16) are on the real line only. The
case where $X$ has genus $\geq 1$ is of special interest as it allows for the
appearance of a smooth (or gas) phase in the large scale limit, in addition to
the more familiar solid and rough (or liquid) phases [39]. We show that the
matrix valued orthogonality leads to scalar orthogonality conditions on $X$.
See [7, 8, 9, 15, 27] for other recent contributions to orthogonality on a
Riemann surface.

Returning to $Z$, we choose a row number $j\in\\{1,\ldots,2q\\}$. Then for
$k=1,\ldots,q$, we consider $Z_{j,k}$ as a function on the $k$th sheet of $X$.
The jump conditions (4.15) and (4.16) show that this defines a meromorphic
function $f_{j}$ on $X$ with a pole at the point(s) at infinity. The entry
$Z_{j,q+k}$ for $k=1,\ldots,q$ is also considered on the $k$th sheet, and the
jump properties (4.15) and (4.16) imply that they define a holomorphic
function $\psi_{j}$ on $X\setminus\gamma_{X}$ (we use $\gamma_{X}$ to denote
the contour on $X$ consisting of a copy of $\gamma$ on each of the sheets)
with the jump

$(\psi_{j})_{+}=(\psi_{j})_{-}+f_{j}\frac{\lambda^{L}}{z^{M+N}}\quad\text{ on
}\gamma_{X}.$ (4.18)

The asymptotic condition (4.12) and the definition (4.14) imply that $f_{j}$
has a pole and $\psi_{j}$ has a zero at each point at infinity. The precise
order of the poles and zeros depends on the possible branching of $X$ at
infinity, and the behavior of $E(z)$ at infinity. In any case the conditions
(4.18) imply certain orthogonality conditions for $f_{j}$ of the form

$\int_{\gamma_{X}}f_{j}\frac{\lambda^{L}}{z^{M+N}}\omega=0,$ (4.19)

for a large class of meromorphic differentials $\omega$ on $X$ with poles at
infinity.

Closer inspection shows that the pole conditions at infinity imply that
$f_{j}$ belongs to a vector space of meromorphic functions of dimension
$\approx qN$ and the orthogonality conditions (4.19) are for $\omega$ in a
vector space of meromorphic differentials of dimension $\geq qN-c$ with a
constant $c$ that is independent of $N,M,L$. Typically (4.19) will not be
enough to characterize $f_{j}$ as an orthogonal meromorphic function on $X$,
as the conditions (4.19) will have to be supplemented by $\approx c$
additional conditions.

Since $c$ is independent of $N,M,L$ we expect that in generic cases the
additional conditions do not influence the behavior of the bulk of the zeros
of $f_{j}$ in the large $N,M,L$ limit. We expect that the zeros will
accumulate on an $S$-curve in an external field on the Riemann surface $X$, as
it is known in the genus zero case. The equilibrium measure in external field
on an $S$-curve has a $g$-function that will be useful in the next
transformation in the steepest descent analysis of the Riemann-Hilbert
problem.

To illustrate, let us take $q=2$ and assume that $X$ has branching at
infinity. Let $g_{1}$ be the restriction of the $g$ function to the first
sheet, and $g_{2}$ to the second sheet. Then we expect that the following
transformation $Z\to U$,

$U=Z\operatorname{diag}\left(e^{-2Ng_{1}},e^{-2Ng_{2}},e^{2Ng_{1}},e^{2Ng_{2}}\right)$
(4.20)

will be the appropriate next step in the steepest descent analysis.

###  4.6 The two-periodic Aztec diamond

The analogous model of the domino tilings of the Aztec diamond with two-
periodic weightings was successfully analyzed in [25] but without the use of
$g$-functions. The transformation $Z\to U$ in [25, formula (5.20)] (note that
$X$ is used in [25] instead of $Z$) is explicit in terms of $\lambda$ and $z$.
Let us show here that the transformation is actually of the form (4.20).

The Riemann surface $X$ in [25] is associated with the equation

$X:\qquad w^{2}=z(z-x_{1})(z-x_{2})$

with real $x_{1},x_{2}$ given by $x_{1}=-\alpha^{2}$, $x_{2}=-\alpha^{-2}$ for
some $\alpha>1$. A point on $X$ is denoted by $p=(w,z)$ and we write $w=w(p)$,
$z=z(p)$, for the projections onto the $w$ and $z$ coordinate. Then $X$ is a
two-sheeted compact Riemann surface with two sheets
$\mathbb{C}\setminus(-\infty,x_{1}]\cup[x_{2},0])$. The first sheet is such
that $w(p)>0$ for $p=(w,z)$ with $z(p)>0$ on the first sheet.

The meromorphic function $\lambda$ on $X$ has a double pole at $p_{1}$ where
$p_{1}$ is the point on the first sheet with $z(p_{1})=1$, and a double zero
at $p_{2}$, the point on the second sheet with $z(p_{2})=1$. There are no
other zeros or poles [25, Lemma 5.2]. The bipolar Green’s function
$G(p,p_{1})$ with pole at infinity and $p_{1}$ is explicit in terms of
$\lambda$, namely for some constant $c$,

$G(p,p_{1})=\frac{1}{4}\log|\lambda(p)|-\frac{1}{2}\log|z(p)-1|+c$ (4.21)

since indeed the right-hand side satisfies the requirements of Proposition 2.1
(a), (b) and (c). Note that (4.21) is finite at $p=p_{2}$ since $\lambda$ has
a double zero at $p_{2}$. Without loss of generality we may assume that the
bipolar Green’s function is taken such that $c=0$ in the above formula.

Let $0<r\leq 1$ and let $\mu_{r}$ be the uniform normalized Lebesgue measure
on the circle $F_{r}:|z-1|=r$ on the first sheet. By conformal invariance it
is the balayage of the point mass $\delta_{p_{1}}$ onto $F_{r}$, which means
that $\int G(p,q)d\mu_{r}(q)=G(p,p_{1})$ for $p\in X$ in the exterior
$\Omega_{\text{ext}}$ of the circle (which includes the entire second sheet).
The equality extends to $F_{r}$, and thus $2\int
G(p,q)d\mu_{r}(q)=\frac{1}{2}\log|\lambda(p)|-\log r$ on the support of
$\mu_{r}$, which means that $\mu_{r}$ is the equilibrium measure of $F_{r}$ in
the external field $-\frac{1}{2}\log|\lambda|$, cf. Proposition 2.8.

We verify that $F_{r}$ has the $S$-property in the external field. Since
$\frac{1}{4}\log|\lambda(p)|+\frac{1}{2}\log|z(p)-1|$ is harmonic in the
interior domain $\Omega_{\text{int}}$ of $F_{r}$ (including at $p_{1}$) and
agrees with $G(p,p_{1})+\log r$ on the circle. We conclude

$\int
G(p,q)d\mu_{r}(q)=\begin{cases}\frac{1}{4}\log|\lambda(p)|-\frac{1}{2}\log|z(p)-1|,&\text{
in }\Omega_{ext},\\\\[5.0pt]
\frac{1}{4}\log|\lambda(p)|+\frac{1}{2}\log|z(p)-1|-\log r,&\text{ in
}\Omega_{int}.\end{cases}$

Then, with appropriate branches of the logarithm

$g(p)=\begin{cases}-\frac{1}{4}\log\lambda(p)+\frac{1}{2}\log(z(p)-1),&\text{
in }\Omega_{ext}\\\\[5.0pt]
-\frac{1}{4}\log\lambda(p)-\frac{1}{2}\log(z(p)-1)-\log r,&\text{ in
}\Omega_{int}\end{cases}$

and indeed $g_{+}+g_{-}+\frac{1}{2}\log\lambda$ is constant on
$\operatorname{supp}(\mu)$, which is the $S$-property in external field by
Definition 2.9. It follows that the transformation in [25, Section 5.4] is
indeed of the form (4.20).

The $2\times k$ periodic Aztec diamond was studied by Berggren [5] using a
Wiener-Hopf factorization technique developed in [6]. The formula (4.21)
applies in the $2\times k$ periodic case as well and it gives rise to an
equilibrium measure in the same way as described above.

###  4.7 Hexagon tilings

In the Aztec diamond example, one is fortunate to be able to find a contour
with the $S$-property via an explicit construction. In the hexagon tiling
models this does not seem to work and that is why we redeveloped the methods
of this paper in order to prove the existence of contours with the
$S$-property in an external field.

The final result of this paper, Theorem 3.13, is restricted to a special
situation of genus one. It will apply to lozenge tilings with periodic
weightings of periods $p=3$ and $q=2$. In that case the spectral curve (4.17)
has genus one and it can be put in the form $X:\quad
w^{2}=(z-x_{1})(z-x_{2})(z-x_{3})$ with $x_{1}<x_{2}<x_{3}\leq 0$. If
$x_{3}=0$ then we are in the situation of (3.19). The external field is

$\varphi(z)=\operatorname{Re}V(z)=-\frac{b}{2}\log|\lambda|+\frac{1+c}{2}\log|z|$
(4.22)

with $b=\lim\frac{L}{N}$, $c=\lim\frac{M}{N}$, $3b>2c$, which is invariant
under the involution (3.20). The condition $3b>2c$ comes from the fact that
$3L=B+C>C=2M$, hence $3\frac{L}{N}>2\frac{M}{N}$, and we take the large $N$
limit such that the strict inequality remains valid. Then $dV$ has residue
$r_{0}=1+c$ at $0$ (due to the branching at $0$). Since $\lambda$ will have a
third order pole at $\infty$ and $z$ has a second order pole there, one may
calculate from (4.22) that $dV$ has a simple pole at $\infty$ with residue
$r_{\infty}=3\frac{b}{2}-2\frac{1+c}{2}=\frac{3b-2c}{2}-1$. Thus the residue
conditions of Theorem 3.11 are satisfied, and thus there is a critical set $F$
with equilibrium measure $\mu^{F}$ that is a critical measure in the external
field $\varphi$. By symmetry we may assume that $F$ and $\mu^{F}$ are
invariant under the involution $\sigma$.

