$D_{z}\cap{\bf rp}({\mathcal{R}})\subseteq\\{z\\}$ and $D_{\zeta}\cap{\bf
rp}({\mathcal{R}})\subseteq\\{\zeta\\}$. Let as before $m(\xi)$ be the
ramification order of $\xi$, so that $D_{z}$ (resp. $D_{\zeta}$) is
(isomorphic to) an $m(z)$-sheeted cyclic covering of $D_{0}$. For $x\in
D_{0}\setminus\widetilde{N}_{0}$, define

$g(x):=m(\zeta)\sum_{\xi\in p^{-1}(x)\cap
D_{z}}m(\xi)ler(\xi)-m(z)\sum_{\xi\in p^{-1}(x)\cap D_{\zeta}}m(\xi)ler(\xi).$

It follows from (3.46) and (A.32) that

$\sum_{\xi\in p^{-1}(x)\cap
D_{z}}m(\xi)ler(\xi)=m(z)g(\mu,\mathbb{D};x)-g\big{(}p_{*}(\nu_{\mathcal{b}D_{z}}),\mathbb{D};x\big{)}\\\
-\sum_{\xi\in p^{-1}(x)\cap
D_{z}}m(\xi)\left(g\big{(}\nu_{\mathcal{b}{\mathcal{R}}\setminus
D_{z}},{\mathcal{R}};\xi\big{)}+h_{\mathcal{R}}(\xi)\right).$

Notice that the last summand above is a continuous, even harmonic function in
$D_{0}$. Hence, the sum itself is finely continuous in
$D_{0}\setminus\widetilde{N}_{0}$. Likewise, the second sum in the definition
of $g$ is finely continuous in $D_{0}\setminus\widetilde{N}_{0}$ and so is
$g$. In particular, $U:=\\{x\in D_{0}\setminus\widetilde{N}_{0}:g(x)\neq 0\\}$
is finely open, and since $p(G_{+})$ is finely open by Lemma A.2 the set
$U\cap p(G_{+})$ is in turn finely open. From the first part of the proof, it
follows that if $x\in(D_{0}\cap p(G_{+}))\setminus p(B)$ then $g(x)=0$, whence
$U\cap p(G_{+})\subseteq p(B)$. Thus, $U\cap p(G_{+})$ must be empty as $p(B)$
is polar, that is, $g\equiv 0$ on $D_{0}\cap p(G_{+})$. Now, it can be readily
checked that

$g(p(z))=m(z)m(\zeta)\big{(}ler(z)-ler(\zeta)\big{)},$

and therefore $ler(z)=ler(\zeta)$ if $z\in G_{+}$, thereby proving the first
assertion of the lemma. The second is then obvious, and the third follows from
Lemma A.2. ∎

######  Lemma 3.11.

The set $G_{-}$ lies schlicht over $\mathbb{D}$. That is, $p:G_{-}\to D_{-}$
is a bijection. Moreover, $G_{-}\cap{\bf rp}({\mathcal{R}})=\varnothing$, and
for each $z\in G_{-}$ we have that

(3.64) $ler(\zeta)=0\text{ \ \ for all \ }\zeta\in
p^{-1}(p(z))\setminus\\{z\\}.$

###### Proof.

Pick $z\in G_{-}\setminus{\bf rp}({\mathcal{R}})$ and let
$D_{0}\subset{\mathcal{R}}\setminus{\bf rp}({\mathcal{R}})$ be a conformal
disk centered at $z$, homeomorphic under $p$ to a Euclidean disk $p(D_{0})$;
note that $p^{-1}(D_{0})\setminus D_{0}$ is open. Define $U:=G_{-}\cap D_{0}$
and $V:=\big{(}p^{-1}(D_{0})\setminus D_{0}\big{)}\cap p^{-1}(p(U))$. Clearly
$U$ and $V$ are disjoint finely open subsets of ${\mathcal{R}}$, by Lemma A.2.
In view of Lemma 3.6, there is a polar set $A_{0}\subset{\mathcal{R}}\setminus
N_{0}$ such that $V\cap G_{-}\subseteq A_{0}$. Thus, $V\cap G_{-}$ must be
empty as otherwise it is finely open. This shows that $G_{-}\setminus{\bf
rp}({\mathcal{R}})$ lies schlicht over $\mathbb{D}$. However, the complement
of a schlicht set is always non-thin at any ramification point by Lemma A.4.
Hence, $G_{-}$ cannot be a fine neighborhood of a ramification point, and
since it is finely open $G_{-}\cap{\bf rp}({\mathcal{R}})=\varnothing$.
Altogether, $G_{-}$ lies schlicht over $\mathbb{D}$ and (3.64) now follows
from this and Lemma 3.10. ∎

###  3.8. Modified Logarithmic Error Function

In this subsection we modify the function $ler(z)$ by clearing out parts of
$\mathbb{D}$ and ${\mathcal{R}}$ from the support of $\mu$ and $\nu$,
respectively. We shall accomplish this via the technique of balayage,
described in Section A.9. Let us start with some preliminary geometric
considerations. Recall that any connected (topological) $1$-manifold embedded
in ${\mathcal{R}}$ is a Jordan curve.

######  Lemma 3.12.

For each $\epsilon>0$ there exists a Jordan curve $J\subset G_{-}$ such that
$p(J)$ is a Jordan curve included in $\\{z:1-\epsilon<|z|<1\\}$ and $p({\bf
rp}({\mathcal{R}}))$ belongs to the interior domain of $p(J)$. Moreover, there
exists a finely connected component of $G_{-}$, say $G_{J}$, such that
$J\subset G_{J}$.

###### Proof.

We may assume that $\epsilon$ is small enough that $p({\bf
rp}({\mathcal{R}}))\subset\mathbb{D}_{1-\epsilon}$. In particular, $p$ is
injective on every conformal disk centered at a point of
$\partial{\mathcal{R}}$ with radius smaller than or equal to $\epsilon$.

Recall that ${\mathcal{R}}$ is a subset of a Riemann surface
${\mathcal{R}}_{*}$ lying over $\overline{\mathbb{C}}$. Define
$G_{-}^{*}:=G_{-}\cup\mathcal{T}\cup\mathcal{D}$, where $\mathcal{D}$ is the
connected component of
$p^{-1}(\overline{\mathbb{C}}\setminus\overline{\mathbb{D}})$ that borders
$\mathcal{T}$. Let us show that for each $\eta\in\mathcal{T}$ there is a disk
$D_{\eta}\subset\\{z:1-\epsilon<|z|<1+\epsilon\\}$, centered at $p(\eta)$ with
radius $r_{\eta}$, such that the circle $\partial D_{\eta}$ is included in
$p(G_{-}^{*})$. In fact, we can pick $r_{\eta}$ so that there exist radii
$r^{\prime}_{\eta}$ arbitrarily close but not equal to $r_{\eta}$ for which
each disk $D^{\prime}_{\eta}$ centered at $p(\eta)$ of radius
$r^{\prime}_{\eta}$ also satisfies $\partial D^{\prime}_{\eta}\subset
p(G_{-}^{*})$. Indeed, like we did to establish Lemma 3.9, let $\widehat{\mu}$
be the lift $\mu$ to ${\mathcal{R}}$ , see (A.33). By definition, $ler(z)\geq
0$ for $z\in{\mathcal{R}}\setminus(G_{-}\cup N_{0})$. Therefore, if
$\eta\in\mathcal{T}$ is a limit point of ${\mathcal{R}}\setminus G_{-}$, then
we get from (3.17), (3.46), the definition of $h_{\mathcal{R}}(z)$, and the
identity $g(\widehat{\mu},{\mathcal{R}};\zeta)=+\infty$ when $\zeta\in N_{0}$
that

$\liminf_{{\mathcal{R}}\setminus
G_{-}\ni\zeta\to\eta}g(\widehat{\mu},{\mathcal{R}};\zeta)\geq
2/\mathrm{cap}_{\mathbb{D}}(K_{f}).$

The claim now follows from Lemma A.5 by taking $1-r_{\eta}$ to be an
accumulation point of $R_{1/\mathrm{cap}_{\mathbb{D}}(K_{f})}$ in
$(1-\epsilon,1)$.

Let $U_{\eta}$ be the connected component of $p^{-1}(D_{\eta})$ containing
$\eta$, which is an open subset of ${\mathcal{R}}_{*}$ satisfying $\partial
U_{\eta}\subset G_{-}^{*}$. Since the collection $\\{U_{\eta}\\}_{\eta}$
covers $\mathcal{T}$, which is compact, it contains a finite subcover, say
$\\{U_{\eta_{i}}\\}$. Replacing $D_{\eta_{i}}$ by some $D^{\prime}_{\eta_{i}}$
as above if needed, we can ensure by (finite) induction on $i$ that $\partial
U_{\eta_{i}}$ and $\partial U_{\eta_{j}}$ may intersect only transversally for
$i\neq j$, and that $\partial U_{\eta_{i}}\cap\partial
U_{\eta_{j}}\cap\partial U_{\eta_{k}}=\varnothing$ if $i,j,k$ are all
distinct. Then $V:=\cup_{i}U_{\eta_{i}}$ is an open neighborhood of
$\mathcal{T}$ with boundary $\partial V$ included in $G_{-}^{*}$ that consists
of a finite union of disjoint Lipschitz-smooth Jordan curves. In fact, there
are exactly two such curves, one in ${\mathcal{R}}$ and $\mathcal{D}$, because
each connected component of $\partial V$ can be continuously deformed into
$\mathcal{T}$ via radial retraction within $V$. We can choose the component
within ${\mathcal{R}}$ to be $J$ since Lipschitz curves are finely connected
in ${\mathcal{R}}$, see Section A.5. ∎

Recall now the finely open subsets $D_{+}$ and $D_{-}$ of $\mathbb{D}$
introduced before Lemma 3.10. Denote by $D_{J}$ the union of all finely
connected components of $D_{+}\cup D_{-}$ that lie entirely within the
interior domain of $p(J)$, where $J$ was introduced in Lemma 3.12. Note that
$D_{J}$ is finely open because so are the fine components of finely open sets,
see Section A.5. Define

(3.65) $D^{\prime}:=D_{J}\cup i(\mathbb{D}\setminus D_{J})\quad\text{and}\quad
V^{\prime}:=p^{-1}(D^{\prime}),$

where $i(\cdot)$ is the subset of finely isolated points. Thus, it follows
from (A.9) and Lemma A.2 that $D^{\prime}$ and $V^{\prime}$ are finely open
while

(3.66) $b\big{(}\mathbb{D}\setminus D^{\prime}\big{)}=\mathbb{D}\setminus
D^{\prime}\quad\text{and}\quad b\big{(}{\mathcal{R}}\setminus
V^{\prime}\big{)}={\mathcal{R}}\setminus V^{\prime},$

where $b(\cdot)$ stands for the base of a set (in particular,
$\mathbb{D}\setminus D^{\prime}$ has no finely isolated points). In other
words, $D^{\prime}$ and $V^{\prime}$ are regular finely open sets, see Section
A.7. Recall from Section A.9 the notation $\sigma^{E}$ for the balayage of the
measure $\sigma$ onto the set $E$.

######  Lemma 3.13.

Let $N_{1}:=p^{-1}(\widetilde{N}_{1})$, where $\widetilde{N}_{1}$ is the
$+\infty$-set of $g\big{(}\mu^{(1)},\mathbb{D};\cdot\big{)}$ and we set

(3.67) $\mu^{(1)}:=\mu^{\mathbb{D}\setminus
D^{\prime}}\quad\text{and}\quad\nu^{(1)}:=\nu^{{\mathcal{R}}\setminus
V^{\prime}}.$

Then $N_{1}\subseteq N_{0}$, $N_{1}\setminus V^{\prime}=N_{0}\setminus
V^{\prime}$, and for every $z\in{\mathcal{R}}\setminus N_{1}$ we can define

$ler^{(1)}(z):=g\big{(}\mu^{(1)},\mathbb{D};p(z)\big{)}-g\big{(}\nu^{(1)},{\mathcal{R}};z\big{)}-h_{\mathcal{R}}(z)$

with values in $[-\infty,+\infty)$. This function satisfies

(3.68)
$ler^{(1)}(z)=\begin{cases}ler(z),&z\in{\mathcal{R}}\setminus(V^{\prime}\cup
N_{1}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ 0,&z\in V^{\prime}\setminus
N_{1}.\end{cases}$

###### Proof.

Since the Green potential of a measure dominates the Green potential of any
balayage of that measure, as explained at the beginning of Section A.9, it
holds that

$g\big{(}\mu^{(1)},\mathbb{D};z\big{)}\leq
g\big{(}\mu,\mathbb{D};z\big{)}<+\infty,\quad z\in{\mathcal{R}}\setminus
N_{0},$

by the very definition of $N_{0}$. Thus, $N_{1}\subseteq N_{0}$ and the upper
equality in (3.68) as well as equality $N_{1}\setminus
V^{\prime}=N_{0}\setminus V^{\prime}$ are consequences of (3.66) and (A.22).
Since $h_{\mathcal{R}}(z)$ is a harmonic function on ${\mathcal{R}}$ and
$\overline{V^{\prime}}\cap\partial{\mathcal{R}}=\varnothing$ by construction
(recall that $p(V^{\prime})$ lies interior to $p(J)$), equation (A.39) yields
that

(3.69) $\int h_{\mathcal{R}}(x)d\delta_{z}^{{\mathcal{R}}\setminus
V^{\prime}}(x)=h_{\mathcal{R}}(z),\quad z\in V^{\prime}.$

Since $V^{\prime}$ is a regular finely open set, Lemma A.6 implies that
$\delta_{z}^{{\mathcal{R}}\setminus V^{\prime}}$ is carried by
$\partial_{\mathsf{f}}V^{\prime}$ and does not charge polar sets. As
$ler(z)=0$ for quasi every $z\in\partial_{\mathsf{f}}V^{\prime}$ by the
definition of $V^{\prime}$, we get from the definition of $ler^{(1)}$, (3.69),
(A.25), Lemma A.7, and (3.46) that

$\displaystyle ler^{(1)}(z)$ $\displaystyle=$
$\displaystyle\int\big{(}g(\mu,\mathbb{D};p(x))-g(\nu,{\mathcal{R}};x)-h_{\mathcal{R}}(x)\big{)}d\delta_{z}^{{\mathcal{R}}\setminus
V^{\prime}}(x)$ $\displaystyle=$ $\displaystyle\int
ler(x)d\delta_{z}^{{\mathcal{R}}\setminus V^{\prime}}(x)=0$

for $z\in V^{\prime}\setminus N_{1}$, which proves the lower equality in
(3.68). ∎

Let $G_{J}$ be as in Lemma 3.12 for some small $\epsilon>0$. Denote by
$G^{*}_{J}$ the union of $G_{J}$ and the annular region delimited by $J$ and
$\mathcal{T}$; the latter is diffeomorphic under $p$ to
$\\{z:1-\epsilon<|z|<1\\}$ if we fix $\epsilon$ small enough. Clearly,
$G^{*}_{J}$ is a fine domain that lies schlicht over $\mathbb{D}$. Let
$\mathcal{G}$ be the collection of all fine domains $G\subset{\mathcal{R}}$
lying schlicht over $\mathbb{D}$ and containing $G_{J}^{*}$. The set
$\mathcal{G}$ is partially ordered by inclusion and every chain in it is
bounded above by the union of its elements. Therefore, by Zorn’s lemma,
$\mathcal{G}$ possesses a maximal element, say $G_{\mathsf{max}}$. Note that
if ${\mathcal{R}}\setminus G_{\mathsf{max}}$ is thin at two points
$\zeta_{1},\zeta_{2}\in{\mathcal{R}}$, then $p(\zeta_{1})\neq p(\zeta_{2})$ as
otherwise $G_{\mathsf{max}}$ could not be schlicht over $\mathbb{D}$ by Lemma
A.2. Hence, it follows from (A.9) and the maximality of $G_{\mathsf{max}}$
that ${\mathcal{R}}\setminus G_{\mathsf{max}}$ is its own base.

######  Lemma 3.14.

Any maximal domain $G_{\mathsf{max}}\in\mathcal{G}$ is a Euclidean domain.
Moreover, no connected component of ${\mathcal{R}}\setminus G_{\mathsf{max}}$
(resp. $\mathbb{D}\setminus p(G_{\mathsf{max}})$) consists of a single point.

###### Proof.

Observe that $G_{\mathsf{max}}$ cannot contain a ramification point of
${\mathcal{R}}$ as it is finely open and lies schlicht over $\mathbb{D}$, see
Lemma A.4. Thus, for every $\zeta\in G_{\mathsf{max}}$ there exists $r_{0}>0$
such that each component of

$p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-p(\zeta)|<r_{0}\big{\\}}\big{)}$

is in one-to-one correspondence with
$\\{z\in\mathbb{D}:~{}|z-p(\zeta)|<r_{0}\\}$ under $p$. Let $V$ be the
component containing $\zeta$. Since the intersection $V\cap G_{\mathsf{max}}$
is finely open, $V\setminus G_{\mathsf{max}}$ is thin at $\zeta$ and therefore
there is $r_{1}\in(0,r_{0})$ such that

$V\cap
p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-p(\zeta)|=r_{1}\big{\\}}\big{)}\subset
G_{\mathsf{max}},$

see Section A.6. Now, as $G_{\mathsf{max}}$ lies schlicht over $\mathbb{D}$,
it holds that

(3.70)
$G_{\mathsf{max}}\cap\left(p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-p(\zeta)|=r_{1}\big{\\}}\big{)}\setminus
V\right)=\varnothing.$

Further, since $G_{\mathsf{max}}$ is finely connected, we necessarily have
that

$G_{\mathsf{max}}\cap\left(p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-p(\zeta)|<r_{1}\big{\\}}\big{)}\setminus
V\right)=\varnothing,$

otherwise $G_{\mathsf{max}}$ would intersect the fine boundary of
$p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-p(\zeta)|<r_{1}\big{\\}}\big{)}\setminus
V$ which is contained in
$p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-p(\zeta)|=r_{1}\big{\\}}\big{)}\setminus
V$ (in fact, equal to it by regularity), thereby contradicting (3.70). The
maximality of $G_{\mathsf{max}}$ now yields that

$p^{-1}\big{(}\big{\\{}z\in\mathbb{D}:~{}|z-\zeta|<r_{1}\big{\\}}\big{)}\cap
V\subset G_{\mathsf{max}},$

hence $\zeta$ belongs to the Euclidean interior of $G_{\mathsf{max}}$ and so
$G_{\mathsf{max}}$ is Euclidean open. Thus, its connected components are open
and therefore finely open. Hence, $G_{\mathsf{max}}$ is a Euclidean domain
since it is finely connected by definition.

To prove the second assertion, assume to the contrary that a point
$\zeta\in{\mathcal{R}}\setminus G_{\mathsf{max}}$ is a connected component of
the latter. _We claim_ that any open neighborhood $W$ of $\zeta$ contains an
open set $O\ni\zeta$ whose boundary $\partial O$ is a smooth Jordan curve
$\mathcal{C}$ contained in $G_{\mathsf{max}}$. To see this, assume with no
loss of generality that $\overline{W}\subset{\mathcal{R}}$, and put
$F:={\mathcal{R}}\setminus G_{\mathsf{max}}\cap\overline{W}$. The latter is
closed in ${\mathcal{R}}$, so there is a smooth function
$h:{\mathcal{R}}\to\mathbb{R}^{+}$ of which $F$ is the zero set, see Section
3.4. Given a sequence $\\{c_{n}\\}$ of regular values of $h$ tending to $0$,
let $O_{n}$ be the connected component containing $\zeta$ of the open set
$\\{z:\,h(z)<c_{n}\\}$. The sets $O_{n}$ form a (strictly) nested sequence of
connected open sets containing $\zeta$, whose intersection is connected and
contained in $F$ whence reduces to $\zeta$. If $O_{n_{k}}\cap\partial
W\neq\varnothing$ for some increasing subsequence of indices $n_{k}$, take
$z_{n_{k}}\in O_{n_{k}}\cap\partial W$ and extract from $\\{z_{n_{k}}\\}$ a
subsequence converging to $z\in\partial W$, which is possible due to
compactness of the latter. On the other hand, since
$\overline{O}_{n_{k+1}}\subset O_{n_{k}}$,
$z\in\cap_{k}\overline{O}_{n_{k}}=\\{\zeta\\}\in\overline{W}\setminus\partial
W$, which is a contradiction. Therefore, $O_{n}\subset W$ for $n$ large enough
and $\partial O_{n}\cap F=\varnothing$ by construction, so we may set
$O=O_{n}$ and $\mathcal{C}:=\partial O_{n}$ for any $n$ large enough, because
$\partial O_{n}$ is a connected component of the level set $h^{-1}(c_{n})$,
and thus it is a smooth Jordan curve. _This proves the claim_.

