A simply connected Greenian Riemann surface is a disc

The kernel at $p_0$. By the given data and [F2], [F3], the envelope $g_0$ is finite, strictly positive and harmonic on $X\setminus \{p_0\}$. The compact case of [F3] is identically infinite, so the given finite envelope forces $X$ to be noncompact. In the centred chart $z:U\to \mathbb{D}$ at $p_0$ there is a harmonic function $h_1$ on $U$ with $g_0=-\log \mathrm{\mid }z\mathrm{\mid }+h_1$ on $U\setminus \{p_0\}$.

The holomorphic map with $\mathrm{\mid }\varphi \mathrm{\mid }=e^{-g_0}$. Apply [F4] with $Y:=X$, $P:=\{p_0\}$, $u:=g_0$, $m_1:=1$ and $h_1$ as in step 1.1: there is a meromorphic function $F:X\to \widehat{\mathbb{C}}$ with $\mathrm{\mid }F\mathrm{\mid }=e^{-g_0}$ on $X\setminus \{p_0\}$, with $F=zG$ near $p_0$ for a holomorphic $G$ satisfying $G(p_0)\mathrm{\ne }0$, and with no zeros or poles outside $p_0$. Since the only local exponent is $m_1=1>0$, the function $F$ has no poles at all; taking values in $\mathbb{C}$ it is a holomorphic map $\varphi :=F:X\to \mathbb{C}$ [F1], with a simple zero at $p_0$ and no other zeros, and $\mathrm{\mid }\varphi (x)\mathrm{\mid }=e^{-g_0(x)}<1$ for every $x\mathrm{\ne }{p}_0$ by the strict positivity in step 1.1. In particular $\varphi (p_0)=0$ and $\varphi (X)\subseteq \mathbb{D}$.

The normalised map at an arbitrary second point. Fix $p_1\in X\setminus \{p_0\}$ and put $a:=\varphi (p_1)$, so $a\in \mathbb{D}\setminus \{0\}$ because $p_1\mathrm{\ne }{p}_0$ and $p_0$ is the only zero of $\varphi$ (step 2.1). By [F8] the map ${\mathrm{\Phi }}_a(w):=\frac{w-a}{1-\bar{a}w}$ is an automorphism of $\mathbb{D}$, so ${\varphi }_1:={\mathrm{\Phi }}_a\circ \varphi :X\to \mathbb{D}$ is holomorphic with $\mathrm{\mid }{\varphi }_1\mathrm{\mid }<1$ on $X$ [F1, F8]. Moreover ${\varphi }_1(p_1)=0$ and ${\varphi }_1(p_0)={\mathrm{\Phi }}_a(0)=-a\mathrm{\ne }0$. The map ${\varphi }_1$ is not constant on any nonempty open subset (such a constancy would make $\varphi ={\mathrm{\Phi }}_a^{-1}\circ {\varphi }_1$ constant there, and then $\varphi$, being holomorphic on the connected surface $X$, would be constant by the clopenness in [F9], contradicting $\varphi (p_0)=0\mathrm{\ne }a=\varphi (p_1)$); hence [F9] makes its zero set $Z_1:={\varphi }_1^{-1}(0)=\{x\in X:\varphi (x)=a\}$ closed, and every point of $Z_1$ has an open neighbourhood meeting $Z_1$ only in that point, so that $Z_1$ is locally finite.