If the support of $\mu^{F}$ does not intersect the bounded real oval $C_{1}$,
then by Theorem 3.13, the support is a union of maximal trajectories of a
quadratic differential $-\upomega$ given by (3.26). Note that by (3.25),
$\upomega$ has double poles at the poles of $dV$, but also at $p_{\infty}$ due
to the simple pole of $\int C(u,q)d\mu(q)$ at $u=p_{\infty}$. Counting
multiplicities, $\upomega$ thus has a total of $8$ poles. Because we are in
genus one, we find that $\upomega$ then also has $8$ zeros (counted with
multiplicity).

The equilibrium measure in external field and its associated $g$-function will
be useful in the steepest descent analysis of the RH problem as we indicated.
The details of the steepest descent analysis, and their consequences for the
random tiling model are under current investigation.

##  5 Proofs of Propositions 3.4 and 3.8

###  5.1 Proof of Proposition 3.4

###### Proof.

Let $\Phi$ denote the flow associated with the $C^{1}$ vector field $h$. We
have

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}V(\Phi(\varepsilon,p))=h(p)dV(p)$
(5.1)

and thus by (3.4)

$\displaystyle\int Vd\mu_{\varepsilon,h}$ $\displaystyle=\int
V(\Phi(\varepsilon,p))d\mu(p)$ $\displaystyle=\int Vd\mu+\varepsilon\int
hdVd\mu+o(\varepsilon)\quad\text{ as }\varepsilon\to 0.$ (5.2)

Since $p\mapsto G(p,q)$ is harmonic, it is locally the real part of a
holomorphic function $\mathcal{G}(p,q)$ and

$C(p,q)=-2\partial_{p}G(p,q)=-\partial_{p}\mathcal{G}(p,q).$

Then for $p\neq q$,

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\mathcal{G}(\Phi(\varepsilon,p),q)=-h(p)C(p,q)$

and by taking the real part

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}G(\Phi(\varepsilon,p),q)=-\operatorname{Re}(h(p)C(p,q)).$

Since $G$ is symmetric in the two variables, we then also have

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}G(p,\Phi(\varepsilon,q))=-\operatorname{Re}(h(q)C(q,p)),$

and by the chain rule

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}G(\Phi(\varepsilon,p),\Phi(\varepsilon,q))=-\operatorname{Re}(h(p)C(p,q)+h(q)C(q,p))$
(5.3)

Then because of (2.9), (3.4) and (5.3).

$E[\mu_{\varepsilon,h}]=\iint
G(\Phi(\varepsilon,p),\Phi(\varepsilon,q))d\mu(p)d\mu(q)\\\
=E[\mu]-\varepsilon\operatorname{Re}\iint(h(p)C(p,q)+h(q)C(q,p))d\mu(p)d\mu(q)+o(\varepsilon)$
(5.4)

as $\varepsilon\to 0$. Combining (5.2) and (5.4) we obtain the limit in part
(a).

If $\mu$ is critical then the expression in part (a) is zero for every $h$,
and this implies that the real part of $D_{V,h}(\mu)$ vanishes. Changing $h$
to $ih$, we find that the imaginary part vanishes as well, and part (b)
follows. ∎

###  5.2 Convergence of perturbed measures in energy norm

The next lemma concerns the convergence of the measures in energy norm; see
(2.11).

######  Lemma 5.1.

Suppose $\mu$ is compactly supported in $X\setminus\\{p_{\infty}\\}$ with
$E[\mu]<+\infty$. Let $h$ be a $C^{1}$ vector field. Then the measures
$\mu_{\varepsilon,h}$ tend to $\mu$ as $\varepsilon\to 0$ in energy norm.

###### Proof.

If $f$ is a continuous function, then

$\int fd\mu_{\varepsilon,h}=\int f(\Phi(\varepsilon,p))d\mu(p)$

and $f(\Phi(\varepsilon,p))\to f(p)$ for every $p$. Since $f$ is bounded on
$K$, we can apply dominated convergence to conclude that $\int
fd\mu_{\varepsilon,h}\to\int fd\mu$ as $\varepsilon\to 0$. This proves the
weak∗ convergence of $(\mu_{\varepsilon,h})_{\varepsilon}$ to $\mu$.

Since $q\mapsto\int G(p,q)d\mu(p)$ is lower semi-continuous, we have by weak∗
convergence

$E[\mu]=\iint G(p,q)d\mu(p)d\mu(q)\leq\liminf_{\varepsilon\to 0}\iint
G(p,q)d\mu(p)d\mu_{\varepsilon,h}(q).$

From (5.4) we conclude $E[\mu_{\varepsilon,h}]\to E[\mu]$, and therefore

$\displaystyle\limsup_{\varepsilon\to 0}E[\mu_{\varepsilon,h}-\mu]$
$\displaystyle=\limsup_{\varepsilon\to
0}\left(E[\mu_{\varepsilon,h}]+E[\mu]-2\iint
G(p,q)d\mu(p)d\mu_{\varepsilon,h}(q)\right)$ $\displaystyle\leq
E[\mu]+E[\mu]-2E[\mu]=0,$

which is the convergence in energy norm. ∎

For later use, we need a uniformity in the limit (3.6) for measures on a fixed
compact.

######  Lemma 5.2.

Let $K\subset X\setminus\\{p_{\infty}\\}$ be a compact set such that
$\varphi=\operatorname{Re}V$ is bounded on $K$. Let $h$ be a $C^{1}$ vector
field. Then for every probability measure $\mu$ on $K$ with $E[\mu]<+\infty$,
we have

$E_{\varphi}[\mu_{\varepsilon,h}]-E_{\varphi}[\mu]-\varepsilon\operatorname{Re}D_{V,h}(\mu)=o(\varepsilon)\quad\text{
as }\varepsilon\to 0,$ (5.5)

where the $o(\varepsilon)$ only depends on $h$ and $K$, but not on $\mu$.

###### Proof.

Let $\Phi$ be the flow associated with the vector field $h$. Since $\varphi$
is bounded on $K$, the set $K$ does not contain any of the poles of $dV$. Then
there is $\varepsilon>0$ small enough such that

$\bigcup_{t\in[-\varepsilon,\varepsilon]}\bigcup_{p\in K}\Phi(t,p)$ (5.6)

is a compact subset of $X\setminus\\{p_{\infty}\\}$ that does not contain any
of the poles of $dV$. Thus $\varphi$ is bounded on the set (5.6).

We already observed in Remark 3.5 that

$(p,q)\mapsto H(p,q)=h(p)C(p,q)+h(q)C(q,p)$ (5.7)

is a well defined function on $(X\setminus\\{p_{\infty}\\})^{2}$. It is
clearly $C^{1}$ function for $p\neq q$, since we assumed that $h$ is $C^{1}$.

Suppose $p,q$ are in a local chart, where $p$ and $q$ correspond to $z$ and
$w$ respectively in the local coordinate. Then

$C(p,q)=\frac{1}{z-w}dz+\widetilde{C}(z,w)dz$

where $\widetilde{C}(z,w)$ is $C^{\infty}$ smooth. Write
$h=h_{1}\frac{\partial}{\partial z}+h_{2}\frac{\partial}{\partial\bar{z}}$
with $C^{1}$ functions $h_{1}$ and $h_{2}$. Then by (5.7)

$\displaystyle H(p,q)$
$\displaystyle=h_{1}(z)\left(\frac{1}{z-w}+\widetilde{C}(z,w)\right)+h_{1}(w)\left(\frac{1}{w-z}+\widetilde{C}(w,z)\right)$
$\displaystyle=\frac{h_{1}(z)-h_{1}(w)}{z-w}+h_{1}(z)\widetilde{C}(z,w)+h_{1}(w)\widetilde{C}(w,z)$

which shows that $H$ is a continuous function in a local chart. Therefore $H$
is uniformly continuous on the compact set (5.6). It follows from the uniform
continuity that

$\sup_{|t|\leq\varepsilon}\sup_{p,q\in K}|H(\Phi(t,p),\Phi(t,q))-H(p,q)|=o(1)$
(5.8)

as $\varepsilon\to 0$, where the $o$-term only depends on $h$ and the compact
set $K$.

Next, by integrating (5.3) and recalling (5.7), we find

$G(\Phi(\varepsilon,p),\Phi(\varepsilon,q))-G(p,q)+\varepsilon\operatorname{Re}H(p,q)\\\
=-\operatorname{Re}\int_{0}^{\varepsilon}\left[H(\Phi(s,p),\Phi(s,p))-H(p,q)\right]ds.$

Combining this with (5.8), we arrive at the estimate

$\left|G(\Phi(\varepsilon,p),\Phi(\varepsilon,q))-G(p,q)+\varepsilon\operatorname{Re}H(p,q)\right|=o(\varepsilon),\quad
p,q\in K,$

as $\varepsilon\to 0$, where the $o$-term only depends on $h$ and $K$. We
integrate with respect to $d\mu(p)d\mu(q)$ where $\mu$ is a probability
measure on $K$ with $E[\mu]<\infty$, to find that as $\varepsilon\to 0$,

$\left|\iint G(p,q)d\mu_{\varepsilon,h}(p)d\mu_{\varepsilon,h}(q)-\iint
G(p,q)d\mu(p)d\mu(q)\right.\\\ \left.+\varepsilon\operatorname{Re}\iint
H(p,q)d\mu(p)d\mu(q)\right|=o(\varepsilon),$ (5.9)

with the $o$-term independent of $\mu$.