If $\zeta\notin{\bf rp}({\mathcal{R}})$, let us pick $W$ so small that $p$ is
injective on each connected component of $p^{-1}(p(W))$. We now argue as
before, observing that ${\mathcal{C}}\subset G_{\mathsf{max}}$ and
$(p^{-1}(p({\mathcal{C}}))\setminus{\mathcal{C}})\cap
G_{\mathsf{max}}=\varnothing$ (by schlichtness), so that
$G_{\mathsf{max}}\cap(p^{-1}(p(O))\setminus O)=\varnothing$ because
$G_{\mathsf{max}}$ is connected and
$(p^{-1}(p({\mathcal{C}}))\setminus{\mathcal{C}})$ is the boundary of
$p^{-1}(p(O))\setminus O$. Thus, by maximality, $G_{\mathsf{max}}$ should
contain $O$ which contradicts the fact that $\zeta\in O$. Finally, if
$\zeta\in{\bf rp}({\mathcal{R}})$ then Lemma A.3 contradicts the existence of
${\mathcal{C}}$.

This proves that no connected component of ${\mathcal{R}}\setminus
G_{\mathsf{max}}$ consists of a single point. To show the same is true of
$\mathbb{D}\setminus p(G_{\mathsf{max}})$, observe that such a component would
consist, by the same reasoning as before, of a $z\in\partial
p(G_{\mathsf{max}})$ lying interior to Jordan curves of arbitrary small
diameter contained in $p(G_{\mathsf{max}})$. For $\gamma:[0,2\pi]\to
p(G_{\mathsf{max}})$ such a parametrized Jordan curve, let $\xi\in
G_{\mathsf{max}}$ satisfy $p(\xi)=\gamma(0)$ ($=\gamma(2\pi)$). Let further
$\ell:[0,2\pi]\to{\mathcal{R}}$ be a continuous lift of $\gamma$ starting at
$\xi$, i.e., $\ell(0)=\xi$ and $p(\ell(t))=\gamma(t)$ for $t\in[0,2\pi]$. Of
necessity $\ell([0,2\pi])\subset G_{\mathsf{max}}$, because $p(\partial
G_{\mathsf{max}})\cap p(G_{\mathsf{max}})=\varnothing$ since
$G_{\mathsf{max}}$ is open and lies schlicht over $\mathbb{D}$, while $p$ is
an open map; then schlichtness again implies that $\ell$ is a parametrized
Jordan curve in $G_{\mathsf{max}}$, and it is the unique lift of $\gamma$ to
$G_{\mathsf{max}}$.

Set $p^{-1}(z)=\\{\zeta_{1},\ldots,\zeta_{\ell}\\}$, and let $D_{z}$ be an
open disk centered at $z$ such that each connected component of
$p^{-1}(D_{z})$ is (isomorphic to) a $m(\zeta_{j})$-sheeted cyclic covering
$D_{\zeta_{j}}$ of $D_{z}$; in addition, we require that $D_{z}$ is so small
that $D_{\zeta_{j}}\subset U_{\zeta_{j}}$ for all $j$, where $U_{\zeta_{j}}$
is as in Lemma A.3. Let $\gamma:[0,2\pi]\to D_{z}$ be a parametrized Jordan
curve containing $z$ in its interior, and $\ell:[0,2\pi]\to G_{\mathsf{max}}$
the associated lift. Necessarily $\ell([0,2\pi])$ is contained in a single
component of $p^{-1}(D_{z})$, say $D_{\zeta_{j}}$, and it must contain
$\zeta_{j}$ in its interior (otherwise $\ell$ would be a unit in
$\pi_{1}(D_{\zeta_{j}}\setminus\zeta_{j})$ and so would be $\gamma=p\circ\ell$
in $\pi_{1}(D_{z}\setminus z)$). Now, if $m(\zeta_{j})>1$, then we contradict
Lemma A.3. Hence, $\ell$ is valued in a $D_{\zeta_{j}}$ such that
$m(\zeta_{j})=1$, which is thus homeomorphic to $D_{z}$ under $p$.

Let $\gamma_{n}$ be a sequence of Jordan curves in $D_{z}$, containing $z$ in
their interior and shrinking to $z$ when $n\to\infty$. Let further $\ell_{n}$
be the corresponding sequence of lifts to $G_{\mathsf{max}}$. By what
precedes, some subsequence $\ell_{m_{n}}$ shrinks to $\zeta_{j}$ in
$D_{\zeta_{j}}$, for some $j$ such that $D_{\zeta_{j}}$ is homeomorphic to
$D_{z}$ under $p$. Moreover, $\ell_{m_{n}}$ contains $\zeta_{j}$ in its
interior. We can now argue as we did to show that no connected component of
${\mathcal{R}}\setminus G_{\mathsf{max}}$ consists of a single point, and
contradict the maximal character of $G_{\mathsf{max}}$. This completes the
proof of the lemma. ∎

Define $D_{\mathsf{max}}:=p(G_{\mathsf{max}})$ and
$V_{\mathsf{max}}:=p^{-1}(D_{\mathsf{max}})$. Notice that both sets are open
by Lemma 3.14 along with openness and continuity of $p$.

######  Lemma 3.15.

Let $N_{2}:=p^{-1}(\widetilde{N}_{2})$, where $\widetilde{N}_{2}$ is the
$+\infty$-set of $g\big{(}\mu^{(2)},\mathbb{D};\cdot\big{)}$ and we set

(3.71) $\mu^{(2)}:=\big{(}\mu^{(1)}\big{)}^{\mathbb{D}\setminus
D_{\mathsf{max}}}\quad\text{and}\quad\nu^{(2)}:=\big{(}\nu^{(1)}\big{)}^{{\mathcal{R}}\setminus
V_{\mathsf{max}}}.$

Then $N_{2}=N_{1}\setminus V_{\mathsf{max}}$, and for
$z\in{\mathcal{R}}\setminus N_{2}$ we can define

(3.72)
$ler^{(2)}(z):=g\big{(}\mu^{(2)},\mathbb{D};p(z)\big{)}-g\big{(}\nu^{(2)},{\mathcal{R}};z\big{)}-h_{\mathcal{R}}(z).$

In this case it holds that

(3.73) $\begin{cases}\displaystyle\limsup_{\zeta\to
z}ler^{(2)}(\zeta)\leq-2/\mathrm{cap}_{\mathbb{D}}(K_{f}),&z\in\mathcal{T},\\\
ler^{(2)}(\zeta)=0,&z\in{\mathcal{R}}\setminus(G_{\mathsf{max}}\cup
N_{2}).\end{cases}$

Moreover, we have that

(3.74) $\big{\|}\mu^{(2)}\big{\|}\begin{cases}=\|\mu\|&\text{if
}\mathrm{supp}(\mu^{(1)})\cap D_{\mathsf{max}}=\varnothing,\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ <\|\mu\|&\text{otherwise}.\end{cases}$

###### Proof.

By Lemma 3.14 the set $\mathbb{D}\setminus D_{\mathsf{max}}$ is closed in
$\mathbb{D}$ and none of its connected components reduces to a point. Hence,
it has no finely isolated points (remember that a connected set cannot be thin
at an accumulation point, see discussion after (A.8)). Thus,
$\mathbb{D}\setminus D_{\mathsf{max}}$ is its own base. Consequently,
${\mathcal{R}}\setminus V_{\mathsf{max}}$ is also its own base by Lemma A.2.
Therefore, we get from (A.22) that

(3.75) $\mu^{(2)}(D_{\mathsf{max}})=\nu^{(2)}(V_{\mathsf{max}})=0.$

This implies that the potentials of $\mu^{(2)}$ and $\nu^{(2)}$ are harmonic
in $D_{\mathsf{max}}$ and $V_{\mathsf{max}}$, respectively. Moreover, since
$\partial D_{\mathsf{max}}\setminus\mathbb{T}$ is separated from $\mathbb{T}$
by the very definition of $G_{\mathsf{max}}$, $\partial
V_{\mathsf{max}}\setminus\partial{\mathcal{R}}$ is necessarily separated from
$\partial{\mathcal{R}}$ and these potentials extend continuously by zero to
$\mathbb{T}$ and $\partial{\mathcal{R}}$ respectively by the regularity of the
latter, see the discussion after (A.12). Therefore, $ler^{(2)}$ extends
harmonically to the whole of $V_{\mathsf{max}}$ and continuously to
$\partial{\mathcal{R}}\setminus\mathcal{T}$ by Lemma 3.8. So, as in Lemma
3.13, the inclusion $N_{2}\subseteq N_{1}\setminus V_{\mathsf{max}}$ follows
directly from (A.21) and the equality $N_{2}=N_{1}\setminus V_{\mathsf{max}}$
is then deduced from (A.22). In addition, the first inequality in (3.73) holds
in view of (3.17), the definition of $h_{\mathcal{R}}$ given after (3.46), and
the fact that $h^{\prime\prime}$ is non-negative, see Lemma 3.5.

Since $N_{1}\setminus V^{\prime}=N_{0}\setminus V^{\prime}$ by Lemma 3.13 we
have that ${\mathcal{R}}\setminus(V^{\prime}\cup V_{\mathsf{max}}\cup
N_{0})={\mathcal{R}}\setminus(V^{\prime}\cup V_{\mathsf{max}}\cup N_{1})$.
Moreover, $ler(z)=0$ on this set by the very definitions of $V^{\prime}$ and
$V_{\mathsf{max}}$. Further,

${\mathcal{R}}\setminus(V_{\mathsf{max}}\cup
N_{2})={\mathcal{R}}\setminus(V_{\mathsf{max}}\cup
N_{1})=({\mathcal{R}}\setminus(V^{\prime}\cup V_{\mathsf{max}}\cup
N_{1}))\cup(V^{\prime}\setminus N_{1}).$

Since $\mathbb{D}\setminus D_{\mathsf{max}}$ and ${\mathcal{R}}\setminus
V_{\mathsf{max}}$ are their own bases, it therefore follows from (A.22) and
(3.68) that

(3.76) $ler^{(2)}(z)=ler^{(1)}(z)=0,\quad
z\in{\mathcal{R}}\setminus(V_{\mathsf{max}}\cup N_{2}).$

To study the values of $ler^{(2)}$ on $V_{\mathsf{max}}\setminus
G_{\mathsf{max}}$, let us show that this is an open set. Indeed, since
$p(G_{\mathsf{max}})=p(V_{\mathsf{max}})=D_{\mathsf{max}}$, for each $\zeta\in
V_{\mathsf{max}}$ there exists a disk $D_{p(\zeta)}\subset D_{\mathsf{max}}$,
centered at $p(\zeta)$, and a point $z\in G_{\mathsf{max}}$ with
$p(z)=p(\zeta)$ such that $D_{\zeta}\subset V_{\mathsf{max}}$ and
$D_{z}\subset G_{\mathsf{max}}$, where $D_{\zeta}$ and $D_{z}$ are the
connected components of $p^{-1}(D_{p(\zeta)})$ that contain $\zeta$ and $z$,
respectively. If $\zeta\neq z$, then $D_{\zeta}\cap
G_{\mathsf{max}}=\varnothing$ since $G_{\mathsf{max}}$ lies schlicht over
$\mathbb{D}$ and therefore $D_{z}\subset V_{\mathsf{max}}\setminus
G_{\mathsf{max}}$ as claimed. Moreover, since ${\mathcal{R}}\setminus
V_{\mathsf{max}}$ is its own base, we get from Lemma A.1 that
${\mathcal{R}}\setminus(V_{\mathsf{max}}\setminus G_{\mathsf{max}})$ is also
its own base.

Pick $z\in V_{\mathsf{max}}\setminus G_{\mathsf{max}}$ and notice that
$V_{\mathsf{max}}\setminus G_{\mathsf{max}}$ consists of at most finitely many
connected components. Let $V_{z}$ be the component containing $z$. As
$G_{\mathsf{max}}$ contains the annular region delimited by $J$ and
$\mathcal{T}$ and lies schlicht over $\mathbb{D}$, it follows that
$\partial(V_{\mathsf{max}}\setminus
G_{\mathsf{max}})\cap\partial{\mathcal{R}}=\partial{\mathcal{R}}\setminus\mathcal{T}$.
Since $h_{\mathcal{R}}\equiv 0$ on $\partial{\mathcal{R}}\setminus\mathcal{T}$
by Lemma 3.8, we get from Lemmas A.6 and A.10 that

(3.77) $\int h_{\mathcal{R}}(x)d\delta_{z}^{{\mathcal{R}}\setminus
V_{z}}(x)=\int
h_{\mathcal{R}}(x)d\delta_{z}^{{\mathcal{R}}\setminus(V_{\mathsf{max}}\setminus
G_{\mathsf{max}})}(x)=h_{\mathcal{R}}(z).$

Since $V_{z}$, $V_{\mathsf{max}}\setminus G_{\mathsf{max}}$, and
$G_{\mathsf{max}}$ are Euclidean open sets, it holds that $\partial
V_{z}\subseteq\partial(V_{\mathsf{max}}\setminus
G_{\mathsf{max}})\subseteq\partial V_{\mathsf{max}}$, a set on which
$ler^{(1)}(z)$ is zero except possibly on $N_{2}$ by (3.76). Consequently, in
view of (3.72), we see upon using Lemma A.7, (A.25), and (3.77) that

$ler^{(2)}(z)=\int ler^{(1)}(x)d\delta_{z}^{{\mathcal{R}}\setminus
V_{z}}(x)=0,$

where the second equality follows from Lemma A.6 as
$\partial_{\mathsf{f}}V_{z}\subseteq\partial V_{z}$ and $ler^{(1)}(x)=0$ quasi
and therefore $\delta_{z}^{{\mathcal{R}}\setminus V_{z}}$-almost everywhere on
$\partial_{\mathsf{f}}V_{z}$.

Finally, we get from (A.24) and (A.38) that $\|\mu^{(2)}\|=\|\mu^{(1)}\|$ when
$\mathrm{supp}(\mu^{(1)})\cap D_{\mathsf{max}}=\varnothing$ and
$\|\mu^{(2)}\|<\|\mu^{(1)}\|$ otherwise as well as that
$\|\mu^{(1)}\|=\|\mu\|$, which proves (3.74). ∎

###  3.9. Projected Logarithmic Error Function

Recall that $z\in\partial G_{\mathsf{max}}$ is called accessible if there
exists a continuous map $\psi$ on $[0,1]$ such that $\psi(t)\in
G_{\mathsf{max}}$ for $t\in[0,1)$ and $\psi(1)=z$.

######  Lemma 3.16.

It holds that $\operatorname*{card}\left(p^{-1}(z)\cap(\partial
G_{\mathsf{max}}\setminus E^{*})\right)\leq 2$ for all except countably many
$z\in\partial D_{\mathsf{max}}$, where $E^{*}\subset\partial G_{\mathsf{max}}$
is the subset of non-accessible points.

###### Proof.

Since $\mathbb{T}\subset\partial D_{\mathsf{max}}$ and
$p^{-1}(\mathbb{T})\cap\partial G_{\mathsf{max}}=\mathcal{T}$, we only need to
consider $z\in\partial D_{\mathsf{max}}\cap\mathbb{D}$. The set $(\partial
D_{\mathsf{max}}\cap\mathbb{D})\setminus p({\bf rp}({\mathcal{R}}))$ can be
covered by countably many open sets of the form $D_{x}$, where $x\in(\partial
D_{\mathsf{max}}\cap\mathbb{D})\setminus p({\bf rp}({\mathcal{R}}))$ and
$D_{x}\subset\mathbb{D}$ is a disk centered at $x$, small enough that each
component of $p^{-1}(D_{x})$ is homeomorphic to $D_{x}$ under $p$. Hence, it
is enough to show that for each such $D_{x}$ and all $z\in\partial
D_{\mathsf{max}}\cap D_{x}$ but countably many, one has
$\operatorname*{card}\left(p^{-1}(z)\cap(\partial G_{\mathsf{max}}\setminus
E^{*})\right)\leq 2$. Fix $D_{x}$ and let $x_{1},\ldots,x_{N}$ denote the
preimages of $x$ under $p$. We write $V_{x_{\ell}}$ for the connected
component of $p^{-1}(D_{x})$ containing $x_{\ell}$, $1\leq\ell\leq N$. Then,
each $z\in D_{x}$ has preimages $z_{1},\ldots,z_{N}$ under $p$, with
$z_{\ell}\in V_{x_{\ell}}$. When $z\in\partial D_{\mathsf{max}}\cap D_{x}$, it
follows from the definition of accessibility that if $z_{\ell}\in\partial
G_{\mathsf{max}}\setminus E^{*}$, then there is a continuous arc
$\psi_{z_{\ell}}:[0,1]\to\overline{G}_{\mathsf{max}}$ such that
$\psi_{z_{\ell}}(t)\in G_{\mathsf{max}}$ for $t\in[0,1)$ and
$\psi_{z_{\ell}}(1)=z_{\ell}$. Moreover, on shortening and reparametrizing the
arc if necessary, we may assume that $\psi_{z_{\ell}}$ is valued in
$V_{x_{\ell}}$. Now, if $p^{-1}(z)\cap(\partial G_{\mathsf{max}}\setminus
E^{*})$ contains 3 distinct points, say $z_{\ell_{j}}$ for $j=\\{1,2,3\\}$,
then $p(\psi_{z_{\ell_{j}}})$ are Jordan arcs having only the point $z$ in
common, because $G_{\mathsf{max}}$ lies schlicht over $\mathbb{D}$ and the
$V_{x_{\ell_{j}}}$ are disjoint. Thus, $T_{z}:=\cup_{j}p(\psi_{z_{\ell_{j}}})$
is a triod, and if we had uncountably many such $z$ then some triple
$(\ell_{1},\ell_{2},\ell_{3})$ would occur uncountably many times. Assigning
different colors to $\psi_{z_{\ell_{1}}}$, $\psi_{z_{\ell_{2}}}$ and
$\psi_{z_{\ell_{3}}}$, we get and uncountable collection of colored triods
$T_{z}$ whose arcs of different colors never meet, again because
$G_{\mathsf{max}}$ lies schlicht over $\mathbb{D}$ and the $V_{x_{\ell_{j}}}$
are disjoint. However, this contradicts the Moore triod theorem [50,
Proposition 2.18], thereby finishing the proof. ∎

Put $K_{\mathsf{max}}:=\mathbb{D}\setminus D_{\mathsf{max}}$. It follows from
Lemma 3.14 that $f$ has a single-valued meromorphic continuation throughout
$D_{\mathsf{max}}\setminus p(E_{f})$ and $K_{\mathsf{max}}$ is a compact
subset of $\mathbb{D}$. Hence, $K_{\mathsf{max}}\in\mathcal{K}_{f}$, where
$\mathcal{K}_{f}$ was defined just before (2.5). Clearly, the measure
$\mu^{(2)}$ is supported on $K_{\mathsf{max}}$.

######  Lemma 3.17.

Let $m_{\zeta}$ be the ramification order of $\zeta\in{\mathcal{R}}$. Define

(3.78) $ler_{\mathsf{pr}}(z):=\sum_{\zeta\in
p^{-1}(z)}m_{\zeta}ler^{(2)}(\zeta),\quad
z\in\mathbb{D}\setminus\widetilde{N}_{2}.$

Then $ler_{\mathsf{pr}}$ is a $\delta$-subharmonic function in $\mathbb{D}$
such that $ler_{\mathsf{pr}}(z)=0$ when $z\in
K_{\mathsf{max}}\setminus\widetilde{N}_{2}$ and

$\limsup_{z\to\zeta}ler_{\mathsf{pr}}(z)\leq-2/\mathrm{cap}_{\mathbb{D}}(K_{f}),\quad\zeta\in\mathbb{T}.$

Moreover, there exist non-negative measures $\mu_{\mathsf{pr}}$ and
$\nu_{\mathsf{pr}}$, supported on $\partial D_{\mathsf{max}}$, and a non-
negative function $h_{\mathsf{pr}}$, harmonic in $\mathbb{D}$, such that
$\mu_{\mathsf{pr}}\leq 2\mu^{(2)}$ and for
$z\in\mathbb{D}\setminus\widetilde{N}_{2}$

(3.79)
$ler_{\mathsf{pr}}(z)=g(\mu_{\mathsf{pr}},\mathbb{D};z)-g(\nu_{\mathsf{pr}},\mathbb{D},z)-h_{\mathsf{pr}}(z)-2/\mathrm{cap}_{\mathbb{D}}(K_{f}).$

###### Proof.

The first two claims of the lemma follow readily from (3.73) and the
computation in (A.11). Let $\widehat{\mu}^{(2)}$ be the pullback of
$\mu^{(2)}$ onto ${\mathcal{R}}$, see (A.33) and (A.34). Then we can
equivalently write

$\displaystyle ler^{(2)}(z)$
$\displaystyle=g\big{(}\widehat{\mu}^{(2)},{\mathcal{R}};z\big{)}-g\big{(}\nu^{(2)},{\mathcal{R}};z\big{)}-h_{\mathcal{R}}(z)$
$\displaystyle=g(\mu_{\mathsf{R}},{\mathcal{R}};z)-g(\nu_{\mathsf{R}},{\mathcal{R}};z)-h_{\mathcal{R}}(z),$

where $\mu_{\mathsf{R}}-\nu_{\mathsf{R}}=\widehat{\mu}^{(2)}-\nu^{(2)}$ and
$\mu_{\mathsf{R}}$, $\nu_{\mathsf{R}}$ are positive mutually singular measures
on $p^{-1}(K_{\mathsf{max}})$, i.e., $\mu_{\mathsf{R}}-\nu_{\mathsf{R}}$ is
the Riesz charge of a $\delta$-subharmonic function $ler^{(2)}$. Clearly,
$\nu_{\mathsf{R}}\leq\nu^{(2)}$, $\mu_{\mathsf{R}}\leq\widehat{\mu}^{(2)}$,
and for any Borel set $B\subset p^{-1}(K_{\mathsf{max}})\setminus{\bf
rp}({\mathcal{R}})$ that lies schlicht over $\mathbb{D}$ it holds that

(3.80) $\mu_{\mathsf{R}}(B)\leq\mu^{(2)}(p(B)).$

Similarly to the computation in (A.11), one can show that $\sum_{\zeta\in
p^{-1}(z)}m_{\zeta}h(\zeta)$ is harmonic in $\mathbb{D}$ when $h$ is harmonic
on ${\mathcal{R}}$. Hence, it follows from the maximum principle for harmonic
functions and Lemma 3.2 that

$\sum_{\zeta\in
p^{-1}(z)}m_{\zeta}h_{\mathcal{R}}(\zeta)=2/\mathrm{cap}_{\mathbb{D}}(K_{f})+h_{\mathsf{pr}}(z),\quad
h_{\mathsf{pr}}(z):=\sum_{\zeta\in
p^{-1}(z)}m_{\zeta}h^{\prime\prime}(\zeta).$

Thus, if we set $\mu_{\mathsf{pr}}=p_{*}(\mu_{\mathsf{R}})$ and
$\nu_{\mathsf{pr}}=p_{*}(\nu_{\mathsf{R}})$, see (A.32), we get (3.79).