Using (5.1) and the fact that $\operatorname{Re}V$ is bounded on (5.6) we find
in an analogous way that for any probability measure $\mu$ on $K$, as
$\varepsilon\to 0$,

$\left|\int\operatorname{Re}Vd\mu_{\varepsilon,h}-\int\operatorname{Re}Vd\mu-\varepsilon\int
hdVd\mu\right|=o(\varepsilon).$ (5.10)

Combining (5.9) and (5.10), and recalling the definitions, we obtain (5.5). ∎

###  5.3 Proof of Proposition 3.8

###### Proof.

Let $F\subset X\setminus\\{p_{\infty}\\}$ be a union of continua such that
$\operatorname{Re}V$ is bounded from below on $F$ and let $\mu^{F}$ be the
equilibrium measure of $F$ in the external field $\varphi$.

Part (b) follows directly from the definition once part (a) has been
established, so we only need to prove part (a).

To establish (3.12) it suffices to show that

$E_{\varphi}(F_{\varepsilon,h})-E_{\varphi}(F)=E_{\varphi}[\mu_{\varepsilon,h}^{F}]-E_{\varphi}[\mu]+o(\varepsilon)$
(5.11)

as $\varepsilon\to 0$, since we already know from Proposition 3.4 that the
limit in the right-hand side of (3.12) exists, and having (5.11) we conclude
that the limit in the left-hand side also exists and the equality (3.12)
holds.

For a $C^{1}$ vector field $h$, and $\varepsilon\in\mathbb{R}$, we are going
to use the four probability measures $\mu^{F}$, $\mu^{F}_{\varepsilon,h}$,
$\mu^{F_{\varepsilon,h}}$ and $(\mu^{F_{\varepsilon,h}})_{-\varepsilon,h}$. To
simplify notation, we write

$\mu=\mu^{F},\quad\mu_{\varepsilon}=\mu^{F}_{\varepsilon,h},\quad\mu^{\varepsilon}=\mu^{F_{\varepsilon,h}},\quad\mu^{\varepsilon}_{-\varepsilon}=(\mu^{F_{\varepsilon,h}})_{-\varepsilon,h}.$

Thus $\mu^{\varepsilon}$ is the equilibrium measure of $F_{\varepsilon,h}$
(see (3.10)) in the external field $\varphi$, and therefore

$E_{\varphi}(F_{\varepsilon,h})=E_{\varphi}[\mu^{\varepsilon}]\leq
E_{\varphi}[\mu_{\varepsilon}],$ (5.12)

since $\mu_{\varepsilon}$ is supported on $F_{\varepsilon,h}$. Since
$\mu^{\varepsilon}_{-\varepsilon}$ is the image of $\mu^{\varepsilon}$ under
the reverse flow associated with $h$, it is supported on $F$, and hence

$E_{\varphi}(F)=E_{\varphi}[\mu]\leq
E_{\varphi}[\mu^{\varepsilon}_{-\varepsilon}],$ (5.13)

Applying Lemma 5.2 to $\mu$ and $\mu^{\varepsilon}$ (with $-\varepsilon$
instead of $\varepsilon$ for the latter measure), we obtain

$\displaystyle E_{\varphi}[\mu_{\varepsilon}]$
$\displaystyle=E_{\varphi}[\mu]+\varepsilon\operatorname{Re}D_{V,h}(\mu)+o(\varepsilon),$
(5.14) $\displaystyle E_{\varphi}[\mu^{\varepsilon}]$
$\displaystyle=E_{\varphi}[\mu^{\varepsilon}_{-\varepsilon}]+\varepsilon\operatorname{Re}D_{V,h}\left(\mu^{\varepsilon}\right)+o(\varepsilon).$

Subtracting the two identities in (5.14) and using the inequalities (5.12) and
(5.13), we get

$0\leq
E_{\varphi}[\mu_{\varepsilon}]-E_{\varphi}[\mu^{\varepsilon}]\leq\varepsilon\operatorname{Re}D_{V,h}(\mu)-\varepsilon\operatorname{Re}D_{V,h}\left(\mu^{\varepsilon}\right)+o(\varepsilon)$
(5.15)

The expression (3.7) shows that $D_{V,h}(\mu^{\varepsilon})$ remains bounded
as $\varepsilon\to 0$ and therefore by (5.15)

$\lim_{\varepsilon\to
0}\left(E_{\varphi}[\mu_{\varepsilon}]-E_{\varphi}[\mu^{\varepsilon}]\right)=0.$
(5.16)

Next, we use the identity

$E[\mu_{\varepsilon}-\mu^{\varepsilon}]=2E_{\varphi}[\mu_{\varepsilon}]+2E_{\varphi}[\mu^{\varepsilon}]-4E_{\varphi}\left[\frac{\mu_{\varepsilon}+\mu^{\varepsilon}}{2}\right]$

and the inequality

$E_{\varphi}(F_{\varepsilon,h})\leq
E_{\varphi}\left[\frac{\mu_{\varepsilon}+\mu^{\varepsilon}}{2}\right],$

which holds since $\frac{\mu_{\varepsilon}+\mu^{\varepsilon}}{2}$ is a
probability measure on $F_{\varepsilon,h}$, to find that

$0\leq E[\mu_{\varepsilon}-\mu^{\varepsilon}]\leq
2E_{\varphi}[\mu_{\varepsilon}]-2E_{\varphi}[\mu^{\varepsilon}].$

Hence in view of (5.16)

$\lim_{\varepsilon\to 0}E[\mu_{\varepsilon}-\mu^{\varepsilon}]=0.$ (5.17)

Since $\mu_{\varepsilon}$ tends to $\mu$ in energy norm, see Lemma 5.1, we
find from (5.17) that $\mu^{\varepsilon}$ also tends to $\mu$ in energy norm,
and in particular in weak∗ sense. Then

$\lim_{\varepsilon\to 0}D_{V,h}(\mu^{\varepsilon})=D_{V,h}(\mu)$ (5.18)

by weak∗ convergence, since the functions that appear in the integrals (3.7)
are bounded and continuous. Using (5.18) in (5.15), we find

$E_{\varphi}[\mu_{\varepsilon}]-E_{\varphi}[\mu^{\varepsilon}]=o(\varepsilon)$

as $\varepsilon\to 0$. Combining this with the equalities in (5.12) and
(5.13), we obtain (5.11), and the proof of Proposition 3.8 is complete. ∎

##  6 Proof of Theorem 3.11

###  6.1 A preliminary lemma

######  Lemma 6.1.

Suppose $F\in\mathcal{F}_{\delta}$, and let $\widetilde{F}$ be a connected
component of $F$.

  1. (a)

If $\widetilde{F}\subset U_{\delta_{0}}(p_{0})$ then $\widetilde{F}$ separates
$p_{0}$ from $\partial U_{\delta_{0}}(p_{0})$.

  2. (b)

If $\widetilde{F}\subset U_{\delta_{\infty}}(p_{\infty})$ then $\widetilde{F}$
separates $p_{\infty}$ from $\partial U_{\delta_{\infty}}(p_{\infty})$.

  3. (c)

There is $\eta>0$, depending only on $\delta$, such that
$\operatorname{diam}(\widetilde{F})\geq\eta$.

###### Proof.

Any contour $\gamma\in\mathcal{S}_{\delta}$ has the properties (a) and (b),
since otherwise $\gamma$ would be homotopic to a point in
$X\setminus\\{p_{0},p_{\infty}\\}$, which would contract the definition of
$\mathcal{S}_{\delta}$ in Definition 3.10 (a).

We next prove that

$\inf_{\gamma\in\mathcal{S}_{\delta}}\operatorname{diam}(\gamma)>0.$ (6.1)

By compactness of $X$ there is an $\eta>0$ such that every $\eta$ neighborhood
of a point is contractible. Thus any $\gamma\in S_{\delta}$ that is not
contractible in $X$ must have $\operatorname{diam}(\gamma)\geq\eta$. Now
suppose $\gamma\in\mathcal{S}_{\delta}$ is contractible in $X$. Then
$X\setminus\gamma$ has a simply connected component that contains at least one
of $p_{0},p_{\infty}$ since otherwise $\gamma$ would be contractible in
$X\setminus\\{p_{0},p_{\infty}\\}$ which again would contradict Definition
3.10 (a). If the simply connected component contains both $p_{0}$ and
$p_{\infty}$ then $\gamma$ will have a minimal diameter. If the simply
connected component contains only one of $p_{0}$ and $p_{\infty}$, then
$\gamma$ separates $p_{0}$ from $p_{\infty}$ and it will have a minimum
diameter since $\gamma$ is outside of $U_{\delta}(p_{0})$ and
$U_{\delta}(p_{\infty})$. Thus (6.1) holds.

Thus properties (a), (b), and (c) hold if $F=\gamma\in\mathcal{S}_{\delta}$.
Then by (3.16) it also holds if $F=\gamma\in\mathcal{T}_{\delta}$. The
properties are preserved under taking closure in the Hausdorff metric and the
lemma follows. ∎

###  6.2 Estimates of $E_{\varphi}(F)$

We start with a proposition, whose proof should be compared to that of [40,
Proposition 5.1].

######  Proposition 6.2.

Suppose $\varphi$ has the behavior (3.14) near $p_{0}$ and the behavior (3.15)
near $p_{\infty}$. Let $\mathcal{F}$, and $\delta_{0},\delta_{\infty}$ be as
in Definition 3.10. Then there is a constant $C$ such that the following hold.

  1. (a)

If $F\in\mathcal{F}$ intersects $\partial U_{\delta}(p_{0})$ for some
$\delta<\delta_{0}/2$ then

$E_{\varphi}(F)\leq(r_{0}-1)\log\delta+C.$ (6.2)

  2. (b)

If $F\in\mathcal{F}$ intersects $\partial U_{\delta}(p_{\infty})$ for some
$\delta<\delta_{\infty}/2$, then

$E_{\varphi}(F)\leq(1+r_{\infty})\log\delta+C.$ (6.3)

###### Proof.

(a) Take $0<\delta<\delta_{0}/2$ and assume $F\in\mathcal{F}$ intersects
$\partial U_{\delta}(p_{0})$. Let $\widetilde{F}$ be a connected component of
$F$ that intersects $\partial U_{\delta}(p_{0})$. Then either $\widetilde{F}$
is fully contained in $U_{2\delta}(p_{0})$ and we put $F_{0}=F$, or it
contains a sub-continuum $F_{0}\subset\widetilde{F}$ that intersects both
$\partial U_{\delta}(p_{0})$ and $\partial U_{2\delta}(p_{0})$ and that is
contained in $\overline{U_{2\delta}(p_{0})}$.

In both cases $z_{0}(F_{0})$ is a continuum in $\mathbb{C}$ and we claim that

$\operatorname{diam}(z_{0}(F_{0}))\geq\delta,$ (6.4)

where $z_{0}$ is the local coordinate at $p_{0}$. The claim (6.4) follows
immediately if $F_{0}$ intersects both $\partial U_{\delta}(p_{0})$ and
$\partial U_{2\delta}(p_{0})$, since in that case the $z_{0}$-images of the
two intersection points are in $z_{0}(F_{0})$ with distance $\geq\delta$. The
claim (6.4) also follows if $F_{0}=\widetilde{F}\subset U_{2\delta}(p_{0})$,
since then $z_{0}(F_{0})$ is a continuum in $\mathbb{C}$ that intersects
$\partial D(0,\delta)$ and separates $p_{0}$ from $p_{\infty}$ by Lemma 6.1.

Let $\omega$ be the equilibrium measure from usual logarithmic potential
theory for the continuum $z_{0}(F_{0})\subset\overline{D(0,2\delta)}$. From
(6.4) it follows that the capacity of $z_{0}(F_{0})$ is at least the capacity
of a straight line segment of length $\delta$, which is $\delta/4$, see e.g.
[45, Theorem 5.3.2]. Therefore

$\iint\log\frac{1}{|z-w|}d\omega(z)d\omega(w)\leq-\log\frac{\delta}{4}.$ (6.5)

Because of (2.7) we have for some constant $C_{1}$

$G(p,q)\leq\log\frac{1}{|z_{0}(p)-z_{0}(q)|}+C_{1},\qquad p,q\in
U_{\delta_{0}}(p_{0})$ (6.6)

Let $\mu$ be the pullback of $\omega$ by the local coordinate $z_{0}$. Then
$\mu$ is a probability measure on $F\cap\overline{U_{2\delta}(p_{0})}\subset
U_{\delta_{0}}(p_{0})$ so that by (6.6), the properties of the pullback
measure, and (6.5)

$\displaystyle\iint G(p,q)d\mu(p)d\mu(q)$
$\displaystyle\leq\iint\log\frac{1}{|z_{0}(p)-z_{0}(q)|}d\mu(p)d\mu(q)+C_{1}$
$\displaystyle=\iint\log\frac{1}{|z-w|}d\omega(z)d\omega(w)+C_{1}$
$\displaystyle\leq-\log\delta+\log 4+C_{1}.$ (6.7)

Because of (3.14) there is $C_{2}$ such that

$\varphi(p)\leq r_{0}\log|z_{0}(p)|+C_{2},\qquad\text{ for }p\in
U_{\delta_{0}}(p_{0}).$

Since $|z_{0}(p)|\leq 2\delta$ for
$z\in\operatorname{supp}(\mu)\subset\overline{U_{2\delta}(p_{0})}$, we thus
find

$\int\varphi(p)d\mu(p)\leq r_{0}\log(2\delta)+C_{2}.$ (6.8)

Combining (6.7) and (6.8) we obtain

$E_{\varphi}[\mu]\leq(r_{0}-1)\log\delta+(r_{0}+2)\log 2+C_{1}+C_{2}.$

Since $\mu$ is a probability measure on $F_{0}\subset F$, we have
$E_{\varphi}(F)\leq E_{\varphi}[\mu]$ and (6.2) follows.

(b) The proof of part (b) is similar. We take $0<\delta<\delta_{\infty}/2$ and
we assume that $F\in\mathcal{F}$ intersects $\partial U_{\delta}(p_{\infty})$.
Let $\widetilde{F}$ be a connected component of $F$ that intersects $\partial
U_{\delta}(p_{\infty})$.

Similar to (6.4) we now find a sub-continuum
$F_{0}\subset\widetilde{F}\cap\overline{U_{2\delta}(p_{\infty})}$ with

$\operatorname{diam}\left(z_{\infty}(F_{0})\right)\geq\delta.$

Let $\omega$ be the equilibrium measure of $z_{\infty}(F_{0})$, and $\mu$ its
pullback under the mapping $p\mapsto z_{\infty}(p)$. Then (6.5) again holds,
and from (2.6) we find for some constant $C_{1}$,

$\displaystyle\iint G(p,q)d\mu(p)d\mu(q)$
$\displaystyle\leq\iint\log\frac{1}{|z_{\infty}(p)^{-1}-z_{\infty}(q)^{-1}|}d\mu(p)d\mu(q)+C_{1}$
$\displaystyle=\iint\log\frac{1}{|z^{-1}-w^{-1}|}d\omega(z)d\omega(w)+C_{1}$
$\displaystyle=\iint\log\frac{|zw|}{|z-w|}d\omega(z)d\omega(w)+C_{1}$
$\displaystyle\leq-\log\frac{\delta}{4}+2\int\log|z|d\omega(z)+C_{1}.$

Since $|z|\leq 2\delta$ for $z\in\operatorname{supp}(\omega)$ we obtain

$\displaystyle\iint G(p,q)d\mu(p)d\mu(q)$ $\displaystyle\leq-\log\delta+\log
4+2\log(2\delta)+C_{1}$ $\displaystyle=\log\delta+4\log 2+C_{1},$ (6.9)

which is the analogue of (6.7), but note the different sign with $\log\delta$.

Because of (3.15) there is $C_{2}$ such that $\varphi(p)\leq
r_{\infty}\log|z_{\infty}(p)|+C_{2}$ for $p\in U_{\delta_{0}}(p_{\infty})$,
which gives us the analogue of (6.8)

$\int\varphi(p)d\mu(p)\leq r_{\infty}\log(2\delta)+C_{2}$ (6.10)

since $\mu$ is supported on $U_{2\delta}(p_{\infty})\subset
U_{\delta_{0}}(p_{\infty})$ and $z_{\infty}(p)\leq 2\delta$ for
$p\in\operatorname{supp}(\mu)$.

Adding (6.9) and (6.10) we find

$E_{\varphi}\left[\mu\right]\leq(1+r_{\infty})\log\delta+(r_{\infty}+4)\log
2+C_{1}+C_{2}.$

Since $\mu$ is a probability measure on $F$ we have $E_{\varphi}(F)\leq
E_{\varphi}[\mu]$ and (6.3) follows. ∎

###  6.3 The max-min problem is well-posed

If $r_{0}>1$ then it follows from (6.2) that $E_{\varphi}(F)$ is small (i.e.,
very negative) in case $F$ comes very close to $p_{0}$. If $r_{\infty}>-1$
then similarly $E_{\varphi}(F)$ is small if $F$ comes close to $p_{\infty}$.
This allows us to prove that the max-min energy problem is well-posed.

######  Lemma 6.3.

Suppose $r_{0}>1$ and $r_{\infty}>-1$. Then the max-min energy problem is
well-posed: that is, $\sup_{F\in\mathcal{F}}E_{\varphi}(F)$ is finite.

###### Proof.

Let $m=\max(r_{0}-1,r_{\infty}+1)>0$ and let $C$ be as in Proposition 6.2. We
fix $\delta=\delta_{0}/4$, so that by Proposition 6.2 (and the standing
assumption $\delta_{0}\leq\delta_{\infty}$),

$E_{\varphi}(F)\leq m\log\delta+C<\infty$ (6.11)

for all $F\in\mathcal{F}$ that intersect $\partial U_{\delta}(p_{0})$ or
$\partial U_{\delta}(p_{\infty})$.