As explained in Lemma 3.15, ${\mathcal{R}}\setminus V_{\mathsf{max}}$ is its
own base and $V_{\mathsf{max}}\setminus G_{\mathsf{max}}$, $G_{\mathsf{max}}$
are disjoint open sets. Hence, ${\mathcal{R}}\setminus G_{\mathsf{max}}$ is
its own base by Lemma A.1. Moreover, the function $ler^{(2)}$ is equal to zero
on this set by (3.73). Hence, we get from the proof of [13, Theorem 2] (that
carries over mutatis mutandis to any hyperbolic surface) the implication:

(3.81) $\delta_{z}^{{\mathcal{R}}\setminus
G_{\mathsf{max}}}(F)=0\quad\Rightarrow\quad(\mu_{\mathsf{R}}+\nu_{\mathsf{R}})(F)=0$

for any $F\subset{\mathcal{R}}\setminus G_{\mathsf{max}}$ and some (therefore
any) $z\in G_{\mathsf{max}}$. In particular, it follows from Lemma A.6 that
$\mathrm{supp}(\mu_{\mathsf{R}}+\nu_{\mathsf{R}})\subseteq\partial
G_{\mathsf{max}}\setminus\mathcal{T}$ and that this measure does not charge
polar sets. Hence, since $G_{\mathsf{max}}$ lies schlicht over
$D_{\mathsf{max}}$, it holds that
$\mathrm{supp}(\mu_{\mathsf{pr}}+\nu_{\mathsf{pr}})\subseteq\partial
D_{\mathsf{max}}\setminus\mathbb{T}$. Moreover, let $\tilde{E}$ be the set of
points $z\in\partial D_{\mathsf{max}}$ such that
$\operatorname*{card}\left(p^{-1}(z)\cap(\partial G_{\mathsf{max}}\setminus
E^{*})\right)>2$, where $E^{*}$ is the set of non-accessible points of
$\partial G_{\mathsf{max}}$. It follows from Lemma 3.16 that $\tilde{E}$ is at
most countable and therefore is not charged by $\mu_{\mathsf{pr}}$. Of course,
the same is true of $p({\bf rp}({\mathcal{R}}))$, as it is a finite set. For
any Borel set $B\subseteq\partial D_{\mathsf{max}}$, let
$\tilde{B}:=B\setminus(\tilde{E}\cup p({\bf rp}({\mathcal{R}})))$. Then, we
obtain

$\displaystyle\mu_{\mathsf{pr}}(B)$
$\displaystyle=\mu_{\mathsf{pr}}(\tilde{B})=\mu_{\mathsf{R}}\big{(}p^{-1}(\tilde{B})\cap\partial
G_{\mathsf{max}}\big{)}$
$\displaystyle=\mu_{\mathsf{R}}\big{(}p^{-1}(\tilde{B})\cap(\partial
G_{\mathsf{max}}\setminus E^{*})\big{)}\leq 2\mu^{(2)}(B),$

where the third equality follows from [13, Corollary 2], which says that the
Riesz charge of a $\delta$-subharmonic function cannot charge points that are
non-accessible from the complement of the base of its zero set, and the last
inequality is due to Lemma 3.16 and (3.80). ∎

###  3.10. Computation of the Logarithmic Error Function

Lemma 3.17 implies that $ler_{\mathsf{pr}}(z)=0$ for $z\in
K_{\mathsf{max}}\setminus\widetilde{N}_{2}$. Hence, it follows from (3.79) and
properties of the Green equilibrium potential that

(3.82)
$g\big{(}\mu_{\mathsf{pr}},\mathbb{D};z\big{)}-g(\nu_{\mathsf{pr}},\mathbb{D};z)-h_{\mathsf{pr}}(z)=2\frac{\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})}{\mathrm{cap}_{\mathbb{D}}(K_{f})}g\big{(}\mu_{\mathbb{D},K_{\mathsf{max}}},\mathbb{D};z\big{)}$

for $z\in K_{\mathsf{max}}\setminus\widetilde{N}_{2}$ (since
$K_{\mathsf{max}}$ is equal to its own base,
$g\big{(}\mu_{\mathbb{D},K_{\mathsf{max}}},\mathbb{D};\cdot\big{)}\equiv
1/\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})$ on $K_{\mathsf{max}}$). Assume
that either $\nu_{\mathsf{pr}}$ is a non-zero measure or $h_{\mathsf{pr}}$ is
strictly positive (harmonic) function in $\mathbb{D}$. Then, since
$K_{\mathsf{max}}$ is separated from $\mathbb{T}$, it would hold that

$g\big{(}\mu_{\mathsf{pr}},\mathbb{D};z\big{)}>2\frac{\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})}{\mathrm{cap}_{\mathbb{D}}(K_{f})}g\big{(}\mu_{\mathbb{D},K_{\mathsf{max}}},\mathbb{D};z\big{)}$

for $z\in K_{\mathsf{max}}\setminus\widetilde{N}_{2}$. Since
$\mathrm{supp}(\mu_{\mathbb{D},K_{\mathsf{max}}})\subseteq K_{\mathsf{max}}$
and $K_{\mathsf{max}}$ is its own base, the Strong Domination Principle, see
Section A.6, implies that the above inequality holds everywhere in
$\mathbb{D}$. Thus, integrating both sides of this inequality against
$\mu_{\mathbb{D},K_{\mathsf{max}}}$ which is a probability measure, we get on
using Tonelli’s theorem on the left-hand side and multiplying by
$\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})$ that

(3.83)
$\|\mu_{\mathsf{pr}}\|>2\frac{\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})}{\mathrm{cap}_{\mathbb{D}}(K_{f})}.$

On the one hand, by Lemma 3.17 and equation (3.74) together with the very
construction of $\mu$, we have that $\|\mu_{\mathsf{pr}}\|\leq
2\|\mu^{(2)}\|\leq 2\|\mu\|\leq 2$. On the other hand holds
$\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})\geq\mathrm{cap}_{\mathbb{D}}(K_{f})$,
see (2.5). These observations clearly show that (3.83) is impossible. Hence,
it is necessarily the case that

(3.84) $\nu_{\mathsf{pr}}=0\quad\text{and}\quad h_{\mathsf{pr}}(z)\equiv
0,\;\;z\in\mathbb{D}.$

Then, one can rewrite (3.82) as

(3.85)
$g\big{(}\mu_{\mathsf{pr}},\mathbb{D};z\big{)}=2\frac{\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})}{\mathrm{cap}_{\mathbb{D}}(K_{f})}g\big{(}\mu_{\mathbb{D},K_{\mathsf{max}}},\mathbb{D};z\big{)}$

for $z\in K_{\mathsf{max}}\setminus\widetilde{N}_{2}$. Using now the Strong
Domination Principle in both directions, we get that (3.85) holds for every
$z\in\mathbb{D}$. Therefore,

(3.86)
$\mu^{(2)}\geq\frac{1}{2}\mu_{\mathsf{pr}}=\frac{\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})}{\mathrm{cap}_{\mathbb{D}}(K_{f})}\mu_{\mathbb{D},K_{\mathsf{max}}}$

which, upon recalling once again (2.5) and the fact that
$\mu_{\mathbb{D},K_{\mathsf{max}}}$ is a probability measure, gives us

(3.87)
$\mathrm{cap}_{\mathbb{D}}(K_{\mathsf{max}})=\mathrm{cap}_{\mathbb{D}}(K_{f})\quad\text{and}\quad\mu^{(2)}=\mu_{\mathbb{D},K_{\mathsf{max}}}.$

In particular $\widetilde{N}_{2}=\varnothing$ and the first equality in
(3.87), combined with the minimality and uniqueness of $K_{f}$, yields that
$K_{\mathsf{max}}=K_{f}$. In addition, as $\|\mu^{(2)}\|=1$ by (3.87), we get
from (3.74) that $\mathrm{supp}(\mu^{(1)})\cap D_{\mathsf{max}}=\varnothing$
and therefore $\mu^{(1)}=\mu^{(2)}$, by (A.22), (3.71), and the fact that
$\mathbb{D}\setminus D_{\mathsf{max}}$ is its own base (because so is
${\mathcal{R}}\setminus G_{\mathsf{max}}$ and we can use Lemma A.2). Moreover,
we also get from (3.74) that $\mathrm{supp}_{\mathsf{f}}(\mu^{(1)})\cap
D_{\mathsf{max}}=\varnothing$ (note that
$\mathrm{supp}_{\mathsf{f}}(\mu^{(1)})$ exists because $\mu^{(1)}$ is
admissible, see Section A.9). Remembering that the fine open set $D^{\prime}$
is regular by (3.66), we get from (A.24) and Lemma A.6 that
$\mathrm{supp}_{\mathsf{f}}(\mu^{(1)})\supset\cup_{t}\partial_{\mathsf{f}}D^{\prime}_{t}$
where $D^{\prime}_{t}$ are the finely connected components of $D^{\prime}$
such that $\mu(D_{t})\neq 0$. However, since $K_{\mathsf{max}}$(= $K_{f}$) has
no fine interior by inspection of (2.17), each
$\partial_{\mathsf{f}}D_{t}^{\prime}$ must intersect $D_{\mathsf{max}}$
whenever $D^{\prime}_{t}$ is nonempty. Hence, $\mu$ cannot charge $D^{\prime}$
as otherwise it would contradict that
$\mathrm{supp}_{\mathsf{f}}(\mu^{(1)})\cap D_{\mathsf{max}}=\varnothing$.
Consequently, since $\mathbb{D}\setminus D^{\prime}$ is its own base, we
conclude in view of (A.22) and (3.67) that $\mu^{(1)}=\mu$. Thus, we obtain
altogether that

(3.88) $\mu_{\mathbb{D},K_{f}}=\mu^{(2)}=\mu^{(1)}=\mu=\mu^{\prime},$

where the last equality comes from the inequalities $\mu\leq\mu^{\prime}$ and
$\|\mu^{\prime}\|\leq 1$, see (3.27). In addition, since
$\mu_{\mathbb{D},K_{f}}$ has finite potential everywhere, we get that
$\widetilde{N}_{0}=N_{0}=\varnothing$ whence also
$\widetilde{N}_{1}=N_{1}=\varnothing$.

From (3.88) one sees that $\mu$ does not charge $D_{\mathsf{max}}$, implying
in view of (3.44) that $ler$ is subharmonic on
$V_{\mathsf{max}}=p^{-1}(D_{\mathsf{max}})$. In particular, since
$g(\mu_{\mathbb{D},K_{\mathsf{max}}},\mathbb{D};\cdot)$ extends continuously
by zero on $\mathbb{T}$ and $h^{\prime}$ by
$2/\mathrm{cap}_{\mathbb{D}}(K_{f})$ on $\mathcal{T}$, see Lemma 3.2), while
$\mathrm{fine}\lim_{z\to\zeta}ler(z)=ler(\zeta)<0$ when $\zeta\in J$ by the
fine continuity of $ler$ and the fact that $J\subset G_{-}$, it follows from
the relative fine boundary maximum principle [20, Theorem 10.8] that $ler<0$
in the annular region $\mathcal{A}(\mathcal{T},J)$ bounded by $\mathcal{T}$
and $J$, since it is bounded above by the fine potential
$g(\widehat{\mu}_{\mathbb{D},K_{\mathsf{max}}},{\mathcal{R}};\cdot)$ there,
see (A.34). Let now $G^{\prime}$ be a finely connected component of $G_{+}$.
Since $G^{+}=p^{-1}(p(G^{+}))$ by Lemma 3.10 and
$\mathcal{A}(\mathcal{T},J)\subset G_{-}$, $p(G^{\prime})$ lies in the
interior of $p(J)$. Hence, $D^{\prime}=p(G^{\prime})\in D_{J}$ and by what
precedes $D^{\prime}\cap K_{f}=\varnothing$ as otherwise
$\mu=\mu_{\mathbb{D},K_{f}}$ would charge $D^{\prime}$ because it is carried
by the whole set $K_{f}$ according to (A.30), i.e., it cannot be carried by
$K_{f}\setminus D^{\prime}$ (which is finely closed). Consequently,
$G^{\prime}\cap p^{-1}(K_{f})=\varnothing$ which implies that $ler\leq 0$ on
$p^{-1}(K_{f})$, and the relative fine boundary maximum principle in turn
implies that $ler\leq 0$ on $V_{\mathsf{max}}$. Moreover, $ler<0$ on
$G_{\mathsf{max}}$ as it is strictly negative on $\mathcal{A}(\mathcal{T},J)$.
Immediately we deduce that $G_{+}=\varnothing$ and $G_{\mathsf{max}}\subseteq
G_{-}$. Furthermore, since $G_{-}$ lies schlicht over $\mathbb{D}$ by Lemma
3.11 while $K_{f}$ has no fine interior, maximality of $G_{\mathsf{max}}$
implies that $G_{\mathsf{max}}=G_{-}$. Altogether, we obtain that
$D^{\prime}=V^{\prime}=\varnothing$ and $G_{-}=G_{\mathsf{max}}$. In
particular, the step of Lemma 3.13 is vacuous and $\nu^{(1)}=\nu$ as well as
$ler=ler^{(1)}$. Moreover, $ler(z)=0$ holds for $z\in{\mathcal{R}}\setminus
G_{\mathsf{max}}$.

Next, by (3.86), (3.88) and the construction of the measures
$\mu_{\mathsf{pr}}$, $\mu_{\mathsf{R}}$ in Lemma 3.17, along with the
discussion after (3.81), one has

$2\mu_{\mathbb{D},K_{f}}=2\mu^{(2)}=\mu_{\mathsf{pr}}=p_{*}(\mu_{\mathsf{R}})\quad\text{and}\quad\mu_{\mathsf{R}}\leq(\widehat{\mu}_{\mathbb{D},K_{f}})_{\mathcal{b}\partial
G_{\mathsf{max}}}.$

Lemma 3.16 now yields that this last inequality is in fact an equality. As
$\mu_{\mathsf{R}}-\nu_{\mathsf{R}}$ is the Riesz charge of $ler^{(2)}$ that
vanishes on ${\mathcal{R}}\setminus G_{\mathsf{max}}$ by (3.73), the
discussion after (3.81) implies
$\nu_{\mathsf{R}}=\nu^{(2)}_{\mathcal{b}\partial G_{\mathsf{max}}}$, and since
$\nu_{\mathsf{pr}}=p_{*}(\nu_{\mathsf{R}})$ we get from (3.84) and (3.71) that

(3.89)
$0=\nu^{(2)}(\overline{G}_{\mathsf{max}})=\nu^{(1)}(\overline{G}_{\mathsf{max}})=\nu(\overline{G}_{\mathsf{max}}),$

where the middle equality holds by (A.24) and Lemma A.6 (because
$\delta_{z}^{{\mathcal{R}}\setminus G_{\mathsf{max}}}$ is a strictly positive
function of $z$ in the regular open set $G_{\mathsf{max}}$), while the last
equality comes from $ler=ler^{(1)}$. Recall that
$\nu=\nu^{\prime\prime}+\nu^{\prime}=\nu^{*}+\tilde{\nu}+\nu^{\prime}$, see
(3.46) and (3.38), where $\nu^{*}$ the vague limit point of $\\{\nu_{n}\\}$ in
${\mathcal{R}}$, see Lemma 3.5. As $ler(z)=0$ for $z\in{\mathcal{R}}\setminus
G_{\mathsf{max}}$ and $ler$ is a $\delta$-subharmonic function, we get from
[13, Theorem 2] that its Riesz charge is supported on $\partial
G_{\mathsf{max}}$. In view of (3.89), it entails that

(3.90)
$(\widehat{\mu}_{\mathbb{D},K_{f}})_{\mathcal{b}\overline{p^{-1}(K_{f})\setminus\partial
G_{\mathsf{max}}}}=\nu=\nu^{\prime\prime}=\nu^{*},$

where $\tilde{\nu}+\nu^{\prime}=0$ since it is a measure supported on $E_{f}$,
see Lemma 3.1 and (3.36), while $\nu$ does not charge polar sets by the first
equality above. Since $\nu=\nu^{(1)}$ and $\nu^{(1)}$ has no mass in
$V_{\mathsf{max}}$ by (3.90), the step of Lemma 3.15 was also vacuous. We thus
get that $ler=ler^{(2)}$, and it follows from (3.84) together with the
construction of $h_{\mathsf{pr}}$ in Lemma 3.17 that
$h_{\mathcal{R}}=h^{\prime}$. Therefore, the inequality in (3.73) is in fact
an equality by Lemma 3.2, and consequently

(3.91)
$ler(z)=\begin{cases}2g(\mu_{\mathbb{D},K_{f},},\mathbb{D};p(z))-2/\mathrm{cap}_{\mathbb{D}}(K_{f}),&z\in
G_{\mathsf{max}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
0,&z\in{\mathcal{R}}\setminus G_{\mathsf{max}},\end{cases}$

as both sides of (3.91) are continuous on $\overline{{\mathcal{R}}}$, equal to
zero on ${\mathcal{R}}\setminus G_{\mathsf{max}}$, equal to
$-2/\mathrm{cap}_{\mathbb{D}}(K_{f})$ on $\mathcal{T}$, and harmonic in
$G_{\mathsf{max}}$ so that the equality in $G_{\mathsf{max}}$ is consequence
of the maximum principle for harmonic functions.

Notice that we started our proof with the limit (3.7) taking place along the
full sequence $\mathbb{N}$ of integers, that was later refined into a
subsequence in Section 3.5, and refined still further in Section 3.7 to a
subsequence $\mathcal{N}$ along which all the above results hold. However, we
could have initiated our argument using any subsequence
$\mathbb{N}_{0}\subset\mathbb{N}$ in (3.7) with the same conclusions holding
along some $\mathcal{N}\subseteq\mathbb{N}_{0}$. Hence, if there existed a
subsequence along which either $\mu^{\prime}\neq\mu_{\mathbb{D},K_{f}}$,
$\nu^{\prime}\neq 0$, $\tilde{\nu}\neq 0$, or $\nu^{*}$ are not as in the
left-hand side of (3.90), or if $h^{\prime\prime}$ were not equal to zero,
then we could use it as $\mathbb{N}_{0}$ in (3.7) to arrive at a
contradiction. Hence, all lemmas of this section hold along the full sequence
$\mathbb{N}$.

This proves (2.15) and the analogous assertion in Theorem 2.5.