Now if $F\in\mathcal{F}$ does not intersect $\partial U_{\delta}(p_{0})$ or
$\partial U_{\delta}(p_{\infty})$, any connected component $\widetilde{F}$ of
$F$ satisfies $\widetilde{F}\subset U_{\delta}(p_{0})$, $\widetilde{F}\subset
U_{\delta}(p_{\infty})$ or $\widetilde{F}\subset
X\setminus(U_{\delta}(p_{0})\cup\partial U_{\delta}(p_{\infty}))$. In the
first case, we can take a $\delta^{\prime}<\delta$ for which
$\widetilde{F}\cap\partial U_{\delta^{\prime}}(p_{0})\neq\emptyset$ and apply
Proposition 6.2 to conclude that

$E_{\varphi}(F)\leq(r_{0}-1)\log\delta^{\prime}+C\leq m\log\delta+C.$

That is, the same upper bound as in (6.11) can be used. Similarly,

$E_{\varphi}(F)\leq(r_{\infty}+1)\log\delta^{\prime}+C\leq m\log\delta+C$

in the second case and the upper bound from (6.11) again works. It remains to
find an upper bound for all $F\in\mathcal{F}$ with $F\subset
X\setminus(U_{\delta}(p_{0})\cup\partial U_{\delta}(p_{\infty}))$.

Let $\Gamma$ be a simple curve connecting $p_{0}$ and $p_{\infty}$ without
containing any further poles of $dV$. Then for all $\varepsilon>0$ small
enough, the fattened set $\Gamma_{\varepsilon}=\\{p\in
X:d(p,\Gamma)\leq\varepsilon\\}$ does not contain any other poles of $dV$
either. We fix such a value of $\varepsilon$ so that $\varepsilon<\delta$ as
well and define $K=\Gamma_{\varepsilon}\setminus(U_{\delta}(p_{0})\cup
U_{\delta}(p_{\infty}))$. The external field $\varphi$ is continuous and hence
bounded on the compact set $K$, say $|\varphi|\leq M$ on $K$ for some
$M\in\mathbb{R}$.

As $K$ is compact, we can cover it with a finite number of coordinate charts
$(U_{j},z_{j})$, $j=1,\ldots,n$. By Lemma A.1, there are $\eta>0$ and $C>0$
such that for every subset $K^{\prime}$ of $K$ with diameter at most $\eta$,
we have $K^{\prime}\subset U_{j}$ for some $j$ and
$\operatorname{diam}(K^{\prime})\leq C\operatorname{diam}(z_{j}(K^{\prime}))$.
Without loss of generality, we may assume $\eta\leq\varepsilon$.

Now take $F\in\mathcal{F}$ with $F\subset
X\setminus(U_{\delta}(p_{0})\cup\partial U_{\delta}(p_{\infty}))$. Then $K\cap
F$ contains a continuum $F_{0}$ with diameter $\eta$, namely a continuum that
connects $\partial\Gamma_{\eta}$ with $\Gamma$. Hence there is a
$j=1,\ldots,n$ such that $F_{0}\subset\tilde{U}_{j}$. Since
$\operatorname{diam}(F_{0})\leq C\operatorname{diam}(z_{j}(F_{0}))$, it
follows that $z_{j}(F_{0})$ is a continuum in $\mathbb{C}$ with (Euclidean)
diameter at least $\frac{\eta}{C}$.

Let $\omega$ be the equilibrium measure of $z_{j}(F_{0})$ (in the usual
logarithmic potential theory in the plane) without external field, and let
$\mu$ be the pullback of $\omega$ by the local coordinate $z_{j}$. By similar
arguments as given in the proof of Proposition 6.2, we obtain

$\iint
G(p,q)d\mu(p)d\mu(q)\leq-\log\left(\frac{\eta}{4C}\right)+C_{j}=-\log\eta+\log
4+\log C+C_{j}$

for some constant $C_{j}$ that only depends on $U_{j}$. Taking
$\tilde{C}=\max_{j}C_{j}$ and using that $|\varphi|\leq M$ on
$\operatorname{supp}(\mu)\subset K$, we thus find

$E_{\varphi}[\mu]=\iint G(p,q)d\mu(p)d\mu(q)+\int\varphi(p)d\mu(p)\\\
\leq-\log\eta+\log 4+\log C+C_{j}+M\leq-\log\eta+2\log 2+\log C+\tilde{C}+M.$

Consequently,

$E_{\varphi}(F)\leq E_{\varphi}(F_{0})\leq E_{\varphi}[\mu]\leq-\log\eta+2\log
2+\log C+\tilde{C}+M.$

As the upper bound is independent of $F$, this concludes the proof. ∎

###  6.4 Continuity of the energy functional

Rakhmanov [44, Theorem 9.8] established the following fundamental result for
the complex plane. His arguments extend to the higher genus case and this was
already done by Chirka [19, Section 2.9] in the unweighted case. We formulate
the continuity of the weighted energy functional for the case of external
field $\varphi=\operatorname{Re}V$ that is of interest in the paper, but it
holds for more general $\varphi$.

######  Proposition 6.4.

The weighted energy functional $F\mapsto E_{\varphi}(F)$ is continuous on
$\mathcal{F}$.

To prepare for the proof of Proposition 6.4 we need three lemmas. The first
lemma provides an estimate for the usual Green’s function $G_{\Omega}$ of
$\Omega=X\setminus F$ with $F\in\mathcal{F}_{\delta}$. Recall that the Green’s
function $(p,q)\mapsto G_{\Omega}(p,q)$ is non-negative for $p,q\in X\times
X$, zero whenever $p\in F$ or $q\in F$ and for a fixed $q\in\Omega$, $p\mapsto
G_{\Omega}(p,q)$ is continuous on $X$, harmonic on $\Omega$, with

$G_{\Omega}(p,q)=-\log|z(p)|+O(1)\text{ as }p\to q$

where $p\mapsto z(p)$ is a local coordinate around $q$. The following estimate
for the Green’s function is contained in [44, Lemma 9.9] for compacts in the
complex plane. The extension to the higher genus case is mentioned in [19, p.
331].

######  Lemma 6.5.

For every $0<\delta<\delta_{0}$ there is a constant $C=C(\delta)>0$ such that
for every $F\in\mathcal{F}_{\delta}$ and every $p\in X\setminus
U_{\delta/2}(p_{\infty})$, one has

$G_{\Omega}(p,p_{\infty})\leq C\sqrt{d_{H}(p,F)},$

where $\Omega=X\setminus F$.

###### Proof.

The argument of [19, p. 331] applies since the diameter of each component of
$F$ is at least $\eta>0$ by Lemma 6.1 (c). ∎

Of course, it is possible that $F\in\mathcal{F}_{\delta}$ contains a pole of
$dV$ where $\varphi\to+\infty$. The next lemma shows that for our purposes, we
may replace $\varphi$ by a continuous external field on
$\mathcal{F}_{\delta}$. For every $m\in\mathbb{R}$, we denote
$\min(\varphi,m)$ by $\varphi_{m}$, which is a continuous function on $F$.

The proof is inspired by the proof of Theorem I.1.3(b) in [46]. However, in
[46], the underlying set of the equilibrium problem is fixed, while here we
have a family of sets. Therefore, we decided to give a full proof.

######  Lemma 6.6.

For every $\delta<\delta_{0}$, there exists an $m\in\mathbb{R}$ such that for
every $F\in\mathcal{F}_{\delta}$, the equilibrium measures of $F$ in the
external field $\varphi$ and $\varphi_{m}=\min(\varphi,m)$ are the same.
Moreover, for the shared equilibrium measure $\mu$, we have
$\operatorname{supp}\mu\subset\\{\varphi\leq m\\}$ and $\varphi=\varphi_{m}$
on $\operatorname{supp}\mu$. Finally, $E_{\varphi}$ and $E_{\varphi_{m}}$
agree as functions on $\mathcal{F}_{\delta}$.

###### Proof.

Let $\delta<\delta_{0}$. Since the max-min energy problem is well-posed (see
Lemma 6.3), the number $M=\sup_{F\in\mathcal{F}_{\delta}}E_{\varphi}(F)$ is
finite.

We write $K=X\setminus(U_{\delta}(p_{0})\cup U_{\delta}(p_{\infty}))$ and note
that every $F\in\mathcal{F}_{\delta}$ is contained in $K$. Because $G$ is
bounded away from $-\infty$ away from $p_{\infty}$ and the external field
$\varphi$ is only $-\infty$ at $p_{0}$ and possibly at $p_{\infty}$ by
assumption, we have

$m_{G}=\min_{p,q\in K}G(p,q)>-\infty\quad\text{ and }\quad
m_{\varphi}=\min_{p\in K}\varphi(p)>-\infty.$

We take $m\geq m_{\varphi}$ such that
$m_{G}+\frac{1}{2}m+\frac{1}{2}m_{\varphi}\geq M+1$ and write $K_{m}=\\{p\in
K:\varphi(p)\leq m\\}$. Note that $m$ is independent of
$F\in\mathcal{F}_{\delta}$.