###  3.11. Convergence in Capacity

In view of Sections 3.2-3.3, to complete the proof of Theorems 2.1 and 2.5, we
only need to establish that

(3.92)
$\frac{1}{2n}\log|(f-N(M_{n}))(z)|\,\stackrel{{\scriptstyle\scriptsize\mathrm{cap}}}{{\rightarrow}}\,g\big{(}\mu_{\mathbb{D},K_{f}},\mathbb{D};z\big{)}-\frac{1}{\mathrm{cap}_{\mathbb{D}}(K_{f})}\quad\text{as}\quad
n\to\infty,\;\;z\in\mathbb{D}\setminus K_{f}.$

Let $K\subset\mathbb{D}\setminus K_{f}$ be compact. Further, let $V$ be an
open neighborhood of $p^{-1}(K_{f})$ on ${\mathcal{R}}$ whose closure is
disjoint from $K$, and $U$ an open subset of ${\mathcal{R}}$ containing $K$
whose closure is disjoint from $\overline{V}\cup\mathcal{T}$. For convenience,
we also assume that $V=p^{-1}(p(V))$. Recall from (3.9)–(3.11) the relation

$\frac{1}{2n}\log|(f-N(M_{n})\circ
p)(z)|=\frac{1}{2}\big{(}h_{n,n}(z)+g(\mu_{n},\mathbb{D};p(z))-g(\nu_{n},\Omega_{n};z)\big{)},\quad
z\in\Omega_{n}.$

We have established in the previous section that the functions $h_{n,n}$
converge to $-h_{{\mathcal{R}}}=-h^{\prime}$ locally uniformly in
${\mathcal{R}}\setminus E_{f}$ and therefore on $K$, see Lemma 3.1 (note that
$\mathcal{N}^{\prime}$ may now be replaced by $\mathbb{N}$ by the last remark
in Section 3.10). Since

$\frac{1}{2}\big{(}-h_{{\mathcal{R}}}(z)+g(\mu_{\mathbb{D},K_{f}},\mathbb{D};p(z))-g(\nu,{\mathcal{R}};z)\big{)}=\frac{1}{2}ler(z)=g(\mu_{\mathbb{D},K_{f}},\mathbb{D};p(z))-\frac{1}{\mathrm{cap}_{\mathbb{D}}(K_{f})}$

for $z\in G_{\mathsf{max}}$ by (3.46) and (3.91), to obtain (3.92) we only
need to establish that for any $a>0$

(3.93)
$\begin{cases}\displaystyle\lim_{n\to\infty}\mathrm{cap}_{\mathbb{D}}\big{(}\\{z\in
p(K):|g(\mu_{n},\mathbb{D};z)-g(\mu_{\mathbb{D},K_{f}},\mathbb{D};z)|>a\\}\big{)}=0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\lim_{n\to\infty}\mathrm{cap}_{\mathbb{D}}\big{(}p\\{z\in
K:|g(\nu_{n},\Omega_{n};z)-g(\nu,{\mathcal{R}};z)|>a\\}\big{)}=0.\end{cases}$

Write $\mu_{n}=\mu_{n,1}+\mu_{n,2}$, where
$\mu_{n,1}:=\mu_{n\mathcal{b}p(V)}$. Notice that
$\mu_{n,1}\stackrel{{\scriptstyle*}}{{\to}}\mu_{\mathbb{D},K_{f}}$ and
therefore the differences
$g(\mu_{n,1},\mathbb{D};\cdot)-g(\mu_{\mathbb{D},K_{f}},\mathbb{D};\cdot)$
converge uniformly to zero on $K$. Recall that the Green potential of a
measure supported in $\mathbb{D}$ can be written as a difference of the
logarithmic potentials of that measure and of its balayage onto $\mathbb{T}$,
see (A.29). In [6, Lemma 21], it was shown that when compactly supported
measures converge weak∗ to the zero measure then their logarithmic potentials
converge to zero in logarithmic capacity on $\mathbb{C}$, see (A.6) for the
definition of logarithmic capacity. Thus, the first relation in (3.93) holds
but in logarithmic capacity. However, since $t\mapsto g_{D}(t,z)+\log|z-t|$ is
uniformly bounded for $z$ on compact subsets of $\mathbb{D}$, the convergence
in logarithmic and Greenian capacities in $\mathbb{D}$ are equivalent. This
finishes the proof of the first limit in (3.93).

The proof of the second limit in (3.93) proceeds similarly, but requires a
more detailed analysis. As we have shown in the previous subsection, the
measures $\nu_{n\mathcal{b}\Omega_{n}}$ converge vaguely to $\nu^{*}$ on
${\mathcal{R}}$, where $\nu^{*}$ stands for the left-hand side of (3.90).
Hence, one has the limiting relation

(3.94)
$\nu_{n,1}:=\nu_{n\mathcal{b}\overline{V}\cap\Omega_{n}}\overset{w*}{\to}\nu^{*}$

on ${\mathcal{R}}$. The functions $g_{\mathcal{R}}(z,w)-g_{\Omega_{n}}(z,w)$
uniformly converge to zero for $w\in V$ and $z\in K$, see (3.56) (we extend
$g_{\Omega_{n}}(z,w)$ by zero to ${\mathcal{R}}\setminus\Omega_{n}$). Thus, as
the measures $\nu_{n,1}$ have uniformly bounded masses (for they converge
weak∗ to a finite measure), we get that

(3.95)
$\lim_{n\to\infty}\big{(}g(\nu_{n,1},{\mathcal{R}};z)-g(\nu_{n,1},\Omega_{n};z)\big{)}=0$

uniformly on $K$. Moreover, since the $g(\nu_{n,1},{\mathcal{R}};\cdot)$ are
positive harmonic functions on $U$ for all $n$ large, they converge uniformly
to $g(\nu^{*},{\mathcal{R}};\cdot)$ on $K$ (they converge pointwise on $U$ by
(3.94) and hence uniformly on $K$ by Harnack’s theorem). That is, we get from
(3.95) that

(3.96) $\lim_{n\to\infty}g(\nu_{n,1},\Omega_{n};z)=g(\nu^{*},{\mathcal{R}},z)$

uniformly on $K$. Next, let $\nu_{n,2}:=\nu_{n\mathcal{b}U}$. As the $\nu_{n}$
converge vaguely to $\nu^{*}$ and
$\overline{U}\cap(\overline{V}\cup\mathcal{T})=\varnothing$, the measures
$\nu_{n,2}$ converge strongly to the zero measure. Subsequently, the measures
$p_{*}(\nu_{n,2})$ strongly converge to zero. As before, it entails that the
potentials $g(p_{*}(\nu_{n,2}),\mathbb{D};p(\cdot))$ as well as
$g(\nu_{n,2},{\mathcal{R}};\cdot)$ converge to zero in capacity on $K$, by
(A.32). As the latter potentials majorize $g(\nu_{n,2},\Omega_{n};\cdot)$,
these in turn converge to zero in capacity on $K$. Finally, define
$\nu_{n,3}:=\nu_{n}-\nu_{n,1}-\nu_{n,2}$. The potentials
$g(\nu_{n,3},\Omega_{n};\cdot)$ form a sequence of positive harmonic functions
in $U$. By Harnack’s theorem there exists a subsequence of indices, say
$\mathcal{N}_{0}$, along which these potentials converge locally uniformly on
$U$ to some $h_{0}$ which is a non-negative harmonic function or $+\infty$.
Now, we can initiate the proof of Theorem 2.5 in Section 3.5 with
$\mathcal{N}_{0}$ instead of $\mathbb{N}$. Then, it would follow from (3.90)
and (3.96) that the function $h^{\prime\prime}$ from Lemma 3.5 must coincide
with $h_{0}$ in $U$. Proceeding with the remainder of the proof we again
inevitably arrive at the conclusion that $h^{\prime\prime}\equiv 0$. Hence,
the potentials $g(\nu_{n,3},\Omega_{n};\cdot)$ converge to $0$ locally
uniformly on $U$ and thus uniformly on $K$. This finishes the proof of the
second limit in (3.93) and, respectively, of (3.92); that is to say, of
Theorem 2.5.

Finally, we need to account for Remark 2.2, in which we claimed that for any
sequence $\\{M_{n}\\}$ of $n$-th root optimal meromorphic approximants to
$f\in\mathcal{F}(\overline{\mathbb{C}}\setminus D)$ and any $\epsilon>0$, it
holds that

(3.97) $\lim_{n\to\infty}\mathrm{cap}\left\\{z\in
K:\frac{1}{2n}\log|(f-M_{n})(z)|-g\big{(}\mu_{D,K_{f}},D;z\big{)}+\frac{1}{\mathrm{cap}_{D}(K_{f})}>\epsilon\right\\}=0$

for every compact set $K\subset\overline{D}\setminus K_{f}$; moreover, if
$|f-M_{n}|^{1/n}$ is asymptotically constant on $T$, i.e., if

(3.98)
$\lim_{n\to\infty}\inf_{z\in\mathcal{T}}|(f-M_{n})(z)|^{1/n}=\lim_{n\to\infty}\sup_{z\in\mathcal{T}}|(f-M_{n})(z)|^{1/n}=\exp\left\\{-\frac{2}{\mathrm{cap}_{D}(K_{f})}\right\\},$

then the limit in (3.97) is supplemented by

(3.99) $\lim_{n\to\infty}\mathrm{cap}\left\\{z\in
K:\frac{1}{2n}\log|(f-M_{n})(z)|-g\big{(}\mu_{D,K_{f}},D;z\big{)}+\frac{1}{\mathrm{cap}_{D}(K_{f})}<-\epsilon\right\\}=0.$

Together, (3.97) and (3.99) establish a full analog of (3.92) but in
logarithmic rather than Greenian capacity, and not just in $D\setminus K_{f}$
but in $\overline{D}\setminus K_{f}$. Notice also that we work directly on the
domain $D$ here, because logarithmic capacity is not preserved by conformal
maps.

Since convergence in logarithmic and Greenian capacities are equivalent on
compact subsets of $D\setminus K_{f}$, we only need to establish (3.97) and
(3.99) for a compact set $K_{\eta}$ of the form

$K_{\eta}:=\big{\\{}z\in\overline{D}:1-\eta\leq\big{|}\phi^{-1}(z)\big{|}\leq
1\big{\\}}$

for some $\eta>0$ sufficiently small, where $\phi:\mathbb{D}\to D$ is a
conformal map. Let $K_{\eta}^{\prime}=\overline{K_{2\eta}\setminus K_{\eta}}$.
We claim that for all $n$ large enough there exists
$r_{n}\in(1-4\eta,1-2\eta)$ such that

(3.100)
$\left|\frac{1}{2n}\log|(f-M_{n})(z)|-g\big{(}\mu_{D,K_{f}},D;z\big{)}+\frac{1}{\mathrm{cap}_{D}(K_{f})}\right|\leq\epsilon,$

for all $z\in T_{r_{n}}$, where $T_{r}:=\phi(\mathbb{T}_{r})$. Indeed, let
$E_{n,2\eta}$ be the set of all $z\in K_{2\eta}^{\prime}$ for which the above
inequality is false. As we have already shown, (3.92) holds for the $n$-th
root optimal approximants $M_{n}$, both in Greenian and logarithmic capacities
on compact subsets of $D\setminus K_{f}$, in particular, on
$K_{2\eta}^{\prime}$. Hence, $\mathrm{cap}(E_{n,2\eta})\to 0$ as $n\to\infty$.
On the other hand, if our claim were false, then $E_{n,2\eta}\cap
T_{r}\neq\varnothing$ for every $r\in[1-4\eta,1-2\eta]$. In this case it must
hold that

$\mathrm{cap}(E_{n,\eta})\geq\frac{\mathrm{cap}([1-4\eta,1-2\eta])}{\|(\phi^{-1})^{\prime}\|_{K_{\eta}^{\prime}}}=\frac{\eta}{2\|(\phi^{-1})^{\prime}\|_{K_{\eta}^{\prime}}}$

because contractive maps, in particular
$\phi^{-1}_{\mathcal{b}K^{\prime}_{\eta}}/\|(\phi^{-1})^{\prime}\|_{K_{\eta}^{\prime}}$
followed by the circular projection, do not increase logarithmic capacity [53,
Theorem 5.3.1]. The above contradiction proves our claim. Clearly, $T_{r_{n}}$
is a Jordan curve homotopic to $T$ in $D\setminus K_{f}$ for all $\eta$ small
enough. By adding the same constant to $f$ and $M_{n}$ we can assume that $f$
is not vanishing on $K_{2\eta}^{\prime}$. Hence, we deduce from (3.100) that
$|f-M_{n}|<|f|$ on $T_{r_{n}}$ for all $n$ large enough. Besides, the same
inequality certainly holds on $T$ and therefore, by Rouché’s theorem, the
number of zeros of $f-M_{n}$ in $K_{2\eta}$ is less that the number of its
poles plus the degree of $f$ on $T\cup T_{r_{n}}$ (which is bounded by a
constant independently of $n$). It now follows from Theorem 2.5 that $f-M_{n}$
has little $o(1)$ poles and zeros in $K_{2\eta}$ as $n\to\infty$. Denote by
$b_{n,\eta}^{pole}$ and $b_{n,\eta}^{zero}$ the Blaschke products in $D$
vanishing at the poles and zeros of $f-M_{n}$ in $K_{2\eta}$, respectively. As
Green potentials of measures converging strongly to zero converge to zero in
capacity in $D$, the functions

$e_{n}(z):=\frac{1}{2n}\log|(f-M_{n})(z)b_{n,\eta}^{pole}(z)/b_{n,\eta}^{zero}(z)|-g(\mu_{D,K_{f}},D;z)+\frac{1}{\mathrm{cap}_{D}(K_{f})}$

still converge to zero in capacity in $D\setminus K_{f}$. Hence, similarly to
(3.100), we can find $r_{n}^{*}\in(1-2\eta,1-\eta)$ such that
$|e_{n}(z)|<\epsilon$ for all $z\in T_{r_{n}^{*}}$. As $\limsup_{n}\sup_{z\in
T}e_{n}(z)\leq 0$ by the $n$-th root optimality of $e_{n}$ and
$\lim_{n}\|e_{n}\|_{T}=0$ if in addition (3.98) holds (asymptotic circularity
of the error), the desired conclusion (3.97) and (3.99) come out by applying
the maximum principle for the harmonic function to $e_{n}$ on $K_{\eta}$,
along with the minimum principle if (3.98) holds.

##  4\. Proof of Theorem 2.6

Since the considerations of Sections 3.2 and 3.3 still apply, we can assume
that $D=\mathbb{D}$, i.e., that $f$ is analytic in
$\overline{\mathbb{C}}\setminus E,$ where $E\subset\mathbb{D}$ is closed
polar, and we may replace $\\{M_{n}\\}$ by the sequence $\\{N(M_{n})\\}$ of
its Nehari modifications. Write $N(M_{n})=h_{n}/b_{n}$, where
$h_{n}\in\mathcal{A}(\mathbb{D})$ and $b_{n}$ is a Blaschke product vanishing
at the poles of $N(M_{n})$ according to their multiplicities. That is,
$b_{n}=q_{n}/\widetilde{q}_{n}$, where $q_{n}\in\mathcal{M}_{n}(\mathbb{D})$
is a polynomial such that $q_{n}M_{n}\in\mathcal{A}(\mathbb{D})$ and
$\widetilde{q}_{n}(z):=z^{n}\overline{q_{n}(1/\bar{z})}$ is the reciprocal
polynomial. We have that

$|f(z)-(h_{n}/b_{n})(z)|=|(b_{n}f)(z)-h_{n}(z)|,\quad z\in\mathbb{T},$

since Blaschke products are unimodular on the unit circle. Thus, $h_{n}$ is in
fact the best Nehari approximant of $b_{n}f$ in $\mathcal{A}(\mathbb{D})$.
Recall that the above expressions converge to zero faster than geometrically
to zero by the very choice of $\\{M_{n}\\}$.

Our goal is to show that $h_{n}/b_{n}$ converge in logarithmic capacity to $f$
in $\overline{\mathbb{D}}\setminus E$ at faster than geometric rate. That is,
we fix a compact set $K\subset\overline{\mathbb{D}}$ disjoint from $E$ and we
will prove that

(4.1) $\lim_{n\to\infty}\mathrm{cap}\big{(}\\{z\in
K:|e_{n}(z)|>a^{n}\\}\big{)}=0\quad\text{for any}\quad a>0,$

where $e_{n}(z):=f(z)-(h_{n}/b_{n})(z)$ is the approximation error. This will
establish convergence in logarithmic capacity on compact subsets of
$\overline{\mathbb{D}}\setminus E$. Subsequently, as we pointed out in Section
3.11, (4.1) yields an analogous claim for the Greenian capacity on any compact
subset of $\mathbb{D}\setminus E$.

Since $\mathrm{cap}(E)=0$, it follows from [53, Theorems 5.5.2 & 5.5.4] that
for each $\eta>0$ there is $k\in\mathbb{N}$ and $p_{k}\in\mathcal{M}_{k}(E)$
(we can take $p_{k}$ to be the $k$-th Fekete polynomial for $E$) such that

(4.2) $E\subset
L_{\eta}:=\big{\\{}\zeta\in\mathbb{C}:|p_{k}(\zeta)|<\eta^{k}\big{\\}}.$

Pick $\eta<\min\\{\mathrm{dist}(K,E)\,,\,1\\}$ to be adjusted later. Of
necessity $K\cap L_{\eta}=\varnothing$, because
$|p_{k}(z)|\geq\mathrm{dist}(K,E)^{k}$ for $z\in K$. Let $\gamma\subset
L_{\eta}$ be a system of closed curves encompassing each point of $E$ exactly
once, and such that $\mathrm{dist}(\gamma,E)<\mathrm{dist}(\mathbb{T},E)/4$.
Then, one has that

(4.3) $f(z)=\int_{\gamma}\frac{f(\xi)}{z-\xi}\frac{d\xi}{2\pi\mathrm{i}},\quad
z\in\overline{\mathbb{C}}\setminus\overline{\mathrm{int}\,\gamma},$

by the Cauchy formula, where $\mathrm{int}\,\gamma$ is the union of the
bounded components of the complement of $\gamma$. Since $\mathbb{P}_{-}$
evaluated at $|z|>1$ coincides with the Cauchy projection having kernel
$(2\pi\mathrm{i}(z-\zeta))^{-1}d\zeta$ on $\mathbb{T}$, the Hankel operator
$\Gamma_{b_{n}f}$ acts on $v\in H^{2}$ by

(4.4)
$\Gamma_{b_{n}f}(v)(z)=\int_{\mathbb{T}}\frac{b_{n}(\zeta)f(\zeta)v(\zeta)}{z-\zeta}\frac{d\zeta}{2\pi\mathrm{i}},\quad|z|>1.$

Inserting (4.3) into (4.4) yields by Fubini’s theorem and the Cauchy formula
for $H^{2}$-functions that

(4.5)
$\Gamma_{b_{n}f}(v)(z)=\int_{\gamma}\frac{b_{n}(\xi)f(\xi)v(\xi)}{z-\xi}\frac{d\xi}{2\pi\mathrm{i}},\quad
z\in\overline{\mathbb{C}}\setminus\overline{\mathrm{int}\,\gamma}.$

Observe that (4.5), initially proven for $|z|>1$, actually defines an analytic
extension to the exterior of $\gamma$ of
$\Gamma_{b_{n}f}(v)=\mathbb{P}_{-}(b_{n}fv)\in H_{-}^{2}$ (and therefore an
extension to $\overline{\mathbb{C}}\setminus E$ since $\gamma$ could be taken
arbitrary close to $E$). Next, recall that a first singular vector of the
Hankel operator $\Gamma_{b_{n}f}$ is an element $v_{0}\in H^{2}$ of unit norm
that maximizes $\|\Gamma_{b_{n}f}(v)\|$ over all $v\in H^{2}$ with
$\|v\|_{2}=1$, and that it always can be chosen to be outer333An outer
function $w\in H^{2}$ is of the form
$w(z)=\alpha\exp\Big{\\{}\int_{\mathbb{T}}\frac{\xi+z}{\xi-z}\log|w(\xi)|\frac{|d\xi|}{2\pi}\Big{\\}}$,
with $w_{\mathcal{b}\mathbb{T}}\in L^{2}(\mathbb{T})$ and $|\alpha|=1$., see
for instance [5, p. 62]. Then, we get from (4.5) and (2.26) (applied with
$n=0$ and $f$ replaced by $fb_{n}$) that

(4.6)
$(e_{n}b_{n})(z)=(b_{n}f-h_{n})(z)=\frac{1}{v_{0}(z)}\int_{\gamma}\frac{b_{n}(\xi)f(\xi)v_{0}(\xi)}{z-\xi}\frac{d\xi}{2\pi\mathrm{i}},\quad
z\in\mathbb{D}\setminus\overline{\mathrm{int}\,\gamma}.$

Note that the right-hand side of (4.6) is analytic in
$\mathbb{D}\setminus\overline{\mathrm{int}\,\gamma}$, since $v_{0}$ is outer
and thus has no zeros in $\mathbb{D}$. Let $B_{k}:=p_{k}/\widetilde{p}_{k}$.
Similarly to (4.3)–(4.6), it holds that

(4.7)
$\mathbb{P}_{-}\left(e_{n}b_{n}B_{k}^{\ell}\right)(z)=\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{b_{n}(\xi)f(\xi)B_{k}^{\ell}(\xi)}{z-\xi}d\xi,\quad
z\in\overline{\mathbb{D}}\setminus\overline{\mathrm{int}\,\gamma},$

where $\ell\in\mathbb{N}$ is such that $\ell k\leq n<(\ell+1)k$ and the right-
hand side again defines an analytic extension of the left-hand side into the
exterior of $\gamma$. Indeed, we can express the left-hand side of (4.7) for
$|z|>1$ as the Cauchy integral of $e_{n}b_{n}B_{k}^{\ell}$ on $\mathbb{T}$
like we did in (4.4) for $\mathbb{P}_{-}(b_{n}fv)$. Since $e_{n}b_{n}$ is
analytic across $\mathbb{T}$ and $B_{k}\in\mathcal{H}(\overline{\mathbb{D}})$,
we then deform the contour of integration into a circle of radius slightly
smaller than $1$, which can be done without changing the value of the integral
by Cauchy’s theorem. Subsequently, we insert (4.6) in this integral and use
Fubini’s theorem and the residue formula as before to get (4.7).

Recall that $\mathrm{dist}(\mathbb{T},E)\geq\mathrm{dist}(K,E)>\eta$ by
construction. Observe also that $|p_{k}|\leq\eta^{k}$ on $\gamma$ by (4.2) and
that $|\widetilde{p}_{k}|\geq\mathrm{dist}(\mathbb{T},E)^{k}$ in $\mathbb{D}$.
Since $|b_{n}|\leq 1$, we get from our choice of $\ell$ that

(4.8)
$\left|\mathbb{P}_{-}\left(e_{n}b_{n}B_{k}^{\ell}\right)(z)\right|\leq\left(\frac{\eta}{\mathrm{dist}(\mathbb{T},E)}\right)^{n-k}\frac{|\gamma|\|f\|_{\gamma}}{\mathrm{dist}(z,\gamma)},\quad
z\in\overline{\mathbb{D}}\setminus\overline{\mathrm{int}\,\gamma},$

where $|\gamma|$ stands for the arclength of $\gamma$. In another connection,
it follows from [49, Lemma] that for any $\varepsilon\in(0,1/3)$ there exists
$W_{n}\subset\mathbb{D}$ such that $\mathrm{cap}(W_{n})\leq 3\varepsilon$ and

$|p_{k}^{\ell}(\zeta)q_{n}(\zeta)|>\varepsilon^{n+\ell
k}\|p_{k}^{\ell}q_{n}\|_{\mathbb{T}},\quad\zeta\in\mathbb{D}\setminus W_{n}.$

As
$|\widetilde{p}_{k}^{\ell}\widetilde{q}_{n}|(z)\leq\|\widetilde{p}_{k}^{\ell}\widetilde{q}_{n}\|_{\mathbb{T}}=\|p_{k}^{\ell}q_{n}\|_{\mathbb{T}}$
for $z\in\mathbb{D}$ by the maximum principle and the definition of the
reciprocal polynomial, we get that
$|(b_{n}B_{k}^{\ell})(z)|\geq\varepsilon^{n+k\ell}$ for
$z\in\mathbb{D}\setminus W_{n}$. Since $\mathbb{P}_{+}+\mathbb{P}_{-}$ is the
identity operator, we can use the analytic continuation provided by (4.7) to
write

(4.9)
$e_{n}(z)=\frac{\mathbb{P}_{+}(e_{n}b_{n}B_{k}^{\ell})(z)+\mathbb{P}_{-}(e_{n}b_{n}B_{k}^{\ell})(z)}{b_{n}(z)B_{k}^{\ell}(z)},\quad
z\in\overline{\mathbb{D}}\setminus\overline{\mathrm{int}\,\gamma}.$

The estimate $|(b_{n}B_{k}^{\ell})(z)|\geq\varepsilon^{n+k\ell}$ for
$z\in\mathbb{D}\setminus W_{n}$, (4.8), and the definition of $\ell$ give us
that

(4.10)
$\left|\frac{\mathbb{P}_{-}(e_{n}b_{n}B_{k}^{\ell})(z)}{b_{n}(z)B_{k}^{\ell}(z)}\right|\leq\frac{1}{\varepsilon^{2n}}\left(\frac{\eta}{\mathrm{dist}(\mathbb{T},E)}\right)^{n-k}\frac{|\gamma|\|f\|_{\gamma}}{\mathrm{dist}(z,\gamma)},\quad
z\in\overline{\mathbb{D}}\setminus(\overline{\mathrm{int}\,\gamma}\cup
W_{n}).$

Now, given $0<\varepsilon<1$ and $0<a<1$, choose $\eta$ in (4.2) so that
$0<\eta<a\varepsilon^{2}\mathrm{dist}(\mathbb{T},E)$. Choice of $\eta$ of
course fixes $k$ in (4.2). Then, since $K$ lies exterior to $\gamma$ because
$K\cap L_{\eta}=\varnothing$, we get from (4.10) that there exists a natural
number $n_{0}=n_{0}(f,K,\varepsilon,\eta,\gamma)$ for which

(4.11)
$\left|\frac{\mathbb{P}_{-}(e_{n}b_{n}B_{k}^{\ell})(z)}{b_{n}(z)B_{k}^{\ell}(z)}\right|<a^{n},\quad
n\geq n_{0},\quad z\in K\setminus W_{n}.$

Next, as $\|e_{n}b_{n}B_{k}^{\ell}\|_{\mathbb{T}}=\|e_{n}\|_{\mathbb{T}}\to 0$
faster than geometrically with $n$ by hypothesis, (4.8) and the triangle
inequality yield that

(4.12)
$\left\|\mathbb{P}_{+}(e_{n}b_{n}B_{k}^{\ell})\right\|_{\mathbb{T}}=\left\|e_{n}b_{n}B_{k}^{\ell}-\mathbb{P}_{-}(e_{n}b_{n}B_{k}^{\ell})\right\|_{\mathbb{T}}\leq
C\left(\frac{\eta}{\mathrm{dist}(\mathbb{T},E)}\right)^{n},\quad n\geq
n^{\prime}_{0},$

for some constant $C=C(f,E,\gamma)$ and some $n^{\prime}_{0}$ depending on $C$
and the speed of approximation of $f$ by $h_{n}/b_{n}$. Subsequently, as in
(4.11), we get from the estimate
$|(b_{n}B_{k}^{\ell})(z)|\geq\varepsilon^{n+k\ell}$ for
$z\in\mathbb{D}\setminus W_{n}$, (4.12), and the maximum modulus principle
that

(4.13)
$\left|\frac{\mathbb{P}_{+}(e_{n}b_{n}B_{k}^{\ell})(z)}{b_{n}(z)B_{k}^{\ell}(z)}\right|<a^{n},\quad
n\geq n_{0}^{\prime\prime},\quad z\in K\setminus W_{n},$

for some natural number
$n_{0}^{\prime\prime}=n_{0}^{\prime\prime}(f,K,\varepsilon,\eta,\gamma)$.
Because $\varepsilon$ and $a$ can be arbitrarily small and
$\mathrm{cap}(W_{n})\leq 3\varepsilon$, (4.1) now follows from (4.9), (4.11)
and (4.13).

Having proven that $M_{n}\overset{\mathrm{cap}}{\to}f$ in $D\setminus E$ at
faster than geometric rate whenever it is a sequence of $n$-th root optimal
meromorphic approximants to $f$, we turn to the construction of rational
functions $R_{k_{n}}\in\mathcal{R}_{k_{n}}(D)$ such that the poles of
$R_{k_{n}}$ are among the poles of $M_{n}$ lying in $V$ and (2.18) holds,
where $V$ any open set such that $E\subset V\subset\overline{V}\subset D$. Let
$B\supset E$ be a closed set contained in $V$ which is regular for the
Dirichlet problem, see Section A.7. Such a $B$ is easily constructed as a
sublevel set, for some small regular value, of a smooth non-negative function
whose zero set is $E$, see, for example, discussion after (3.8). Then the
Green equilibrium potential $G(z):=g(\mu_{D,B},D;z)$ is harmonic in
$D\setminus B$, continuous on $D$, strictly less than the constant
$1/\mathrm{cap}_{D}(B)$ on $D\setminus B$ and equal to that constant on $B$,
see (2.12) or Section A.4 for a more detailed discussion. Since
$\partial_{z}G(z)$ is holomorphic in $D\setminus B$, the critical points of
$G$ are isolated and cannot accumulate in $D\setminus B$, so we can find an
interval $[t_{1},t_{2}]\subset(0,1/\mathrm{cap}_{D}(B))$ that is free of
critical values and such that $G^{-1}([t_{1},t_{2}])\subset V$. For any
$t\in[t_{1},t_{2}]$, $\gamma(t):=G^{-1}(t)$ is a 1-dimensional compact
manifold, i.e., a finite union of disjoint real analytic closed curves
$\gamma_{1,t},\ldots,\gamma_{N,t}$, none of which lies interior to another (by
the maximum principle), and such that
$B\subset\mathrm{int}\,\gamma(t)\subset\overline{\mathrm{int}\,\gamma(t)}\subset
V$. Note that $N$ is independent of $t\in[t_{1},t_{2}]$ since any such $t$ is
a regular value; note also that the total length
$|\gamma(t)|=\sum_{j=1}^{N}|\gamma_{j,t}|$ is bounded above independently of
$t$, say by a constant $L$, because the gradient $\nabla G$ is normal to
$\gamma_{j,t}$ at its every point and therefore the divergence formula implies
for any $t\in[t_{1},t_{2}]$:

$t|\gamma(t)|\min_{z\in G^{-1}(t)}\|\nabla G(z)\|\leq
t_{2}|\gamma(t_{2})|\max_{z\in G^{-1}(t_{2})}\|\nabla
G(z)\|-\int_{G^{-1}([t,t_{2}])}\|\nabla G\|^{2}dxdy.$

Pick $a>0$, set $K:=\max\\{\|\nabla G(z)\|:z\in G^{-1}([t_{1},t_{2}])\\}$ and
let $n_{a}\in\mathbb{N}$ be so large that

(4.14) $\mathrm{cap}\bigl{(}\\{z\in
G^{-1}([t_{1},t_{2}]):|f(z)-M_{n}(z)|>a^{n}\\}\bigr{)}<\frac{t_{2}-t_{1}}{4K},\quad
n\geq n_{a}.$

Such a $n_{a}$ exists by the first part of the proof. Let $A_{n}$ be the set
whose capacity is estimated in (4.14). Assume for the moment that for each
$t\in[t_{1},t_{2}]$ there exists $z\in G^{-1}(t)\cap A_{n}$. Then the image of
$A_{n}$ under $G/K:G^{-1}([t_{1},t_{2}])\to\mathbb{R}$ is equal to the
interval $[t_{1}/K,t_{2}/K]$ whose capacity is $(t_{2}-t_{1})/(4K)$. However,
since contractive maps do not increase the logarithmic capacity [53, Theorem
5.3.1], the capacity of $G(A_{n})/K$ should be strictly smaller than
$(t_{2}-t_{1})/(4K)$ by (4.14). Hence, for each $n\geq n_{a}$ there is
$t_{n}\in[t_{1},t_{2}]$ for which

(4.15)
$\left|\int_{\gamma(t_{n})}\frac{f(\xi)-M_{n}(\xi)}{z-\xi}\frac{d\xi}{2\pi\mathrm{i}}\right|\leq\frac{La^{n}}{2\pi\mathrm{dist}(T,\overline{V})},\quad
z\in T.$

Pick a positive sequence $\\{a_{k}\\}$ converging to $0$ and, without loss of
generality, arrange things so that $n_{a_{k}}<n_{a_{k+1}}$. Define

$J_{n}(z):=\int_{\gamma(t_{n_{a_{k}}})}\frac{M_{n}(\xi)}{z-\xi}\frac{d\xi}{2\pi\mathrm{i}},\quad
z\in D\setminus\overline{V},\quad n_{a_{k}}\leq n<n_{a_{k+1}}.$

Clearly, $J_{n}$ is a rational function retaining the singular part of $M_{n}$
inside the system of arcs $\gamma(t_{n_{a_{k}}})$, and it is of type
$(k_{n}-1,k_{n})$ where $k_{n}\leq n$ is the number of the poles of $M_{n}$
inside this system of arcs, counting multiplicities. If we put
$R_{k_{n}}:=J_{n}$, then since $a_{k}\to 0$ we get from (4.15) and the Cauchy
formula that (2.18) holds, as desired.

##  Appendix A Potential Theory on a Riemann Surface

Even though the proof of Theorem 2.5 in Sections 3.3–3.10 was carried out for
$D=\mathbb{D}$, this appendix is written for a general Jordan domain $D$ since
specializing $D$ to the unit disk would only shorten the proofs of Lemmas A.5
and A.10 by a couple of paragraphs but otherwise would not lead to any further
simplifications.

###  A.1. Subharmonic Functions

Let $d$ be the differential and ∗ the conjugation operators on a connected
Riemann surface. The Laplacian $\Delta:=d{}^{*}d$ takes smooth functions to
$2$-forms. If $U$ is an open subset of the surface and $u:U\to\mathbb{R}$ a
locally integrable function (against the area-form ${}^{*}{\bf 1}$, where
${\bf 1}$ is the constant unit function, or equivalently against the Lebesgue
measure $(\mathrm{i}/2)dz\wedge d\bar{z}$ in any system of local coordinates
$z$, $\overline{z}$), the distributional Laplacian $\Delta u$ is the
$0$-current acting on a smooth compactly supported functions $\varphi$ on $U$
by $\int u\Delta\varphi$. When $\Delta u=0$, one says that $u$ is harmonic on
$U$, and such functions are in fact smooth (even real analytic) by Weyl’s
lemma [18, Theorem 24.9]. Subharmonic functions on $U$ are defined as upper-
semicontinuous functions $u:U\to[-\infty,\infty)$ such that, if $V$ is open in
$U$ and $h:V\to\mathbb{R}$ is harmonic, then $u-h$ is either constant or fails
to have a maximum in $V$. On open subsets of $\mathbb{C}$, this definition
coincides with the usual one; see [53, Definition 2.2.1 & Theorem 2.4.1]. A
superharmonic function is the negative of a subharmonic function. A difference
of two subharmonic functions is sometimes called a $\delta$-subharmonic
function.

Harmonicity and subharmonicity are local properties: $u$ is harmonic (resp.
subharmonic) on $U$ if and only if its restriction to every open subset is, or
equivalently if and only if $u\circ\varphi^{-1}$ is harmonic (resp.
subharmonic) on the open set $\varphi(V\cap U)\subset\mathbb{C}$ whenever
$(V,\varphi)$ is a local chart. Thus, standard facts regarding such functions
on open subsets of a Riemann surface follow from their planar counterparts,
using charts. In particular, the _integrability theorem_ [53, Theorem 2.5.1]
states that a subharmonic function which is not identically $-\infty$ is
locally integrable, and therefore it has a distributional Laplacian. Hence,
two subharmonic functions that coincide almost everywhere (with respect to
area measure) are in fact equal, for either they are both identically
$-\infty$ or they have the same distributional Laplacian, and so their
difference is harmonic; this is the _weak identity principle_. The following
is a variant of Harnack’s theorem [53, Theorem 1.3.10] and of [53, Theorem
2.4.6].

######  Harnack’s Theorem.

A sequence of harmonic functions on $U$ that is bounded below has a
subsequence that converges locally uniformly on $U$, either to $+\infty$ or to
a harmonic function. For an increasing sequence, convergence holds along the
full sequence. A decreasing sequence of subharmonic functions converges
pointwise to a subharmonic function.

A locally integrable function $u$ is subharmonic if and only if $\Delta u$ is
a Radon measure on the surface; that is, $\Delta u$ is a positive linear form
on continuous functions with compact support. Indeed, as this statement is
local, it reduces to its planar analog. The “only if” part follows from [53,
Section 3.7]. As to the “if” part, let $W\subset\mathbb{C}$ be open and $\nu$
be a finite positive Borel measure carried by $W$. The logarithmic potential
of $\nu$, i.e., $V^{\nu}(z):=\int\log|z-t|^{-1}d\nu(t)$, is superharmonic on
$\mathbb{C}$ with distributional Laplacian $-\nu$, so if $u$ is locally
integrable on $W$ with $\Delta u=\nu$ there, then
$h:=u+V^{\nu}_{\mathcal{b}W}$ is harmonic and therefore
$u=-V^{\nu}_{\mathcal{b}W}+h$ is subharmonic in $W$ (as in the main text, for
a set $E$ (that may require further qualification), a subscript $\mathcal{b}E$
indicates “restriction to $E$”).

###  A.2. Green Functions

Throughout, $\Omega$ will be a subdomain of some ambient algebraic Riemann
surface ${\mathcal{R}}_{*}$ such that $p(\Omega)$ is a bounded domain in
$\mathbb{C}$, where $p$ stands for the canonical projection; in particular the
results apply to $\Omega={\mathcal{R}}$ with $p(\Omega)=D$, see beginning of
Section 2.1. Since the lift to $\Omega$ of a positive non-constant
superharmonic function on $p(\Omega)$ is again positive, non-constant, and
superharmonic, $\Omega$ is hyperbolic and as such possesses Green functions
[15, Theorem IV.3.7]. Notice also that there exists a subdomain
$\Omega^{\prime}\subset{\mathcal{R}}_{*}$ with $p(\Omega^{\prime})$ bounded
such that $\overline{\Omega}\subset\Omega^{\prime}$, where an overline (as in
$\overline{\Omega}$) always denotes the closure in ${\mathcal{R}}_{*}$.

Recall that the Green function for $\Omega$ with pole at $w$, denoted by
$g_{\Omega}(\cdot,w)$, is the unique function that is harmonic and positive in
$\Omega\setminus\\{w\\}$ with a logarithmic singularity at $w$ and whose
largest harmonic minorant is identically zero. By a logarithmic singularity at
$w$, it is meant that if $(V,\varphi)$ is a coordinate chart on $\Omega$ such
that $w\in V$ and $\varphi(w)=0$, then
$g_{\Omega}(\varphi^{-1}(\cdot),w)+\log|\cdot|$ is harmonic on $\varphi(V)$.
Obviously $g_{\Omega}(\cdot,w)>0$ everywhere on $\Omega$, by the minimum
principle for harmonic functions. Recall also that if a superharmonic function
on $\Omega$ is not identically $+\infty$ and has a harmonic minorant, then it
has the largest one whose construction can be carried out as in the Euclidean
case [29, Theorem 4.3.5], because Poisson modifications can be performed
locally.

Clearly, $g_{\Omega}(\cdot,w)$ is superharmonic and $\Delta
g_{\Omega}(\cdot,w)=-\delta_{w}$, where $\delta_{w}$ is the Dirac mass at $w$.
Moreover, $g_{\Omega}$ is symmetric in that $g_{\Omega}(z,w)=g_{\Omega}(w,z)$
[15, Theorem IV.3.10]. Symmetry entails that $g_{\Omega}(z,w)$ is separately
harmonic in $z$ and $w$, and therefore jointly harmonic on
$\\{(z,w)\in\Omega\times\Omega:\,z\neq w\\}$ [36, p. 561]; in particular,
$(z,w)\mapsto g_{\Omega}(z,w)$ is continuous off the diagonal. Note that

(A.1) $g_{\Omega}(z,w)\leq g_{\Omega^{\prime}}(z,w),\quad
z,w\in\Omega\subseteq\Omega^{\prime},$

because $g_{\Omega}(\cdot,w)-g_{\Omega^{\prime}}(\cdot,w)$ is a harmonic
minorant of $g_{\Omega}(\cdot,w)$ and therefore must be non-positive. Thus, if
$F_{1}$, $F_{2}$ are relatively closed subsets of $\Omega$ with
$\overline{F}_{1}\cap\overline{F}_{2}=\varnothing$, we deduce that
$(z,w)\mapsto g_{\Omega}(z,w)$ is bounded on $F_{1}\times F_{2}$ because
$g_{\Omega^{\prime}}(z,w)$ is continuous on the compact set
$\overline{F}_{1}\times\overline{F}_{2}$ whenever
$\overline{\Omega}\subset\Omega^{\prime}$. We also remark that to each
$w\in\Omega$ there is an open set $V\ni w$ and a constant $C=C(V)$ such that

(A.2) $\int_{V}g_{\Omega}(z,w^{\prime})\,{}^{*}{\bf 1}(z)<C,\qquad
w^{\prime}\in V,$

a result that follows by uniformization from the corresponding fact on the
disk [29, Theorem 4.4.12]. Moreover, if a sequence of open sets $\Omega_{n}$
increases to $\Omega$ as $n\to\infty$, it follows from (A.1) that
$g_{\Omega}(\cdot,w)_{\mathcal{b}\Omega_{n}}-g_{\Omega_{n}}(\cdot,w)$ is a
decreasing sequence of positive harmonic functions that must converge locally
uniformly in $\Omega$, by Harnack’s theorem; as the limit is necessarily a
non-negative harmonic minorant of $g_{\Omega}(\cdot,w)$, it must be
identically zero.