Moreover, if $(p,q)\notin K_{m}\times K_{m}$, then
$\frac{1}{2}(\varphi_{m}(p)+\varphi_{m}(q))\geq\frac{1}{2}(m+m_{\varphi})$.
Hence

$G(p,q)+\frac{1}{2}(\varphi_{m}(p)+\varphi_{m}(q))\geq
m_{G}+\frac{1}{2}m+\frac{1}{2}m_{\varphi}\geq M+1,\quad(p,q)\notin K_{m}\times
K_{m}.$ (6.12)

We now turn to the proof of the main statement. Let $F\in\mathcal{F}_{\delta}$
and write $F_{m}=F\cap K_{m}=\\{p\in F:\varphi(p)\leq m\\}$. Take $\mu$ as the
equilibrium measure of $F$ in the external field $\varphi_{m}$ and let
$\mu^{F}$ be the equilibrium measure of $F$ in the external field $\varphi$.
Then $E_{\varphi_{m}}[\mu]\leq E_{\varphi_{m}}[\mu^{F}]\leq
E_{\varphi}[\mu^{F}]=E_{\varphi}(F)<M+1$, so that $\mu(F_{m})>0$ by (6.12).
Hence $\tilde{\mu}=\mu|_{F_{m}}/\mu(F_{m})$ is well-defined. Moreover, by
(6.12),

$\displaystyle E_{\varphi_{m}}[\mu]$ $\displaystyle=\left(\iint_{F_{m}\times
F_{m}}+\iint_{(X\times X)\setminus(F_{m}\times
F_{m})}\right)[G(p,q)+\frac{1}{2}(\varphi_{m}(p)+\varphi_{m}(q))]d\mu(p)d\mu(q)$
$\displaystyle\geq\mu(F_{m})^{2}E_{\varphi_{m}}[\tilde{\mu}]+(M+1)(1-\mu(F_{m})^{2}).$

Since $E_{\varphi_{m}}[\mu]<M+1$, the above leads to the contradiction
$E_{\varphi_{m}}[\tilde{\mu}]<E_{\varphi_{m}}[\mu]$ unless $\mu(F_{m})=1$, in
which case $E_{\varphi_{m}}[\mu]\geq E_{\varphi_{m}}[\tilde{\mu}]$. But then
$\mu=\tilde{\mu}$ by the uniqueness of the equilibrium measure, so that
$\operatorname{supp}(\mu)\subset F_{m}$.

Because $\operatorname{supp}(\mu)\subset F_{m}$, we also have
$\varphi=\varphi_{m}$ on $\operatorname{supp}(\mu)$. It directly follows that
$E_{\varphi_{m}}[\mu]=E_{\varphi}[\mu]$. Since $\varphi_{m}\leq\varphi$ on
$F$, we have

$E_{\varphi}(F)\leq
E_{\varphi}[\mu]=E_{\varphi_{m}}[\mu]=E_{\varphi_{m}}(F)\leq E_{\varphi}(F).$

Consequently, we have equality throughout the equation and the equilibrium
measures of $F$ in the external fields $\varphi$ and $\varphi_{m}$ are the
same. The final statement follows directly, which concludes the proof. ∎

The third lemma gives an estimate on the harmonic extension. Suppose $\varphi$
is a continuous function on $F$. Then there is a harmonic function
$\widetilde{\varphi}$ on $X\setminus F$ with $\widetilde{\varphi}=\varphi$ on
$F$. It is simply the solution of the Dirichlet problem. In our setting
$\varphi$ itself is harmonic, wherever it is finite.

Since $\varphi$ is not necessarily continuous on $F$, we replace $\varphi$ by
$\varphi_{m}=\min(\varphi,m)$ from the lemma above. The following is an
estimate on $|\widetilde{\varphi}_{m}-\varphi_{m}|$ near $F$ that is due to
Rakhmanov [44, Lemma 9.7] in the genus zero case (in a much more precise form,
actually). In the higher genus case we can follow the same proof as all the
arguments are local in nature.

######  Lemma 6.7.

For every $\delta<\delta_{0}$, $\varepsilon>0$ and $m\in\mathbb{R}$, there is
$\eta>0$ such that for every $F\in F_{\delta}$ and every $p\in X$ with
$d_{H}(p,F)<\eta$ one has

$\left|\widetilde{\varphi}_{m}(p)-\varphi_{m}(p)\right|\leq\varepsilon,$

where $\widetilde{\varphi}_{m}$ is the harmonic extension of
$\varphi_{m}=\min(\varphi,m)$ to $X\setminus F$.

###### Proof of Proposition 6.4..

It is enough to show that $F\mapsto E_{\varphi}(F)$ is continuous on
$\mathcal{F}_{\delta}$ for every $0<\delta<\delta_{0}$. We fix $\delta$ and
use Lemma 6.6 to take an $m\in\mathbb{R}$ so that
$E_{\varphi}(F)=E_{\varphi_{m}}(F)$ for all $F\in\mathcal{F}_{\delta}$.

Let $\varepsilon>0$. Then by Lemmas 6.5 and 6.7 there is $\eta>0$ such that
for any $F\in\mathcal{F}_{\delta}$ and $p\in X$ with $d_{H}(p,F)<\eta$ we have
$G_{\Omega}(p,p_{\infty})\leq\frac{\varepsilon}{3}$ and
$|\widetilde{\varphi}_{m}(p)-\varphi_{m}(p)|\leq\frac{\varepsilon}{3}$ where
$\Omega=X\setminus F$ and $\widetilde{\varphi}_{m}$ is the harmonic extension
of $\varphi_{m}$ from $F$ to $X\setminus F$.

Now take $F_{1},F_{2}\in\mathcal{F}_{\delta}$ such that
$d_{H}(F_{1},F_{2})<\eta$. Let $\mu_{1}=\mu^{F_{1}}$ be the equilibrium
measure of $F_{1}$ in the external field $\varphi$ and let $\mu_{2}$ be the
balayage of $\mu_{1}$ to $F_{2}$. Then $\mu_{2}$ is a probability measure on
$F_{2}$ such that

$\int\varphi_{m}d\mu_{2}=\int\widetilde{\varphi}_{m}d\mu_{1},$

where $\widetilde{\varphi}_{m}$ is the harmonic extension of $\varphi_{m}$
relative to $F_{2}$. Thus

$\left|\int\varphi_{m}d\mu_{2}-\int\varphi_{m}d\mu_{1}\right|=\left|\int(\widetilde{\varphi}_{m}-\varphi_{m})d\mu_{1}\right|\leq\frac{\varepsilon}{3}$
(6.13)

since $|\widetilde{\varphi}_{m}-\varphi_{m}|\leq\frac{\varepsilon}{3}$ on
$F_{1}$.

Using the Green’s function $G_{\Omega}$ for the open set $\Omega=X\setminus
F_{2}$, it follows that

$\iint G(p,q)d\mu_{2}(p)d\mu_{2}(q)\\\ =\iint
G(p,q)d\mu_{1}(p)d\mu_{1}(q)-\iint G_{\Omega}(p,q)d\mu_{1}(p)d\mu_{1}(q)+2\int
G_{\Omega}(p,p_{\infty})d\mu_{1}(p)\\\ \leq\iint
G(p,q)d\mu_{1}(p)d\mu_{1}(q)+2\int G_{\Omega}(p,p_{\infty})d\mu_{1}(p),$

where the first identity is the analogue of [44, Lemma 9.6]. Hence

$\iint G(p,q)d\mu_{2}(p)d\mu_{2}(q)\leq\iint
G(p,q)d\mu_{1}(p)d\mu_{1}(q)+\frac{2\varepsilon}{3}$ (6.14)

since $G_{\Omega}(p,p_{\infty})\leq\frac{\varepsilon}{3}$ for $p\in F_{1}$.
Consequently, by (6.14), (6.13) and Lemma 6.6,

$\displaystyle E_{\varphi}(F_{2})=E_{\varphi_{m}}(F_{2})$
$\displaystyle\leq\iint G(p,q)d\mu_{2}(p)d\mu_{2}(q)+\int\varphi_{m}d\mu_{2}$
$\displaystyle\leq\iint
G(p,q)d\mu_{1}(p)d\mu_{1}(q)+\int\varphi_{m}d\mu_{1}+\varepsilon$
$\displaystyle=E_{\varphi_{m}}(F_{1})+\varepsilon=E_{\varphi}(F_{1})+\varepsilon.$

By symmetry, we also have $E_{\varphi}(F_{1})\leq
E_{\varphi}(F_{2})+\varepsilon$ and hence
$|E_{\varphi}(F_{1})-E_{\varphi}(F_{2})|\leq\varepsilon$. This shows that
$F\mapsto E_{\varphi}(F)$ is uniformly continuous on $\mathcal{F}_{\delta}$
and concludes the proof. ∎

###  6.5 Proof of Theorem 3.11

Since an extremal set stays away from $p_{0}$ and $p_{\infty}$ in case both
$r_{0}>1$ and $r_{\infty}>-1$, the continuity of the weighted energy
functional allows us to prove Theorem 3.11.

###### Proof of Theorem 3.11..

Let $m=\max(r_{0}-1,r_{\infty}+1)>0$ and let $\delta_{0}$, $\delta_{\infty}$
and $C$ be as in Proposition 6.2. Pick an arbitrary $F_{0}\in\mathcal{F}$, and
take $\delta\in(0,1)$ with $2\delta<\min(\delta_{0},\delta_{\infty})$ small
enough such that $F_{0}\in\mathcal{F}_{\delta}$ and
$m\log\delta+C<E_{\varphi}(F_{0})$.