###  A.3. Green Potentials

A Green potential in $\Omega$ is a non-negative superharmonic function whose
largest harmonic minorant is identically zero. Given $\sigma$, a Radon measure
in $\Omega$, let us put

(A.3) $g(\sigma,\Omega;z):=\int g_{\Omega}(z,w)d\sigma(w).$

This is a superharmonic function of $z\in\Omega$ which is either identically
$+\infty$, or locally integrable with distributional Laplacian $-\sigma$ by
Fubini’s theorem. If $g(\sigma,\Omega;\cdot)$ is not identically $+\infty$,
using the monotone convergence and Fubini’s theorem, the proof of [29, Lemma
4.3.6] carries over to integrals instead of sums to show that the largest
harmonic minorant of $g(\sigma,\Omega;\cdot)$ is the integral against
$d\sigma(w)$ of the largest harmonic minorants of the $g_{\Omega}(\cdot,w)$,
namely zero. Thus, $g(\sigma,\Omega;\cdot)$ is a Green potential. Conversely,
it follows from the Riesz representation theorem stated below that every Green
potential has the form (A.3). Notice that if $\sigma(\Omega)<\infty$, then
$g(\sigma,\Omega;\cdot)\not\equiv+\infty$, for if $V\subset\Omega$ is as in
(A.2) and $W$ is a nonempty open set with $\overline{W}\subset V$, then

$\int_{W}g(\sigma,\Omega;z){}^{*}{\bf
1}(z)<C\sigma(V)+C^{\prime}\sigma(\Omega\setminus V)\int_{W}{}^{*}{\bf 1}$

by Fubini’s theorem, where $C^{\prime}$ is an upper bound for
$g_{\Omega}(z,w)$ on $(\Omega\setminus V)\times\overline{W}$.

######  Riesz Representation Theorem.

Let $u\not\equiv+\infty$ be a superharmonic function on $\Omega$ that has a
harmonic minorant. Then $u=g(\sigma,\Omega;\cdot)+h$, where $h$ is the largest
harmonic minorant of $u$ and $\sigma:=-\Delta u$.

###### Proof.

Assume first that $\sigma(\Omega)<\infty$. Then,
$g(\sigma,\Omega;\cdot)\not\equiv+\infty$ and therefore
$h:=u-g(\sigma,\Omega;\cdot)$ is harmonic on $\Omega$. Clearly, $h$ is a
minorant of $u$. Since the largest harmonic minorant of
$g(\sigma,\Omega;\cdot)$ is zero, $h$ is the largest harmonic minorant of $u$.
If $\sigma(\Omega)=\infty$, pick $\Omega_{m}$ to be an increasing exhaustion
of $\Omega$ by relatively compact open sets. Put
$\sigma_{m}:=\sigma_{\mathcal{b}\Omega_{m}}$, which are finite measures on
$\Omega_{m}$ because $\sigma$ is a Radon measure. By what precedes,
$u_{\mathcal{b}\Omega_{m}}=g(\sigma_{m},\Omega_{m};\cdot)+h_{m}$, where
$h_{m}$ is the largest harmonic minorant of $u_{\mathcal{b}\Omega_{m}}$. As
the functions $g_{\Omega_{m}}(\cdot,w)$ increase locally uniformly to
$g_{\Omega}(\cdot,w)$ while $h_{m}$ decrease and are bounded below by any
harmonic minorant of $u$, we get by monotone convergence and Harnack’s theorem
that $u=g(\sigma,\Omega;\cdot)+h$, where $h$ is harmonic and necessarily
$g(\sigma,\Omega;\cdot)\not\equiv+\infty$. We now conclude the proof as in the
first case. ∎

This version of the Riesz representation theorem featuring the weak-Laplacian
may be compared to the more abstract formulation for Green spaces (of which
$\Omega$ is a special case) in [8, Section VI.7], that does not refer to the
Laplacian; see also the planar statement of [53, Theorem 4.5.4].

The previous considerations allow us to simplify in our case the notion of
admissibility of a measure given in [19, Section 1] and [20, Section I.3].
According to that definition, a measure $\sigma$ is _admissible_ if it is
integrable against continuous Green potentials with compactly supported
Laplacian. Fubini’s theorem immediately implies that
$g(\sigma,\Omega;\cdot)\not\equiv+\infty$ if $\sigma$ is admissible. In the
present Greenian context the condition
$g(\sigma,\Omega;\cdot)\not\equiv+\infty$ is also sufficient for (and
therefore equivalent to) admissibility of $\sigma$. Indeed, if $\nu$ is
compactly supported in $\Omega$ with continuous potential, let $V$ be an open
set such that $\mathrm{supp}\,\nu\subset V\subset\overline{V}\subset\Omega$.
By continuity, there exists $C$ such that $g(\nu,\Omega;z)\leq C$,
$z\in\Omega$. Since $g(\sigma,\Omega;\cdot)\not\equiv+\infty$, the same is
true for the potential of $\sigma_{\mathcal{b}\Omega\setminus V}$. As this
potential is harmonic in $V$ and is not equal identically to $+\infty$ there,
it is finite and locally bounded in $V$. Hence,
$g(\sigma_{\mathcal{b}\Omega\setminus V},\Omega;z)\leq C^{\prime}$,
$z\in\mathrm{supp}\,\nu$. Therefore, it follows from Fubini’s theorem that

$\int g(\nu,\Omega;z)d\sigma(z)\leq C\sigma(V)+C^{\prime}\nu(\Omega).$

###  A.4. Capacities

Given two Radon measures $\sigma_{1}$ and $\sigma_{2}$ on $\Omega$, we put

$(\sigma_{1},\sigma_{2})_{\Omega}:=\int
g(\sigma_{1},\Omega;z)d\sigma_{2}(z)=\int
g(\sigma_{2},\Omega;z)d\sigma_{1}(z),$

which is either a non-negative number or $+\infty$. The _Green energy_ of
$\sigma$ is defined as $I_{\Omega}(\sigma):=(\sigma,\sigma)_{\Omega}$. The
_Greenian capacity_ relative to $\Omega$ of a compact set $K\subset\Omega$ is
a non-negative number

(A.4)
$\mathrm{cap}_{\Omega}(K):=\frac{1}{\inf_{\mu\in\mathcal{P}(K)}I_{\Omega}(\mu)},$

where $\mathcal{P}(K)$ is the set of Borel probability measures on $K$. The
_Greenian capacity_ of a Borel set $B$ is given by

(A.5) $\mathrm{cap}_{\Omega}(B):=\sup_{K\subset
B}\mathrm{cap}_{\Omega}(K)=\inf_{U\supset B}\mathrm{cap}_{\Omega}(U),$

where the supremum is taken over all compact subsets of $B$, the infimum is
taken over all open sets containing $B$, and the equality is due to a theorem
by Choquet [8, Section VIII.4]. When $K$ is compact and
$\mathrm{cap}_{\Omega}(K)>0$, there exists a unique
$\mu_{\Omega,K}\in\mathcal{P}(K)$, called the _Green equilibrium measure_ of
$K$ in $\Omega$, to meet the infimum in (A.4). It is characterized by the fact
that for some constant $C$($=1/\mathrm{cap}_{\Omega}(K)$), the _Green
equilibrium potential_ $g(\mu_{\Omega,K},\Omega;z)$ satisfies
$g(\mu_{\Omega,K},\Omega;z)\leq C$ for $z\in\Omega$ with
$g(\mu_{\Omega,K},\Omega;z)=C$ for $z\in K\setminus E$, where $E$ has Greenian
capacity zero; this can be shown as in the Euclidean case [55, Theorems
II.5.11 & II.5.12].

In the case of an arbitrary set $B$, the infimum in (A.5) introduces the
_outer Greenian capacity_ of $B$ and will serve as a definition of
$\mathrm{cap}_{\Omega}(B)$. However, it may no longer match the supremum (the
latter defines the _inner Greenian capacity_ of $B$).

When $\Omega\subset\mathbb{C}$, another notion of capacity is instrumental in
this paper, namely the _logarithmic capacity_ defined for a compact set
$K\subset\mathbb{C}$ as

(A.6) $\mathrm{cap}(K):=\exp\left\\{-\inf_{\mu\in\mathcal{P}(K)}\int
V^{\mu}(z)d\mu(z)\right\\},$

where $V^{\mu}(z)$ is the logarithmic potential of $\mu$ defined earlier in
Section A.1. The logarithmic capacity of a Borel subset of $\mathbb{C}$ and
the outer logarithmic capacity of an arbitrary subset are defined via the same
process as for Greenian capacity, based on the analog of (A.5), see [53, 55]
(note that in [53], $\mathrm{cap}(E)$ denotes the inner logarithmic capacity
and potentials carry a sign opposite to the current one). If $K$ is compact
and $\mathrm{cap}(K)>0$, then there is a unique $\mu_{K}\in\mathcal{P}(K)$,
called the _logarithmic equilibrium measure_ of $K$, that realizes the infimum
in (A.6). It is characterized by the fact that for some constant
$C$($=-\log\mathrm{cap}(K)$), the _logarithmic equilibrium potential_
$V^{\mu_{K}}(z)$ is at most $C$ for $z\in K$ and in fact equal to $C$ on $K$
except possibly for a subset of logarithmic capacity zero.

Both the Greenian and logarithmic capacities are right continuous on compact
sets, meaning that

$\mathrm{cap}(\cap_{j=1}^{\infty}K_{j})=\lim_{n}\mathrm{cap}(\cap_{j=1}^{n}K_{j})\quad\text{and}\quad\mathrm{cap}_{\Omega}(\cap_{j=1}^{\infty}K_{j})=\lim_{n}\mathrm{cap}_{\Omega}(\cap_{j=1}^{n}K_{j})$

if the $K_{j}$ are compact; see [53, Theorem 5.1.3(a)] for the logarithmic
case, the Greenian one being argued the same way with an obvious adaptation of
[53, Lemma 3.3.3]. In addition, the (outer) Greenian and logarithmic
capacities are left continuous:

$\mathrm{cap}(\cup_{j=1}^{\infty}E_{j})=\lim_{n}\mathrm{cap}(\cup_{j=1}^{n}E_{j})\quad\text{and}\quad\mathrm{cap}_{\Omega}(\cup_{j=1}^{\infty}E_{j})=\lim_{n}\mathrm{cap}_{\Omega}(\cup_{j=1}^{n}E_{j});$

for the logarithmic capacity this follows from [53, Theorem 5.1.3(b)] combined
with Choquet’s theorem, and the Greenian case can be handled similarly,
compare to [8, Section VIII.4].

One form of the _domination principle_ for Green potentials says that if
$g(\sigma,\Omega;\cdot)\leq v$ on $\mathrm{supp}\,\sigma$ (the support of
$\sigma$) for some superharmonic function $v$, then
$g(\sigma,\Omega;\cdot)\leq v$ everywhere on $\Omega$; in fact, we shall state
a stronger version in Section A.6. It implies the _continuity theorem_ ,
saying that if the restriction of $g(\sigma,\Omega;\cdot)$ to
$\mathrm{supp}\,\sigma$ is continuous at $z_{0}\in\mathrm{supp}\,\sigma$ then
$g(\sigma,\Omega;\cdot)$ is continuous at $z_{0}$. When $\sigma$ is a positive
Borel measure with compact support such that $g(\sigma,\Omega;z)<+\infty$ for
$\sigma$-a.e. $z$, there is an increasing sequence of measures $\sigma_{k}$
supported on $\mathrm{supp}\,\sigma$, having continuous Green potentials and
converging to $\sigma$ in the strong (total variation) sense, such that
$g(\sigma_{k},\Omega;\cdot)$ increases pointwise to $g(\sigma,\Omega;\cdot)$
on $\Omega$. The proof is mutatis mutandis the same as for logarithmic
potentials [55, Lemma I.6.10], using the continuity theorem for Green
potentials. In particular, if $K$ is compact with
$\mathrm{cap}_{\Omega}(K)>0$, we find upon letting $\sigma$ be the Green
equilibrium distribution that there exist nonzero positive measures supported
on $K$ whose Green potentials are continuous.

When $\Omega\subset\mathbb{C}$, a subset of $\Omega$ has (outer) Greenian
capacity zero if and only if it has (outer) logarithmic capacity zero. Indeed,
it is enough to verify this claim on compact sets since capacity is left
continuous and a set of outer (Greenian or logarithmic) capacity zero is
contained in a Borel (even $G_{\delta}$) set of capacity zero, by definition.
Moreover, by the increasing character of $g_{\Omega}(z,w)$ with $\Omega$, we
may assume that $\Omega$ is simply connected. Then the result follows by
comparing the logarithmic kernel $\log(1/|z-w|)$ with the Green kernel
$g_{\Omega}(z,w)=\log|(1-\varphi(z)\overline{\varphi(w)})/(\varphi(z)-\varphi(w))|$,
where $\varphi$ is a conformal map $\Omega\to\mathbb{D}$. A property holding
pointwise except on a set of outer Greenian capacity zero (equivalently:
logarithmic capacity zero if $\Omega\subset\mathbb{C}$) is said to hold _quasi
everywhere_.

###  A.5. Fine Topology

A basis for the fine topology on $\Omega$ is given by all sets of the form

(A.7) $\cap_{i=1}^{m}\big{\\{}z\in B:v_{i}(z)<\alpha_{i}\big{\\}},$

where $B\subseteq\Omega$ is open, $v_{i}$ are superharmonic functions on $B$,
and $\alpha_{i}$ are constants. Consequently, all superharmonic functions
$\Omega\to(-\infty,+\infty]$ are finely continuous (equivalently: all
subharmonic functions $\Omega\to[-\infty,+\infty)$ are finely continuous), and
the fine topology is the coarsest with this property because, by the Riesz
representation theorem and the monotonicity of Green functions with respect to
the domain, each set of the form (A.7) contains one for which $v_{i}$ are
Green potentials. In particular, we may as well require in (A.7) that $v_{i}$
be defined and superharmonic on the whole of $\Omega$. Hence, the present
definition modeled after [29, Definition 6.5.1] (which deals with the
Euclidean case) is equivalent to [8, Definition I.1]. It is known that the
fine topology on $\Omega$ is locally connected [20, Corollary to Theorem
9.11], and that the fine connected components of a finely open set are finely
open [19, Corollary 1]. Moreover, Lipschitz curves are finely connected [19,
Theorem 7], so that Euclidean domains are fine domains as well.

As a general convention, we use the prefix “fine” to signify that a notion is
understood with respect to the fine topology. This way we distinguish the
latter from the classical, Euclidean topology (more precisely: the one induced
on $\Omega$ by the Euclidean topology of charts). The fine boundary of a set
$S$ is denoted by $\partial_{\mathsf{f}}S$, and its fine closure by
$\mathrm{clos}_{\mathsf{f}}$.

A set $E\subset\Omega$ is called _polar_ (in $\Omega$) if there is a
superharmonic function $u\not\equiv+\infty$ on $\Omega$ such that
$u(z)=+\infty$ for $z\in E$. Superharmonic functions not identically $+\infty$
are locally integrable, therefore polar sets have area measure zero, see
Section A.1. By definition a polar set is contained in a $G_{\delta}$ polar
set, and every $G_{\delta}$ polar set arises as the $+\infty$-set of a
superharmonic function [8, Section VI.9]. If $U$ is a fine domain and $E$ is
polar, then $U\setminus E$ is again a fine domain [19, Theorem 6]. In fact,
polar sets are exactly the sets of zero outer Greenian capacity (equivalently:
of zero outer logarithmic capacity if $\Omega\subset\mathbb{C}$) defined in
Section A.4; this is justified in Section A.9, but we take it presently for
granted (we stress that [53] defines polar sets as having inner capacity zero,
thereby making for a larger class of non-Borel polar sets). One consequence
is: if $\Omega^{\prime}\supset\Omega$ is hyperbolic and $E$ is polar in
$\Omega$, then it is polar in $\Omega^{\prime}$ as well; indeed, since
$g_{\Omega^{\prime}}(\cdot,w)\geq g_{\Omega}(\cdot,w)$, it is clear that $E$
has zero outer Greenian capacity in $\Omega^{\prime}$ if it does in $\Omega$.
Conversely, it is obvious from the definition that $E$ is polar in $\Omega$ if
it is polar in $\Omega^{\prime}\supset\Omega$, therefore we may speak of a
polar set without specifying a hyperbolic subset of ${\mathcal{R}}_{*}$ in
which $E$ is contained.

A countable union of polar sets is polar, for if $E_{k}$ is included in the
$+\infty$-set of a superharmonic function $u_{k}\not\equiv+\infty$ while
$K\subset\Omega$ is compact and of positive Lebesgue measure, then there are
$t_{k}>0$ such that $u:=\sum_{k}t_{k}u_{k}$ is summable on $K$ (therefore
$u\not\equiv+\infty$) and is superharmonic with value $+\infty$ at each point
of $\cup_{k}E_{k}$. So, if $E\subset\Omega$ is polar and
$p:{\mathcal{R}}_{*}\to\mathbb{C}$ is the natural projection, then $p(E)$ is
polar. Indeed, for $V\subset\Omega$ a domain such that $p:V\to p(V)$ is a
homeomorphism, $\nu\circ p^{-1}$ is superharmonic on $p(V)$ when $\nu$ is
superharmonic on $V$, and $\Omega\setminus\mathbf{rp}({\mathcal{R}}_{*})$ can
be covered with countably many such domains while
$\mathbf{rp}({\mathcal{R}}_{*})$ is finite. Conversely, if
$V\subset\mathbb{C}$ is a bounded open set and $E\subset V$ is polar, then
$p^{-1}(E)$ is polar because $v\circ p$ is superharmonic as soon as $v$ is
superharmonic on $V$.

If $u\not\equiv+\infty$ is superharmonic and finite on a polar set $E$, then
$E$ has outer $\Delta u$-measure zero [8, Section VI.9, item $\beta)$].
Consequently a Radon measure $\sigma$ of finite Green energy cannot charge a
polar set $E$, for we may assume $\sigma$ has compact support (since $\Omega$
is $\sigma$-compact) and as in Section A.4 there is an increasing sequence of
measures $\sigma_{k}$ converging strongly to $\sigma$ with
$g(\sigma_{k},\Omega;\cdot)$ continuous, whence
$\sigma(E)=\lim_{k}\sigma_{k}(E)=0$.

######  Removability Theorem.

If $E\subset\Omega$ is a (relatively) closed polar set while $u$ is
superharmonic on $\Omega\setminus E$ and locally bounded below in a
neighborhood of $E$, then $u$ extends in a unique manner to a superharmonic
function on $\Omega$. Moreover, if $u$ is harmonic in $\Omega\setminus E$ and
locally bounded in a neighborhood of $E$, then $u$ extends harmonically to
$\Omega$.

###### Proof.

The proof of the first statement carries over to hyperbolic Riemann surfaces
from its planar version, see [53, Theorem 3.6.1]. When $u$ is harmonic in
$\Omega\setminus E$ and locally bounded in a neighborhood of $E$, it extends
both to a subharmonic and a superharmonic function on $\Omega$ by the first
part. Since $\Delta u$ does not depend on the extension because $E$ has
Lebesgue measure zero, we deduce that $\Delta u=0$. ∎

The removability theorem implies the following result.

######  Generalized Minimum Principle.

If $u$ is superharmonic and bounded below on some open set
$U\subset\overline{U}\subset\Omega$, and if moreover $\liminf_{U\ni
z\to\xi}u(z)\geq 0$ for quasi every $\xi\in\partial U$, then $u\geq 0$ in $U$.

###### Proof.

Given $\varepsilon>0$, let $E_{\varepsilon}:=\\{\zeta\in\partial
U:\,\liminf_{z\to\xi}u(z)\leq-\varepsilon\\}$. Then $E_{\varepsilon}$ is a
closed polar set and the function $w:\Omega\to(-\infty,+\infty]$ given by
$\min(u,-\varepsilon)$ on $U$ and $-\varepsilon$ on $\Omega\setminus(U\cup
E_{\varepsilon})$ is superharmonic on $\Omega\setminus E_{\varepsilon}$, by
the glueing theorem, see [53, Theorem 2.4.5] for a planar version of this
local result. As $w$ is bounded below, it extends to a superharmonic function
on $\Omega$. Because $U\cup E_{\varepsilon}\subset\overline{U}\subset\Omega$,
it holds that $\liminf_{z\to\xi}w(z)=-\varepsilon$ for any
$\xi\in\partial\Omega$. Thus, $w\geq-\varepsilon$ in $\Omega$ by the classical
minimum principle contained in the very definition of superharmonic functions.
∎

The hypothesis “$u$ is bounded below” can be relaxed somewhat: it is enough to
assume that $u\geq-g$ where $g$ is a _semi-bounded potential_ , meaning that
it is the increasing pointwise limit of a sequence of locally bounded
potentials, see [20, Theorem 9.1]. Note that $g(\sigma,\Omega;\cdot)$ is semi-
bounded when it is finite $\sigma$-a.e., for we may assume $\sigma$ has
compact support (as $\Omega$ is a countable union of compact sets) and then
appeal to properties of the measures $\sigma_{k}$ in Section A.4. In fact,
$g(\sigma,\Omega;\cdot)$ is semi-bounded if and only if it is finite
$\sigma$-a.e. in $\Omega$, which is also if and only if $\sigma$ does not
charge polar sets, see [20, Section I.2.6, Theorem].