Since $\mathcal{F}_{\delta}$ is compact in the Hausdorff distance, and
$E_{\varphi}$ is continuous on $\mathcal{F}_{\delta}$, there is
$F\in\mathcal{F}_{\delta}$ where $E_{\varphi}$ takes its maximum on
$\mathcal{F}_{\delta}$. If
$F^{\prime}\in\mathcal{F}\setminus\mathcal{F}_{\delta}$, then either
$F^{\prime}$ intersects $\partial U_{\delta^{\prime}}(p_{0})$ for some
$\delta^{\prime}\leq\delta$, or $F^{\prime}$ intersects $\partial
U_{\delta^{\prime}}(p_{\infty})$ for some $\delta^{\prime}\leq\delta$. In the
first case, we find by part (a) of Proposition 6.2

$E_{\varphi}(F^{\prime})\leq(r_{0}-1)\leq\log\delta^{\prime}+C\leq(r_{0}-1)\log\delta+C\\\
\leq m\log\delta+C\leq E_{\varphi}(F_{0})\leq E_{\varphi}(F).$

Similarly, in the second case we use part (b) and we also find
$E_{\varphi}(F^{\prime})\leq E_{\varphi}(F)$.

The continuum $F$ satisfies the conditions of Corollary 3.9 and the theorem
follows. ∎

##  7 Proof of Theorem 3.13

###  7.1 Preparation for the proof

By Proposition 3.4 a critical measure $\mu$ satisfies (3.24) for every $C^{1}$
vector field $h$. As indicated in Section 3, we use the $(2,-1)$-Cauchy kernel
defined in Proposition 3.12 to extract relevant information from (3.24).

In the genus zero case one has (1.4) and $h$ is a function (not a vector
field). For the proof of Theorem 1.2 one takes

$h(s)=\frac{1}{z-s}$ (7.1)

with a fixed $z\in\mathbb{C}\setminus\operatorname{supp}\mu$. Then

$\frac{h(s)-h(t)}{s-t}=\frac{1}{(z-s)(z-t)}$ (7.2)

and

$\iint\frac{h(s)-h(t)}{s-t}d\mu(s)d\mu(t)=\left[\int\frac{d\mu(s)}{z-s}\right]^{2}$
(7.3)

which is a crucial step in the proof of Theorem 1.2, see e.g. [41, Proof of
Lemma 5.1] or [40, Proof of Proposition 3.7].

We assume that $X$ has the form (3.19) as discussed in Section 3.5 with
bounded real oval $C_{1}$. We have an analogue of (7.3) in case $\mu$ is
invariant under the involution (3.20) and its support does not intersect
$C_{1}$.

######  Proposition 7.1.

Suppose $\mu$ is a compactly supported measure on
$X\setminus(\\{p_{\infty}\\}\cup C_{1})$ that is invariant under the
involution $\sigma$ from (3.20). Then there exists $a\in C_{1}$ such that

$\iint\left(C(p,q)C^{(2,-1)}(u,p;a)+C(q,p)C^{(2,-1)}(u,q;a)-C(u,p)C(u,q)\right)\\\
d\mu(p)d\mu(q)=0,\qquad u\in X.$ (7.4)

###### Proof.

We compare the expressions

$C(p,q)C^{(2,-1)}(u,p;a)+C(q,p)C^{(2,-1)}(u,q;a)$ (7.5)

and

$C(u,p)C(u,q).$ (7.6)

Both (7.5) and (7.6) are, for fixed $p,q$, and $a$, meromorphic quadratic
differentials in $u$, with simple poles in $u=p$, $u=q$ and a double pole at
$u=p_{\infty}$. The residue at $u=p$ is $C(p,q)$ for both expressions, and the
residue at $u=q$ is $C(q,p)$, again the same for both of them. These two poles
disappear if we take the difference, and the only pole is the double pole at
$u=p_{\infty}$. Then the left-hand side of (7.4) is also a meromorphic
quadratic differential in $u$ with a possible double pole at $u=p_{\infty}$
only.

Suppose $a\in X\setminus(\\{p_{\infty}\\}\cup\operatorname{supp}(\mu))$ is a
zero of the meromorphic differential $\int C(u,q)d\mu(q)$. Then it is a double
zero of $\iint C(u,p)C(u,q)d\mu(p)d\mu(q)=\left(\int C(u,p)d\mu(p)\right)^{2}$
and since both $C^{(2,-1)}(u,p;a)$ and $C^{(2,-1)}(u,q;a)$ have a double zero
at $u=a$ as well, see Proposition 3.12 (b), we find that (7.4) has a double
zero at $u=a$. It has at most a double pole at $u=p_{\infty}$, and there are
no other poles and zeros (since $X$ has genus one), unless it vanishes
identically. The only non-zero meromorphic quadratic differentials on $X$ that
have double zeros and a double pole at $p_{\infty}$ are
$z\frac{dz^{2}}{w^{2}}$, $(z-z_{1})\frac{dz^{2}}{w^{2}}$,
$(z-z_{2})\frac{dz^{2}}{w^{2}}$ and their scalar multiples. Thus if
$a\not\in\\{p_{0},p_{1},p_{2}\\}$ where $p_{0}=(0,0)$, $p_{1}=(0,z_{1})$,
$p_{2}=(0,z_{2})$ then (7.4) holds.

We now show that the differential $\int C(u,q)d\mu(q)$ has exactly two zeros
on $C_{1}$: if at least one of them is not at the branch point, then the proof
follows by the above discussion; we then will consider the special case where
both zeros are at a branch point. Now, let $\mu$ be as in the statement of the
proposition. Then

$G^{\mu}(u):=\int G(u,q)d\mu(q)$

is real analytic on the cycle $C_{1}$, since the support of $\mu$ is disjoint
from $C_{1}$. Since $\mu$ is invariant under $\sigma$, we also have
$G^{\mu}(\sigma(u))=G^{\mu}(u)$ which implies that $\frac{\partial}{\partial
y}G^{\mu}(u)=0$ for $u\in C_{1}$, if we use $z=x+iy$ as the local coordinate
at a point $u=(w,z)$ on $C_{1}$. By compactness, $G^{\mu}$ attains a maximum
and a minimum on $C_{1}$. If an extremum is attained at $a\in C_{1}$, then we
also have $\frac{\partial}{\partial x}G^{\mu}(u)=0$ at $u=a$. Hence by (3.2)
we have that $a$ is a zero of

$\int C(u,q)d\mu(q)=-\left(\partial_{x}-i\partial_{y}\right)\int
G(u,q)d\mu(q).$

Thus, by what we already proved, if $G^{\mu}$ attains an extremum on $C_{1}$
at a point $a\not\in\\{p_{1},p_{2}\\}$ then (7.4) holds.

Next, we consider the special case $\mu=\delta_{q}$ with $q\in
C_{2}\setminus\\{p_{0},p_{\infty}\\}$ on the unbounded cycle. On the complex
torus the extrema of $G(u,q)$ are attained at the zeros of

$u\mapsto\frac{\theta_{1}^{\prime}(u-v)}{\theta_{1}(u-v)}-\frac{\theta_{1}^{\prime}(u)}{\theta_{1}(u)}$
(7.7)

where $v$ is the image of $q$ under the Abel map, see (3.3) with
$\operatorname{Im}v=0$. The elliptic function (7.7) has two simple poles at
$u=0$ and $u=v$. By Abel’s theorem the two zeros add up to $v$ (modulo
$\Lambda=\mathbb{Z}+\tau\mathbb{Z}$). Since $v\neq\frac{1}{2}$, (since
$v=\frac{1}{2}$ corresponds to $q=p_{0}$ in $X$), the two zeros cannot be
$\frac{1}{2}\tau$ and $\frac{1}{2}+\frac{1}{2}\tau$. On $X$ this means that
$C(u,q)$ does not vanish at both $p_{1}$ and $p_{2}$.

Finally, consider the case that $G^{\mu}$ attains its extrema on $C_{1}$ only
at the branch points $p_{1}$ and $p_{2}$, so that $\int C(u,p)d\mu(p)$
vanishes at both $p_{1}$ and $p_{2}$. Pick $q$ on the unbounded cycle, $q\neq
p_{0}$, $q\neq p_{\infty}$ and consider $\mu_{t}=\mu+t\delta_{q}$. Since
$C(u,q)$ does not vanish at both $p_{1}$ and $p_{2}$, we find that for $t>0$,

$\int C(u,p)d\mu_{t}(p)=\int C(u,p)d\mu(p)+tC(u,q)$

is not zero at both $p_{1},p_{2}$, and therefore it will have a zero somewhere
else on the cycle. The zero depends on $t>0$, say $a_{t}\in C_{1}$, and the
identity (7.4) holds for $\mu_{t}$ and $a_{t}$. Letting $t\to 0+$ and by using
a continuity and compactness argument, we find that (7.4) holds for $t=0$ as
well, where $a\in C_{1}$ is any limit point of $(a_{t})_{t>0}$ as $t\to 0+$. ∎

If $\int|C(u,q)|d\mu(q)<\infty$, which is the case for $u$-a.e. on $X$, then
(7.4) can be rewritten to

$\iint\left(C(p,q)C^{(2,-1)}(u,p;a)+C(q,p)C^{(2,-1)}(u,q;a)\right)d\mu(p)d\mu(q)\\\
=\left[\int C(u,q)d\mu(q)\right]^{2}.$ (7.8)

The identity (7.8) can be viewed as a genus one analogue of (7.3). Note
however that we do not have an analogue of the divided difference identity
(7.2).

###  7.2 Proof of Theorem 3.13

###### Proof.

Suppose $\mu$ is a critical measure in the external field
$\varphi=\operatorname{Re}V$ that is invariant under the involution $\sigma$.
Suppose $\operatorname{supp}(\mu)\subset X\setminus(\\{p_{\infty}\\}\cup
C_{1})$. Let $a\in C_{1}$ be as in Proposition 7.1.