###  A.6. Thinness

Fine topology can also be introduced via the notion of thinness. A set
$E\subset\Omega$ is said to be _thin_ at $\zeta\in\Omega$ if $\zeta$ is not a
fine limit point of $E$. Equivalently, $E$ is thin at $\zeta$ if and only if
either $\zeta\notin\overline{E}$ or there exists a function $v$, superharmonic
in a neighborhood of $\zeta$, such that

(A.8) $\liminf_{E\ni z\to\zeta,z\not=\zeta}v(z)>v(\zeta);$

see [29, Theorem 6.6.3] for a proof of this equivalence in the Euclidean
setting, which applies to hyperbolic Riemann surfaces as well and also shows
that $v$ in (A.8) may be taken superharmonic on the whole of $\Omega$. Hence,
the above definition of thinness (which is local) matches [8, Definition I.2]
(whose local character is not immediate, see [8, Theorem VII.1]). Setting
$\liminf$ over the empty set to $+\infty$ by convention, (A.8) may still be
regarded as characterizing thinness at $\zeta\notin\overline{E}$, upon letting
$v\equiv 0$. Note that when the limit inferior in (A.8) is taken over a full
Euclidean neighborhood of $\zeta$, superharmonicity of $v$ implies that the
inequality gets replaced by an equality. Clearly, $E$ is thin at $\zeta$ if
and only if for some (hence any) chart $(V,\varphi)$ with $\zeta\in V$, the
planar set $\varphi(V\cap E)$ is thin at $\varphi(\zeta)$, and a countable
union of thin sets at $\zeta$ is again thin at $\zeta$.

A set $V$ is a fine neighborhood of $\zeta\in V$ if and only if the complement
of $V$ is thin at $\zeta$, see [8, Theorem I.3]. In particular, if $V$ is
finely open and $Z$ is polar, then $V\setminus Z$ is finely open. The points
of $E\subset\Omega$ at which $E$ is thin form a polar set, and $E$ is thin at
each of its points if and only if it is polar [8, Theorem VII.7 & Corollary].
One consequence of $E$ being thin at $\zeta$ is that, locally in a chart
$(V,\varphi)$ with $\zeta\in V$, there are arbitrary small circles centered at
$\varphi(\zeta)$ which do not meet $\varphi(E\cap V)$, see [29, Theorem
6.7.9]; in the same vein, $\varphi(V\setminus E)$ contains a segment of the
form $[\varphi(\zeta),\varphi(\zeta)+re^{i\theta}]$ with $r=r(\theta)>0$ for
quasi-every direction $e^{i\theta}$ in $\mathbb{T}$, see [53, Theorem 5.4.3].
In particular, a connected set cannot be thin at an accumulation point and
therefore polar sets are totally disconnected.

The _base_ $b(E)$ of a set $E$ is the set of points in $\Omega$ at which $E$
is non-thin, and $E$ is called a base if $b(E)=E$. It is known that $b(E)$ is
a finely closed $G_{\delta}$ set, see [8, Proposition VII.8]. We record the
following, elementary fact.

######  Lemma A.1.

Let $E,F\subset\Omega$ be disjoint finely open sets such that
$\Omega\setminus(E\cup F)$ is a base. Then $\Omega\setminus E$ and
$\Omega\setminus F$ are bases as well.

###### Proof.

If $x\in(\Omega\setminus E)\cap(\Omega\setminus F)=\Omega\setminus(E\cup F)$,
the latter set is non-thin at $x$ by assumption and therefore so is
$\Omega\setminus E$. If now $x\in(\Omega\setminus E)\cap F=F$, then the fine
openness of $F$ implies that $\Omega\setminus F$ is thin at $x$ and so is
$E\subset\Omega\setminus F$. However, if $\Omega\setminus E$ were also thin at
$x$ in this case, then $\Omega=E\cup(\Omega\setminus E)$ would be thin at $x$
which is impossible. Hence, $\Omega\setminus E$ is non-thin at any of its
points, therefore it is a base. ∎

The notion of a base generates a strong form of the domination principle, see
[8, Theorem VIII.4].

######  Strong Domination Principle.

Let $v$ be a non-negative superharmonic function in $\Omega$ such that $v\geq
g(\sigma,\Omega;\cdot)$ quasi everywhere on a set $E$ such that
$\sigma(\Omega\setminus b(E))=0$. Then $v\geq g(\sigma,\Omega;\cdot)$
everywhere in $\Omega$.

The fine closure of $E$ is equal to $\mathrm{clos}_{\mathsf{f}}(E)=b(E)\cup
i(E)$, where $i(E)$ is the set of finely isolated points of $E$, see [8,
Proposition V.10]. The finely closed sets are precisely those for which
$b(E)\subseteq E$. Note that $b(E)\cap i(E)=\varnothing$ and therefore, if $V$
is finely open, we have that

(A.9) $b(\Omega\setminus V)=\Omega\setminus V^{\prime},\quad V^{\prime}:=V\cup
i(\Omega\setminus V),$

where we observe that $V^{\prime}$ is in turn finely open. For any set $E$,
the fine boundary $\partial_{\mathsf{f}}E$ is finely closed and, as shown in
[20, Lemma 12.3], it holds that

(A.10) $i(\partial_{\mathsf{f}}E)=i(E)\cup i(\Omega\setminus
E)\quad\text{and}\quad b(\partial_{\mathsf{f}}E)=b(E)\cap b(\Omega\setminus
E).$

The next lemma connects fine topologies in $D$ and ${\mathcal{R}}$ (defined at
the beginning of Section 2.1).

######  Lemma A.2.

The map $p:{\mathcal{R}}\to D$ is finely open and finely continuous, that is,
$p(V)$ and $p^{-1}(U)$ are finely open when $V\subset{\mathcal{R}}$ and
$U\subset D$ are finely open. Moreover, $i({\mathcal{R}}\setminus
p^{-1}(U))=p^{-1}(i(D\setminus U))$.

###### Proof.

Let $V\subset{\mathcal{R}}$ be finely open and $\zeta\in V$. Denote by
$\tilde{O}$ a Euclidean disk centered at $p(\zeta)$ of small enough radius so
that $O$, the connected component of $p^{-1}(\tilde{O})$ containing $\zeta$,
contains no ramification points of ${\mathcal{R}}$ except possibly $\zeta$
itself (if $m(\zeta)>1$) and $O\cap p^{-1}(p(\zeta))=\\{\zeta\\}$. Since
$E:=O\setminus V$ is thin at $\zeta$, there exists a superharmonic function
$v$ in $O$ for which (A.8) takes place. _We claim_ that

(A.11) $\tilde{v}(z):=\begin{cases}\sum_{w\in O\cap
p^{-1}(z)}v(w),&z\in\tilde{O}\setminus\\{p(\zeta)\\},\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ m(\zeta)v(\zeta),&z=p(\zeta),\end{cases}$

is superharmonic on $\tilde{O}$. Indeed, by shrinking $\tilde{O}$ if needed,
we may assume that $v$ is the increasing limit of a sequence of continuous
superharmonic functions $v_{n}$ on $O$, see [53, Corollary 2.7.3] for a proof
of this fact in the planar case that carries over to ${\mathcal{R}}$ using
local charts. Define $\tilde{v}_{n}$ similarly to (A.11), only replacing $v$
with $v_{n}$. Clearly, $\tilde{v}_{n}$ is superharmonic on
$\\{z\in\tilde{O}:z\neq p(\zeta)\\}$. Since it is bounded around $p(\zeta)$ by
the continuity of $\nu_{n}$, the restriction
$\tilde{v}_{n\mathcal{b}\tilde{O}\setminus p(\zeta)}$ uniquely extends to a
superharmonic function on $\tilde{O}$ by the Removability Theorem. Of
necessity, the value at $p(\zeta)$ of this extension is given by

$\liminf_{z\to p(\zeta),z\neq
p(\zeta)}\tilde{v}_{n}(z)=m(\zeta)v_{n}(\zeta)=\tilde{v}_{n}(p(\zeta)),$

where the first equality comes from the continuity of $v_{n}$ at $\zeta$.
Hence, $\tilde{v}_{n}(z)$ is superharmonic on $\tilde{O}$ and so is its
increasing limit $\tilde{v}$. _This proves the claim_. In another connection,
the lower semicontinuity of $v$ shows that the analog of (A.8) holds for
$\tilde{v}$ when the limit inferior is taken along $p(E)$. As
$\tilde{O}\setminus p(V)\subseteq p(E)$, we get that $D\setminus p(V)$ is thin
at $p(\zeta)$ so that $p(V)$ is finely open, as claimed.

To show the second claim, observe that the lift of a function from $D$ to
${\mathcal{R}}$ preserves superharmonicity and that ${\mathcal{R}}\setminus
p^{-1}(U)=p^{-1}(D\setminus U)$. The identity $i({\mathcal{R}}\setminus
p^{-1}(U))=p^{-1}(i(D\setminus U))$ is now straightforward. ∎

In Lemma A.3 below, we single out for easy reference a basic geometric fact,
used at several places in the paper. We say that a continuous injective map
$\gamma:\mathbb{T}\to{\mathcal{R}}$ is a _parametrized Jordan curve_ in
${\mathcal{R}}$ and we simply call the image $\gamma(\mathbb{T})$ a (non-
parametrized) Jordan curve. On a hyperbolic Riemann surface ${\mathcal{R}}$,
any Jordan curve $\Gamma$ homotopic to a point is uniquely the boundary of a
(topological) disk $O\subset{\mathcal{R}}$ [64, Theorem 2.4]; we say that $O$
is the _interior_ of $\Gamma$, and we write $O=\mathrm{int}\,\Gamma$. That
$\Gamma$ is homotopic to a point in particular holds if it is included in a
simply connected open set $U$ which is the domain of a chart. Recall also that
$E\subset{\mathcal{R}}$ is called _schlicht_ over $U\subset\mathbb{C}$ if
$U\supset p(E)$ and the restriction $p_{\mathcal{b}E}:E\to U$ is injective.

######  Lemma A.3.

Let $E\subset{\mathcal{R}}$ be schlicht over $D$ and
$\xi\in\mathbf{rp}({\mathcal{R}})$. There exists a neighborhood $U_{\xi}$ of
$\xi$ such that no Jordan curve in $U_{\xi}$ contains $\xi$ in its interior
and simultaneously is contained in $E$.

###### Proof.

Let $U$ be a simply connected domain of a chart containing $\xi$, so that
every Jordan curve $\mathbb{T}\to U$ is homotopic to a point. Let $O\subset D$
be a disk centered at $p(\xi)$ and $O_{\xi}$ the connected component of
$p^{-1}(O)$ containing $\xi$, with $O$ small enough that $O_{\xi}\subset U$
and $O_{\xi}\cap\mathbf{rp}({\mathcal{R}})=\\{\xi\\}$; this is possible since
${\mathcal{R}}_{*}$ is compact. Then $O_{\xi}$ is a neighborhood of $\xi$
which is (isomorphic via a biholomorphic map fixing $\xi$ to) an
$m(\xi)$-sheeted cyclic covering of $O$. So, if we identify the homotopy
groups $\pi_{1}(O_{\xi}\setminus\\{\xi\\})$ and
$\pi_{1}(O\setminus\\{p(\xi)\\})$ with the infinite cyclic group generated by
the symbol $a$, the induced morphism
$p_{*}:\pi_{1}(O_{\xi}\setminus\\{\xi\\})\to\pi_{1}(O\setminus\\{p(\xi)\\})$
is the map $a\mapsto a^{m(\xi)}$. Now, a parametrized Jordan curve
$\gamma:\mathbb{T}\to O_{\xi}$ containing $\xi$ in its interior is a generator
of the fundamental group of $O_{\xi}\setminus\\{\xi\\}$. Hence, $p\circ\gamma$
is the $m(\xi)$-power of a generator of the fundamental group of
$O\setminus\\{p(\xi)\\}$, in particular it has winding number $\pm m(\xi)$
with respect to $p(\xi)$. However, if $\gamma$ is valued in $E$, then
$p\circ\gamma:\mathbb{T}\to O$ is a parametrized Jordan curve because $p$ is
injective on $E$, and therefore it has winding number $\pm 1$ with respect to
$p(\xi)$. This contradicts the assumption that $m(\xi)>1$, showing that
$U_{\xi}:=O_{\xi}$ satisfies our requirements. ∎

The next lemma, used in the proof of Lemma 3.11, depends on Lemma A.3.

######  Lemma A.4.

If $E\subset{\mathcal{R}}$ is schlicht over $D$ and
$\xi\in\mathbf{rp}({\mathcal{R}})$, then ${\mathcal{R}}\setminus E$ is non-
thin at $\xi$.

###### Proof.

Let $(U_{\xi},\varphi)$ be a chart around $\xi$, with $U_{\xi}$ as in Lemma
A.3 and $\varphi(\xi)=0$. If ${\mathcal{R}}\setminus E$ is thin at $\xi$, then
there is a circle $\mathbb{T}_{r}:=\\{|z|=r\\}\subset\varphi(U_{\xi})$ such
that $\varphi^{-1}(\mathbb{T}_{r})\subset E$. As
$\varphi^{-1}:\mathbb{T}_{r}\to U_{\xi}$ is a Jordan curve in $E$ that
contains $\xi$ in its interior, this contradicts Lemma A.3. ∎

###  A.7. Regularity

Thinness is intimately connected to the notion of a regular boundary point
with respect to the Dirichlet problem. Given a Euclidean open set $U$ with
$\overline{U}\subset\Omega$, a point $\zeta\in\partial U$ is called _regular_
if for any continuous function $\psi$ on $\partial U$ it holds that

$\lim_{U\ni z\to\zeta}H_{\psi}(z)=\psi(\zeta),$

where $H_{\psi}(z)$ is the Perron-Wiener-Brelot solution of the Dirichlet
problem on $U$ with boundary data $\psi$, see [8, Section VI.6, item
$\gamma)$] for a description of the Perron-Wiener-Brelot process; other
boundary points are called _irregular_. When all its boundary points are
regular, we say that $U$ itself is regular. It is known that $\zeta\in\partial
U$ is regular if and only if

(A.12) $\lim_{U\ni z\to\zeta}g_{U}(z,w)=0$

for some (and then any) $w$ in each connected component of $U$. Moreover,
$\zeta$ is irregular if and only if the complement of $U$ is thin at $\zeta$
[8, Theorem VII.13]. This entails that regularity is a local notion, in
particular each point of $\partial U$ is regular as soon as the latter is
locally connected, as follows from the analogous property in a Euclidean space
[53, Theorem 4.2.2].

Let $\sigma$ be a finite measure, compactly supported in $U$. As $g_{U}(z,w)$
is bounded for $w\in\mathrm{supp}\,\sigma$ and $z$ outside of a neighborhood
of the latter, see Section A.2, we get from (A.12) and the dominated
convergence theorem that $g(\sigma,U;\cdot)$ extends continuously by zero to
the set of regular points of $\partial U$. When $\sigma$ is not compactly
supported, a weaker result is stated in Section A.8 (Lemma A.5).

Regular points of _finely open_ sets are defined analogously: when $U$ is
finely open, $\zeta\in\partial_{\mathsf{f}}U$ is said to be regular if
$\Omega\setminus U$ is non-thin at $\zeta$. By (A.10), the set of regular
points is then $b(\partial_{\mathsf{f}}U)$, and if $\partial_{\mathsf{f}}U$ is
its own base one says that $U$ is regular, see [20, Section IV.12].

The following results are the natural analogs, for Green potentials on regular
open sets of hyperbolic surfaces, of their logarithmic counterparts in the
plane, see [55, Theorems I.6.8 & I.6.9].

######  Principle of Descent and Lower Envelope Theorem.

Let $U$ be a regular open set with compact closure
$\overline{U}\subset\Omega$. If $\sigma_{n}$ are positive measures on $U$ with
uniformly bounded masses that converge weak∗ to some measure $\sigma$ as
$n\to\infty$, then

  1. (1)

[Principle of Descent]

(A.13) $\liminf_{n\to\infty}g(\sigma_{n},U;z_{n})\geq g(\sigma,U;z),\quad
z_{n}\to z\in U.$

  2. (2)

[Lower Envelope Theorem]

(A.14) $\liminf_{n\to\infty}g(\sigma_{n},U;z)=g(\sigma,U;z)\quad\text{ for
quasi every}\quad z\in U.$

###### Proof.

The arguments are a minor variation of those used in [55]. For $M>0$ and $z\in
U$, observe from (A.12) that $\varphi_{M,z}(w):=\min\\{M,g_{U}(z,w)\\}$ is
continuous on $\overline{U}$ and zero on $\partial U$, locally uniformly with
respect to $z$. Thus, $\varphi_{M,z}$ lies in the closure of the space
$C_{c}(U)$ of continuous functions on $U$ with compact support endowed with
the $\sup$ norm. Moreover, $|\varphi_{M,z_{n}}-\varphi_{M,z}|$ is arbitrary
small on $U$ for $n$ large enough, by the minimum principle and the continuity
of Green functions off the diagonal. Hence, as
$\sigma_{n}\overset{w*}{\to}\sigma$, we get that
$\lim_{n}\int\varphi_{M,z_{n}}d\sigma_{n}=\int\varphi_{M,z}d\sigma$ and
consequently, by monotone convergence, we deduce (A.13) from the relations

$g(\sigma,U;z)=\lim_{M\to\infty}\int\varphi_{M,z}d\sigma=\lim_{M\to\infty}\lim_{n}\int\varphi_{M,z_{n}}d\sigma_{n}\leq\liminf_{n}g(\sigma_{n},U;z_{n}).$

Next, assume to the contrary that $g(\sigma,U;z)<\liminf_{n}g(\sigma_{n},U,z)$
for $z\in K$, where $K\subset U$ is such that $\mathrm{cap}_{U}(K)>0$.
Clearly, we may suppose that $K$ is compact and so we can find a nonzero
measure $\sigma_{*}$, supported on $K$, such that $g(\sigma_{*},U;\cdot)$ is
continuous on $U$, see Section A.4. Then by Fatou’s lemma it holds that

(A.15) $\int
g(\sigma,U;z)d\sigma_{*}(z)<\int\liminf_{n}g(\sigma_{n},U;z)d\sigma_{*}(z)\leq\liminf_{n}\int
g(\sigma_{n},U;z)d\sigma_{*}(z).$

Moreover, as $g(\sigma_{*},U;\cdot)$ extends continuously by zero on $\partial
U$, see discussion after (A.12), it lies in the closure of $C_{c}(U)$.
Therefore, by Fubini’s theorem,

$\lim_{n}\int g(\sigma_{n},U;z)d\sigma_{*}(z)=\lim_{n}\int
g(\sigma_{*},U;z)d\sigma_{n}(z)=\int g(\sigma_{*},U;z)d\sigma(z)=\int
g(\sigma,U;z)d\sigma_{*}(z),$

thereby contradicting (A.15). ∎

###  A.8. Superlevel Sets of Green Potentials

In this section we restrict attention to a planar simply connected domain $D$,
which is the interior of a Jordan curve $T$. In this case, any conformal map
$\phi:\mathbb{D}\to D$ extends to a homeomorphism from $\overline{\mathbb{D}}$
onto $\overline{D}$ [50, Theorem 2.6], that we continue to denote with $\phi$.
Clearly, such a domain $D$ is regular. As mentioned in the previous
subsection, if $\sigma$ is a finite Borel measure compactly supported in $D$,
then $g(\sigma,D;\cdot)$ continuously extends by zero to $T$. If $\sigma$ is
not compactly supported this may not hold, but when $F$ is a relatively closed
subset of $\mathbb{D}$ with a limit point $\xi\in\mathbb{T}$, it was shown in
[39] that

(A.16) $\lim_{\epsilon\to
0}\mathrm{cap}_{\mathbb{D}}(F\cap\\{|z-\xi|<\epsilon\\})>0\quad\Rightarrow\quad\liminf_{F\ni
z\to\xi}g(\sigma,\mathbb{D};z)=0,$

and if the rightmost limit holds for every finite measure $\sigma$, then the
implication can be reversed. This result is in fact stated in [39] with
$g(\sigma,\mathbb{D};z)$ replaced by $(1-|z|)g(\nu,\mathbb{D};z)$ where $\nu$
is any measure whose Green potential is not identically $+\infty$, but the
latter condition is equivalent to saying that the measure
$d\sigma(z):=(1-|z|)d\nu(z)$ is finite and then convergence to zero along $F$
of the limit inferior of $(1-|z|)g(\nu,\mathbb{D};z)$ and of
$g(\sigma,\mathbb{D};z)$ are equivalent, see [39, Section 3, Lemma]. It is
also pointed out in [39, Equation (2.5)] that the leftmost limit in (A.16) can
be equivalently replaced by
$\mathrm{cap}_{\mathbb{D}}(F\cap\\{|z-\xi|<\epsilon\\})=\infty$ for every
$\epsilon>0$ (note that instead of the Greenian capacity
$\mathrm{cap}_{\mathbb{D}}(E)$ that we use, [39] employs the hyperbolic
capacity $\exp\\{-1/\mathrm{cap}_{\mathbb{D}}(E)\\}$). In Lemma A.5 below, we
derive a useful consequence of (A.16). With the notation of this lemma, we
stress that a stronger conclusion in fact holds quasi everywhere, namely
$U_{\epsilon}$ is thin at quasi every point of $T$ (this can be deduced from
general properties of balayage covered in Section A.9). The interest of Lemma
A.5 lies with the fact that its conclusion holds at _every_ point of $T$.