We take $h(p)=C^{(2,-1)}(u,p;a)$ which is a $C^{1}$ vector field with a double
pole at $p=a$ and a simple pole at $p=u$. Thus $h$ is a $C^{1}$ vector field
on $\operatorname{supp}(\mu)$ if $u\not\in\operatorname{supp}(\mu)$, as it is
already assumed that $a\not\in\operatorname{supp}(\mu)$. Thus by Proposition
3.4 (b) we have $D_{V,h}(\mu)=0$ provided that
$u\not\in\operatorname{supp}(\mu)$.

With an approximation argument as in [40, Lemma 3.5] we can extend the
equality $D_{V,h}(\mu)=0$ from $u\not\in\operatorname{supp}(\mu)$ to any $u\in
X\setminus\\{p_{\infty}\\}$ for which

$\int\frac{d\mu(q)}{d(u,q)}<+\infty.$

which holds for a.e. $u$ on $X$.

From (3.7) we thus have

$\iint\left(C(p,q)C^{(2,-1)}(u,p;a)+C(q,p)C^{(2,-1)}(u,q;a)\right)d\mu(p)d\mu(q)\\\
=\int C^{(2,-1)}(u,q;a)dV(q)d\mu(q),\quad\text{$u$-a.e.\ on $X$.}$ (7.9)

Using (7.4) we get from (7.9) that

$\left[\int C(u,q)d\mu(q)\right]^{2}=\int
C^{(2,-1)}(u,q;a)dV(q)d\mu(q),\quad\text{$u$-a.e.\ on $X$,}$ (7.10)

which is easily seen to be (3.26) with $\upomega$ given by (7.9).

Note that $\upomega$ is indeed a meromorphic quadratic differential on $X$,
since the pole at $u=q$ cancels out in the difference

$C(u,q)dV(u)-C^{(2,-1)}(u,q;a)dV(q)$

and therefore the integral transform in (3.25) indeed extends analytically
across the support $\Sigma$ of the measure $\mu$.

For the proof of (b) we can follow the proof of Proposition 3.8 in [40,
Proposition 3.8], since all arguments are local in nature.

From (3.25) and working in a local coordinate around $u\in\Sigma$, we have

$\left(\int C(u,q)d\mu(q)-\frac{dV(u)}{2}\right)_{\pm}=\pm Q(u)^{1/2}du$

where $\pm$ denote the boundary limits on $\Sigma$. Thus

$\left(\int C(u,q)d\mu(q)-\frac{dV(u)}{2}\right)_{+}=-\left(\int
C(u,q)d\mu(q)-\frac{dV(u)}{2}\right)_{-}$

or put otherwise

$\left(\int C(u,q)d\mu(q)\right)_{+}+\left(\int
C(u,q)d\mu(q)\right)_{-}-dV(u)=0\qquad\text{ on }\Sigma$

as $dV$ does not have a jump on $\Sigma$. If
$g(u)=\int\mathcal{G}(u,p)d\mu(p)$, then

$g_{+}(u)+g_{-}(u)+V(u)=\text{const}\quad\text{ on }D\cap\Sigma.$

Taking the real part we obtain part (c). Taking the imaginary part and using
the Cauchy-Riemann equations, we obtain part (d). ∎

##  Appendix A Existence and symmetry of the bipolar Green’s function

Let $X$ be a Riemann surface and let $p_{\infty}$ be a distinguished point at
infinity. Our goal is to prove Proposition 2.1 and in particular part (d) that
gives us the symmetry $G(p,q)=G(q,p)$ of the bipolar Green’s function. For
this proof we rely on Riemannian geometry where the existence of a symmetric
Green’s function for the Laplacian is known.

###  A.1 Riemann surfaces as Riemannian manifolds

We first recall some relevant notions from Riemannian geometry. The main
reference in this section is [36]. As shown in Lemma 2.3.3 in [36], $X$ admits
a conformal Riemannian metric $\rho$, which is given in the local coordinate
$(U,z)$ by

$\rho_{U}(z)^{2}dzd\overline{z},\quad\rho_{U}(z)>0,$

where $\rho_{U}$ is smooth and which transforms correctly under a holomorphic
change of local coordinates. This turns $X$ into a Riemannian manifold. The
Riemannian metric allows for the definition of the length of a curve
$\gamma\subset U$ and area of a measurable set $B\subset U$, namely

$\ell(\gamma):=\int_{\gamma}\rho_{U}(z)|dz|,\quad\operatorname{area}(B):=\frac{i}{2}\int_{B}\rho_{U}(z)^{2}dzd\overline{z};$

see also [36, p. 21]. The length of general curves is computed by splitting
the curve into a final number of pieces, each of which is contained in a
single coordinate chart; a similar method works for the area. Note that
$\operatorname{area}(X)$ is finite as $X$ is compact.

The Riemannian metric $\rho$ globally defines an area form (i.e., a nowhere-
vanishing $2$ form) $dA$ (the dependence on $\rho$ is suppressed in the
notation) in the local coordinate $(U,z)$ by

$\frac{i}{2}\rho_{U}(z)^{2}dzd\overline{z}.$

The area form $dA$ will be used to integrate functions.

The distance $d:X\times X\to[0,+\infty)$ between two points $p$ and $q$ can
then be defined as

$d(p,q):=\inf\left\\{\ell(\gamma)\mid\text{$\gamma:[0,1]\to X$ is a curve with
$\gamma(0)=p$ and $\gamma(1)=q$}\right\\}.$ (A.1)

The metric topology on $X$ defined by $d$ coincides with the original topology
on $X$. This is because in a coordinate chart $(U,z)$, the induced distance
$d(z(p),z(q))$ is equivalent to the Euclidean distance in $z(U)$ in the sense
that for a fixed compact set $K\subset U$,

$c|z(p)-z(q)|\leq d(z(p),z(q))\leq C|z(p)-z(q)|,\quad p,q\in K,$ (A.2)

where $c>0$ and $C>0$ are the minimum and maximum of $\rho_{U}$ on $K$
respectively. See also [1, Theorem 1.18].

######  Lemma A.1.

Let $K\subset X$ be compact and let $\\{U_{j}\\}_{j=1}^{n}$ be a finite open
cover of $K$ with coordinate charts $(U_{j},z_{j})$. Then there exists an
$\eta>0$ and a $C>0$ such that for every subset $K^{\prime}$ of $K$ with
diameter at most $\eta$, $K^{\prime}\subset U_{j}$ for some $j$ and

$\operatorname{diam}(K^{\prime})\leq C\operatorname{diam}(z_{j}(K^{\prime})).$

Note that we have the diameter with respect to $d$ (see (A.1)) on the left and
the standard Euclidean diameter on the right.

###### Proof of Lemma A.1.

Let $K\subset X$ be compact and let $\\{U_{j}\\}_{j=1}^{n}$ be a finite open
cover of $K$ with coordinate charts $(U_{j},z_{j})$ be given. By the
properties of the metric topology, we can find open sets $\tilde{U}_{j}$ that
are compactly contained in $U_{j}$ (that is, the closure of $\tilde{U}_{j}$ is
compact and contained in $U_{j}$) such that
$K\subset\cup_{j=1}^{n}\tilde{U}_{j}$. Hence by (A.2), for every
$j=1,\ldots,n$, there are numbers $C_{j}>0$ such that

$d(p,q)\leq C_{j}|z_{j}(p)-z_{j}(q)|,\quad p,q\in\tilde{U}_{j}.$

Let $\eta>0$ be a Lebesgue number for the open cover
$\\{\tilde{U}_{j}\\}_{j=1}^{n}$ of $K$ (so every subset of $K$ with diameter
at most $\eta$ is contained in some $\tilde{U}_{j}$) and take
$C=\max_{j}C_{j}>0$. Suppose that $K^{\prime}\subset K$ has diameter
$\operatorname{diam}(K^{\prime})\leq\eta$. Then $K^{\prime}$ lies in some
$\tilde{U}_{j}$. Moreover,

$\displaystyle\operatorname{diam}(K^{\prime})$ $\displaystyle=\sup_{p,q\in
K^{\prime}}d(p,q)\leq C_{j}\sup_{p,q\in K^{\prime}}|z_{j}(p)-z_{j}(q)|$
$\displaystyle\leq C\sup_{p,q\in
K^{\prime}}|z_{j}(p)-z_{j}(q)|=C\operatorname{diam}(z_{j}(K^{\prime})),$

which concludes the proof. ∎

The Riemannian metric $\rho$ moreover defines the Laplace-Beltrami operator
$\Delta$ on functions $f\in C^{2}(X)$ by locally setting

$\Delta f=\frac{4}{\rho_{U}(z)^{2}}\frac{\partial}{\partial
z}\frac{\partial}{\partial\overline{z}}f,$

see [36, Definition 2.3.3] and also [18, Section 1.1]. Note that $\Delta f$ is
a function on $X$. Moreover,

$\Delta f=-\star d\star df$

where $\star$ denotes the Hodge star operator on $k$-forms (see [36, Section
5.2]). The $2$-form $d\star df$ is independent of $\rho$ (and is sometimes
taken as a definition for the Laplacian, see e.g. [26]) and hence $\Delta$
only depends on $\rho$ through the application of the Hodge star operator on
$d\star df$.

###  A.2 Proof of Proposition 2.1

Because $X$ can be turned into a compact Riemannian manifold, it carries a