######  Lemma A.5.

Let $\sigma$ be a finite measure in $D$, and for $\epsilon>0$ set
$U_{\epsilon}:=\\{z\in D:g(\sigma,D;z)>\epsilon\\}$. Let
$\xi\in\overline{U}_{\epsilon}\cap T$ and $\phi:\mathbb{D}\to D$ be a
conformal map such that $\phi(1)=\xi$. Then there exists a closed set
$R_{\epsilon}\subset[0,1]$ such that $R_{\epsilon}\cap(1-\delta,1)$ is non-
polar for any $\delta\in(0,1)$, and for each $r\in R_{\epsilon}$ one has
$U_{\epsilon}\cap\phi(\\{z\in\mathbb{D}:|1-z|=1-r\\})=\varnothing$.

###### Proof.

Suppose initially that $D=\mathbb{D}$. Without loss of generality, we can
assume that $\xi=1$. First, we shall show that

(A.17) $\lim_{\delta\to 0}\mathrm{cap}_{\mathbb{D}}(U_{\epsilon}\cap
D_{\delta})=0,$

where $D_{\delta}:=\\{z\in\mathbb{D}:|z-1|<\delta\\}$. For
$\zeta\in\mathbb{D}$, let
$S(\zeta):=\\{z\in\mathbb{D}:g_{\mathbb{D}}(z,\zeta)\geq\log 2\\}$. It was
shown in [39, Section 3, Lemma] that there exists
$\delta_{0}=\delta(\epsilon)>0$ for which

$\int_{\mathbb{D}\setminus
S(\zeta)}g_{\mathbb{D}}(z,\zeta)d\sigma(z)\leq\epsilon/2,\quad|\zeta|>1-\delta_{0}.$

In particular, this inequality holds for $\zeta\in D_{\delta_{0}}$. Hence, for
any compact subset $F\subset U_{\epsilon}\cap D_{\delta_{0}}$, it holds when
$\zeta\in F$ that

(A.18)
$h(\zeta):=\int_{S(\zeta)}g_{\mathbb{D}}(z,\zeta)d\sigma(z)=g(\sigma,\mathbb{D};\zeta)-\int_{\mathbb{D}\setminus
S(\zeta)}g_{\mathbb{D}}(z,\zeta)d\sigma(z){\color[rgb]{0,0,1}>}\epsilon/2.$

Assume to the contrary that the limit in (A.17) is larger that $2\eta>0$ (the
limit must exist as $\mathrm{cap}_{\mathbb{D}}(U_{\epsilon}\cap D_{\delta})$
decreases with $\delta$). Since $U_{\epsilon}$ is an open set, we get from
(A.5) that for any $\delta>0$ there exists a compact set $F_{\delta}\subset
U_{\epsilon}\cap D_{\delta}$ for which
$\mathrm{cap}_{\mathbb{D}}(F_{\delta})\geq\eta$. This entails that there exist
a sequence $\delta_{n}\to 0$ and disjoint compact sets $F_{n}\subset
U_{\epsilon}\cap D_{\delta_{n}}$ with
$\mathrm{cap}_{\mathbb{D}}(F_{n})\geq\eta$. Let $\nu_{n}$ be the Green
equilibrium distribution on $F_{n}$ and $F_{n}^{*}:=\\{z\in S(\zeta):\text{for
some }\zeta\in F_{n}\\}$. In view of (A.18),

$\epsilon/2\leq\int_{F_{n}}h(\zeta)d\nu_{n}(\zeta)\leq\frac{1}{\eta}\sigma(F_{n}^{*})\to
0,$

where the second inequality and the fact that $\lim_{n}\sigma(F_{n}^{*})\to 0$
can be established as in the proof of [39, Theorem 1] (compare to p. 486 of
that reference). This contradiction proves (A.17).

Let $\mathsf{T}:D_{1}\to(0,1)$ be defined by $z\mapsto\mathsf{T}z:=1-|1-z|$
and put $V_{\epsilon}:=U_{\epsilon}\cap D_{1}$. Denoting by
$\mathsf{T}V_{\epsilon}$ the set $\\{\mathsf{T}\zeta:\,\zeta\in
V_{\epsilon}\\}$, we claim that

(A.19) $\lim_{\delta\to 0}\mathrm{cap}_{\mathbb{D}}(\mathsf{T}V_{\epsilon}\cap
D_{\delta})=0.$

Before proving (A.19), let us show why it implies the lemma. For this,
consider $R_{\epsilon}:=[0,1]\setminus\mathsf{T}V_{\epsilon}$, which is a
closed set. If the conclusion of the lemma were not true, there would exist
$\delta_{0}>0$ such that $R_{\epsilon}\cap(1-\delta_{0},1)$ is polar. By
definition of $\mathsf{T}$ this would imply that
$\mathrm{cap}_{\mathbb{D}}(\mathsf{T}V_{\epsilon}\cap
D_{\delta})=\mathrm{cap}_{\mathbb{D}}((1-\delta,1))=\infty$ for any
$\delta<\delta_{0}$ (the last equality follows at once from the definition of
the Greenian capacity), which contradicts (A.19).

We are now left to demonstrate (A.19). Assume to the contrary that it does not
hold, i.e., there exists $\eta>0$ such that for any $\delta>0$ there is a
compact set $F_{\delta}\subset\mathsf{T}V_{\epsilon}\cap D_{\delta}$ for which
$\mathrm{cap}_{\mathbb{D}}(F_{\delta})\geq\eta$. The previous inequality means
that there exists a probability measure $\mu_{\delta}$ supported on
$F_{\delta}$ such that

$\int\int
g_{\mathbb{D}}(x,y)d\mu_{\delta}(x)d\mu_{\delta}(y)\leq\frac{1}{\eta}.$

Since $U_{\epsilon}$ is open, so is $V_{\epsilon}$ and one easily sees that
each $z\in V_{\epsilon}\cap D_{\delta}$ has a neighborhood, say $O_{z}$, whose
closure is contained in $V_{\epsilon}\cap D_{\delta}$ and whose circular
projection $\mathsf{T}O_{z}$ is an open subinterval of $(0,1)$. These
subintervals form an open cover of $F_{\delta}$, which necessarily has a
finite subcover, say $\mathsf{T}O_{z_{1}},\ldots,\mathsf{T}O_{z_{N}}$. The
closure $K_{\delta}$ of $O_{z_{1}}\cup\cdots\cup O_{z_{N}}$ is a compact
subset of $V_{\epsilon}\cap D_{\delta}$, and clearly
$F_{\delta}\subset\mathsf{T}K_{\delta}$. In particular, there exists a
probability measure $\nu_{\delta}$ on $K_{\delta}$ such that
$\nu_{\delta}\mathsf{T}^{-1}=\mu_{\delta}$, see for example [53, Theorem
A.4.4]. Then, by Fubini’s theorem, it holds that

$\frac{1}{\eta}\geq\int\int
g_{\mathbb{D}}(x,y)d\mu_{\delta}(x)\mu_{\delta}(y)=\int\int
g_{\mathbb{D}}(\mathsf{T}z,\mathsf{T}w)d\nu_{\delta}(z)d\nu_{\delta}(w),$

and if we can show that $g_{\mathbb{D}}(\mathsf{T}z,\mathsf{T}w)\geq
g_{\mathbb{D}}(z,w)$ then we will deduce from the above estimate that

$\mathrm{cap}_{\mathbb{D}}(U_{\epsilon}\cap
D_{\delta})=\mathrm{cap}_{\mathbb{D}}(V_{\epsilon}\cap
D_{\delta})\geq\mathrm{cap}_{\mathbb{D}}(K_{\delta})\geq\eta>0\quad\text{for
any}\ \delta>0,$

which of course contradicts (A.17). Hence, the proof has been reduced to the
verification of $g_{\mathbb{D}}(\mathsf{T}z,\mathsf{T}w)\geq
g_{\mathbb{D}}(z,w)$ for $z,w\in D_{1}$, that we now carry out. Since
$g_{\mathbb{D}}(z,w)=\log|(1-z\bar{w})/(z-w)|$ and $\mathsf{T}$ is real-
valued, we need to show that

(A.20)
$E:=|1-\mathsf{T}z{\mathsf{T}w}|^{2}|z-w|^{2}-|1-z\bar{w}|^{2}|\mathsf{T}z-\mathsf{T}w|^{2}\geq
0.$

Set $a\xi:=1-z$ and $b\eta:=1-w$, where $a,b\in(0,1)$ and $|\xi|=|\eta|=1$
with $\mathrm{Re}\xi,\mathrm{Re}\eta\in(0,1)$. Then

$\displaystyle E$ $\displaystyle=$
$\displaystyle|a+b-ab|^{2}|a\xi-b\eta|^{2}-|a\xi+b\bar{\eta}-ab\xi\bar{\eta}|^{2}|a-b|^{2}$
$\displaystyle=$ $\displaystyle(S+2abU)(T-2abV)-(S+2abW)(T-2ab)$

where $S:=a^{2}+b^{2}+(ab)^{2}$, $T:=a^{2}+b^{2}$, $U:=1-a-b$,
$V:=\mathrm{Re}(\xi\bar{\eta})$, and $W:=\mathrm{Re}(\xi\eta-a\eta-b\xi)$.
Therefore,

$\displaystyle E$ $\displaystyle=$ $\displaystyle
2ab\big{(}S(1-V)+T(U-W)+2ab(W-UV)\big{)}$ $\displaystyle=$ $\displaystyle
2ab\big{(}(S+2abU)(1-V)+(T-2ab)(U-W)\big{)}$ $\displaystyle=$ $\displaystyle
2ab\big{(}(a+b-ab)^{2}(1-V)+(a-b)^{2}(U-W)\big{)}.$

Because $V\leq 1$, the above expression can be estimated from below as

$\displaystyle E$ $\displaystyle\geq$ $\displaystyle 2ab(a-b)^{2}(1-V+U-W)$
$\displaystyle=$ $\displaystyle
2ab(a-b)^{2}\big{(}2-\mathrm{Re}(\xi\bar{\eta}+\xi\eta)-a(1-\mathrm{Re}\eta)-b(1-\mathrm{Re}\xi)\big{)},$

and since $1-\mathrm{Re}\eta$, $1-\mathrm{Re}\xi$, $1-a$ as well as $1-b$ are
all positive, it therefore holds that

$\displaystyle E$ $\displaystyle\geq$ $\displaystyle
2ab(a-b)^{2}\big{(}\mathrm{Re}\eta+\mathrm{Re}\xi-\mathrm{Re}(\xi\bar{\eta}+\xi\eta)\big{)}$
$\displaystyle=$ $\displaystyle
2ab(a-b)^{2}\big{(}\mathrm{Re}\eta+\mathrm{Re}\xi-2\mathrm{Re}\eta\mathrm{Re}\xi\big{)}$
$\displaystyle=$ $\displaystyle
2ab(a-b)^{2}\big{(}\mathrm{Re}\eta(1-\mathrm{Re}\xi)+\mathrm{Re}\xi(1-\mathrm{Re}\eta)\big{)}.$

As $\mathrm{Re}\xi,\mathrm{Re}\eta\in[0,1]$, this establishes (A.20) and
completes the proof of the lemma when $D=\mathbb{D}$.

Finally, it remains to reduce the case of a general domain $D$ to the one of
the unit disk. Using [53, Theorem A.4.4] once more, let $\nu$ be a finite
measure in $\overline{\mathbb{D}}$ such that $\nu\phi^{-1}=\sigma$. Then

$\displaystyle g(\sigma,D;\phi(z))$ $\displaystyle=\int
g_{D}(\phi(z),\zeta)d\sigma(\zeta)=\int g_{D}(\phi(z),\phi(w))d\nu(w)$
$\displaystyle=\int g_{\mathbb{D}}(z,w)d\nu(w)=g(\nu,\mathbb{D};z),\quad
z\in\mathbb{D},$

by conformal equivalence of Green functions. Since
$\phi:\overline{\mathbb{D}}\to\overline{D}$ is a bijection, the superlevel set
$\\{z\in\mathbb{D}:g(\nu,\mathbb{D};z)>\epsilon\\}$ is equal to
$\phi^{-1}(U_{\epsilon})$, from which the desired result follows. ∎

###  A.9. Balayage

Let $v$ be a non-negative superharmonic function on $\Omega$ and $E$ be a
subset of $\Omega$. The _balayage function_ (or _regularized reduction_) of
$v$ relative to $E$, denoted by $\mathcal{B}_{v}^{E}$, is the lower semi-
continuous regularization of

(A.21) $\inf\big{\\{}u|~{}u\text{ is superharmonic and positive in
}\Omega,~{}u\geq v\text{ on }E\big{\\}},$

see [29, Section 5.3] for an account on $\mathbb{R}^{n}$ that carries over to
$\Omega$ without change; in fact, $\mathcal{B}_{v}^{E}$ coincides with the
infimum in (A.21) except perhaps on a polar set where lower semi-continuous
regularization may modify the value. The balayage function
$\mathcal{B}_{v}^{E}$ is superharmonic in $\Omega$, harmonic in
$\Omega\setminus\overline{E}$, and equal to $v$ on $b(E)$ [8, Section VIII.1].
Clearly, $\mathcal{B}_{v}^{E}\leq v$ everywhere, since $v$ qualifies as one of
the functions $u$ in (A.21). The balayage function $\mathcal{B}_{v}^{E}$ does
not change if $E$ gets replaced by $b(E)$ or by
$\mathrm{clos}_{\mathsf{f}}(E)$; in fact, it remains invariant if $E$ is
altered by a polar set.

When $E$ is compact, the Strong Domination Principle and properties of the
Green equilibrium potential $g(\mu_{\Omega,E},\Omega;\cdot)$ imply that
$\mathcal{B}_{1}^{E}=\mathrm{cap}_{\Omega}(E)g(\mu_{\Omega,E},\Omega;\cdot)$.
Thus, it follows from the left continuity of $\mathcal{B}_{1}^{E}$ with
respect to $E$, see [8, Section VI.10 e)], and the monotone convergence
theorem that the outer Greenian capacity of an arbitrary set $E\subset\Omega$
is the mass of $\Delta\mathcal{B}_{1}^{E}$ (in fact, this is the way the outer
capacity is defined in [8, Section VIII.4]). From this and [8, Theorem
VIII.12], we deduce in particular that $E$ is polar if and only if
$\mathrm{cap}_{\Omega}(E)=0$, justifying a claim made in Section A.5.

If $v$ is the Green potential of a positive Borel measure $\sigma$, then
$\mathcal{B}_{v}^{E}$ is a Green potential as well [8, Section VI.11] and the
measure $\sigma^{E}$ such that
$\mathcal{B}_{v}^{E}=g(\sigma^{E},\Omega;\cdot)$ is called the _balayage_ of
$\sigma$ relative to $E$. The measure $\sigma^{E}$ is characterized as the
unique measure satisfying

(A.22) $\sigma^{E}\big{(}\Omega\setminus b(E)\big{)}=0\quad\text{and}\quad
g(\sigma,\Omega;z)=g\big{(}\sigma^{E},\Omega;z\big{)},\quad z\in b(E),$

see [8, Theorem VIII.3]. From (A.22), one deduces at once that

(A.23) $(\sigma^{E})^{F}=\sigma^{F},\qquad F\subset E\subset\Omega.$

Moreover, it holds by [8, Section VI.12, Equation (13)] that

(A.24) $\sigma^{E}(B)=\int\delta_{x}^{E}(B)d\sigma(x),\qquad
B\quad\mathrm{Borel},$

while it follows from [8, Section VI.12, Equation (9)] and Fubini’s theorem
that

(A.25) $g\big{(}\sigma^{E},\Omega;z\big{)}=\int
g(\sigma,\Omega;x)d\delta_{z}^{E}(x).$

Since $g(\sigma^{E},\Omega;\cdot)\leq g(\sigma,\Omega;\cdot)$ for any Radon
measure $\sigma$ by the just discussed properties of balayage, it follows from
Fubini’s theorem that $\int\nu d\sigma^{E}\leq\int\nu d\sigma$ for any Green
potential $\nu$. Since any non-negative superharmonic function $\nu$ is an
increasing limit of potentials, see [10, Lemma 1.1], monotone convergence
yields that $\int\nu d\sigma^{E}\leq\int\nu d\sigma$ remains valid for such
functions as well, see also [20, Section I.3]. In particular, the mass of
$\sigma^{E}$ cannot exceed the mass of $\sigma$.

The _fine support_ of a Radon measure $\sigma$, denoted by
$\mathrm{supp}_{\mathsf{f}}\,\sigma$ when it exists, is the smallest finely
closed carrier of $\sigma$. A sufficient condition for its existence is that
$\sigma$ does not charge polar sets, in which case
$\mathrm{supp}_{\mathsf{f}}\,\sigma$ is its own base, see [8, Theorem VII.12].
If $\sigma$ is admissible, meaning that
$g(\sigma,\Omega;\cdot)\not\equiv+\infty$, see Section A.3, and if
$\sigma(F)=0$ for some polar set $F$, then $\sigma^{E}(F)=0$ for any
$E\subset\Omega$ [19, Theorem 1] (as usual, $\sigma(F)$ means the outer
$\sigma$-measure of $F$ when the latter is not Borel). In particular, if
$\sigma$ is admissible and $\sigma(b(E))=0$, then $\sigma^{E}$ does not charge
polar sets since it is carried by $b(E)$. Thus,
$\mathrm{supp}_{\mathsf{f}}\,\sigma^{E}$ exists in this case. An important
special case is handled by the following lemma, the first item of which
follows from the preceding discussion.

######  Lemma A.6.

[19, Corollary 1 to Theorem 4] [20, Corollaries 2 & 3 to Theorem 12.7]

  * (i)

If $W$ is finely open and either $z\in W$ or $z\in i(\Omega\setminus W)$, then
$\delta_{z}^{\Omega\setminus W}$ does not charge polar sets.

  * (ii)

Let $V$ be a regular finely open set, $z\in V$, and $V_{z}$ the fine component
of $V$ containing $z$. Then $V_{z}$ is regular, and $\Omega\setminus V_{z}$ is
largest among all the bases $B$ such that
$\delta_{z}^{B}=\delta_{z}^{\Omega\setminus V}$. Moreover, the fine support of
$\delta_{z}^{\Omega\setminus V}$ exists and

(A.26) $\mathrm{supp}_{\mathsf{f}}\,\delta_{z}^{\Omega\setminus
V}=\partial_{\mathsf{f}}V_{z}\subset\partial_{\mathsf{f}}V.$

  * (iii)

Let $U$ be a fine domain and $z\in U$ or $z\in i(\Omega\setminus U)$. Then the
fine support of $\delta_{z}^{\Omega\setminus U}$ exists and

(A.27) $\mathrm{supp}_{\mathsf{f}}\,\delta_{z}^{\Omega\setminus
U}=b(\partial_{\mathsf{f}}U)=b(\Omega\setminus U)\cap\partial_{\mathsf{f}}U.$

Let now $O\subset\Omega$ be (Euclidean) open, $z\in O$, and $O_{z}$ the
connected component of $O$ containing $z$. If we let $V$ be the regular finely
open set obtained by adjoining to $O$ the polar set $i(\Omega\setminus O)$ and
$V_{z}$ the fine component containing $z$, then we get from [19, Theorem 6]
(see discussion there) that $O_{z}=V_{z}\setminus i(\Omega\setminus O)$. Thus,
since the balayage function remains the same if the set relative to which it
is defined is altered by a polar set, Lemma A.6 (ii) implies that

$\delta_{z}^{\Omega\setminus O}=\delta_{z}^{\Omega\setminus
V}=\delta_{z}^{\Omega\setminus V_{z}}=\delta_{z}^{\Omega\setminus O_{z}}$

and the latter is carried by the regular points of $\partial O_{z}$. Moreover,
if $O$ has compact closure in $\Omega$, then $\delta_{z}^{\Omega\setminus O}$
is a probability measure, and for $h$ a harmonic function in $O$ with
continuous extension to $\overline{O}$:

(A.28) $h(z)=\int h\,d\delta_{z}^{\Omega\setminus O},\qquad z\in O.$

Indeed, (A.28) follows from [8, Section VI.12, application 1] since $h$ is the
Perron-Wiener-Brelot solution of the Dirichlet problem444In fact, the right-
hand side of (A.28) is the Perron-Wiener-Brelot solution of the Dirichlet
problem on $O$ with boundary data $h$ as soon as the latter is summable
against $\delta_{z}^{\Omega\setminus O}$ for one (and then any) $z$ in each
component of $O$.  in $O$ with boundary data $h_{\mathcal{b}\partial O}$; see
[8, Section VI.6, item $\gamma)$]. Equality in (A.28) shows that the measure
$\delta_{z}^{\Omega\setminus O}$ does not depend on $\Omega$, provided that
the latter is hyperbolic and compactly contains $\overline{O}$. It is called
the _harmonic measure_ for $O$ (at $z$). More general versions of (A.28),
involving the fine Dirichlet problem and cases where $\overline{O}$ is non-
compact, are stated in Theorem A.8 and Lemma A.9 further below.

When $\Omega\subset\mathbb{C}$ and $\overline{O}$ is compact in $\Omega$, it
follows from [55, Chapter II, Theorem 5.1] that