Marshall's maximum-principle inequality. Let $v\in {\mathcal{F}}_{p_1}(X)$ and $\epsilon >0$, and put $u_{\epsilon }:=v+(1+\epsilon )\log \mathrm{\mid }{\varphi }_1\mathrm{\mid }$ on $X\setminus Z_1$; note that $X\setminus Z_1\subseteq X\setminus \{p_1\}$ because ${\varphi }_1(p_1)=0$. We show $u_{\epsilon }\le 0$ on $X\setminus Z_1$. (i) $u_{\epsilon }$ is subharmonic on $X\setminus Z_1$: the restriction of $v$ is subharmonic there, $(1+\epsilon )\log \mathrm{\mid }{\varphi }_1\mathrm{\mid }$ is harmonic on $X\setminus Z_1$ by [F10] and [F7], and a nonnegative multiple of a harmonic function is subharmonic while sums of subharmonic functions are subharmonic [F7]. (ii) Let $\widehat{u}:=\max (u_{\epsilon },0)$ on $X\setminus Z_1$ and $\widehat{u}:=0$ on $Z_1$. Near a point $z\in Z_1$ with $z\mathrm{\ne }{p}_1$, the function $v$ is upper semicontinuous and finite at $z$, hence bounded above on a neighbourhood of $z$ [F7], while $\log \mathrm{\mid }{\varphi }_1(x)\mathrm{\mid }\to -\mathrm{\infty }$ as $x\to z$ by [F10]; near $z=p_1$ the unit pole condition $v\le -\log \mathrm{\mid }w\mathrm{\mid }+C$ in a centred chart $w$ at $p_1$ [F2] and the factorisation ${\varphi }_1=w^{k}G$ with $k\ge 1$ and $G(p_1)\mathrm{\ne }0$ [F9] give $u_{\epsilon }\le (k(1+\epsilon )-1)\log \mathrm{\mid }w\mathrm{\mid }+C^{\mathrm{\prime }}$, which tends to $-\mathrm{\infty }$ because $k(1+\epsilon )>1$. Hence $u_{\epsilon }<0$, and so $\widehat{u}=0$, on a neighbourhood of every point of $Z_1$; consequently $\widehat{u}$ is upper semicontinuous on $X$, and near every point of $Z_1$ it is the constant $0$, hence subharmonic there, so by the locality of subharmonicity [F6] the function $\widehat{u}$ is subharmonic on $X$. (iii) The function $\widehat{u}$ is nonnegative, and it vanishes on the nonempty open set $X\setminus K$, where $K$ is a compact support of $v$ [F2]; since $X$ is noncompact by step 1.1, $X\setminus K$ is nonempty. The upper semicontinuous $\widehat{u}$ is bounded above on compact $K$: its open strict sublevels at positive integer thresholds cover $K$, so a finite subcover gives a finite upper bound; outside $K$ it is zero. Let $M:={\sup }_X\widehat{u}<\mathrm{\infty }$. If $M>0$ then for every $n\ge 0$ the set $F_n:=\{x\in K:\widehat{u}(x)\ge M-1\mathrm{/}(n+1)\}$ is a nonempty closed subset of $K$, these sets decrease, and compactness of $K$ [F12] gives a point $x^{\ast }\in {\bigcap }_n{F}_n$ (otherwise the increasing open sets $K\setminus F_n$ would cover the compact $K$ with no finite subcover), that is, $\widehat{u}(x^{\ast })=M$; the chartwise strong maximum principle [F7] applied on the connected surface $X$ then makes $\widehat{u}\equiv M$ constant, contradicting $\widehat{u}=0$ on $X\setminus K\mathrm{\ne }\mathrm{\varnothing }$. Hence $M=0$ and $u_{\epsilon }\le 0$ on $X\setminus Z_1$. (iv) Fix $p\in X\setminus Z_1$. Since $u_{\epsilon }(p)\le 0$ for every $v\in {\mathcal{F}}_{p_1}(X)$ and every $\epsilon >0$, the definition of the envelope as a supremum [F2] gives $g_X(p,p_1)\le -(1+\epsilon )\log \mathrm{\mid }{\varphi }_1(p)\mathrm{\mid }$; letting $\epsilon \downarrow 0$ yields $g_X(p,p_1)\le -\log \mathrm{\mid }{\varphi }_1(p)\mathrm{\mid }<+\mathrm{\infty }$.

The complement of $Z_1$ is connected. $Z_1$ is closed and locally finite by step 3.1, and it has empty interior because every point of $Z_1$ has a neighbourhood meeting $Z_1$ only in that point; hence $X\setminus Z_1$ is dense in $X$ [F12]. Suppose $X\setminus Z_1=A\bigsqcup B$ with $A,B$ nonempty disjoint open subsets of $X\setminus Z_1$. Since $Z_1$ is closed, $A$ and $B$ are open in $X$ [F12]; since $X\setminus Z_1$ is dense, $X=\overline{A\cup B}=\bar{A}\cup \bar{B}$ [F12], and because $X$ is connected [F1] the two nonempty closed sets $\bar{A},\bar{B}$ cannot be disjoint, so there is $z\in \bar{A}\cap \bar{B}$. The point $z$ lies in $Z_1$: it is not in $A$ (else the open set $A$ would meet $B$, as $z\in \bar{B}$), and symmetrically not in $B$; so $z\in Z_1$. Choose a chart $\psi :W\to \mathbb{C}$ of $X$ at $z$ with $\psi (W)$ an open ball and $W\cap Z_1=\{z\}$, possible because $Z_1$ is locally finite and charts can be shrunk [F1, F12]. Then $W\setminus \{z\}=W\setminus Z_1=(A\cap W)\bigsqcup (B\cap W)$, both parts are nonempty because $z$ lies in the closure of both $A$ and $B$, and both are open in $W\setminus \{z\}$; so $W\setminus \{z\}$ would be disconnected. But $\psi$ carries $W\setminus \{z\}$ homeomorphically onto the punctured ball $\psi (W)\setminus \{\psi (z)\}$, which is connected by [F11]. This contradiction shows that no separation exists, so $X\setminus Z_1$ is connected.

Every pole has a finite kernel. The complement $X\setminus Z_1$ is nonempty, because $Z_1$ is locally finite and no neighbourhood of a point of a surface consists of a single point [F1, F12]; so step 4.1 exhibits a point where the envelope with pole $p_1$ is finite. By the dichotomy of [F3] the envelope with pole $p_1$ is then finite everywhere, so $X$ admits a finite canonical Green kernel at $p_1$; since $p_1\in X\setminus \{p_0\}$ was arbitrary, this holds at every point of $X$. In particular, for every $p_1\mathrm{\ne }{p}_0$ both kernels $g_X(\cdot ,p_0)$ and $g_X(\cdot ,p_1)$ are finite, and [F5] gives the symmetry $g_X(p_0,p_1)=g_X(p_1,p_0)$.

The harmonic difference and its value at $p_0$. Fix $p_1\in X\setminus \{p_0\}$ and let $Z_1$ and ${\varphi }_1$ be as in step 3.1. By steps 4.1 and 5.1, $h:=g_X(\cdot ,p_1)+\log \mathrm{\mid }{\varphi }_1\mathrm{\mid }$ is defined on $X\setminus Z_1$, satisfies $h\le 0$ there, and is harmonic there, because $g_X(\cdot ,p_1)$ is harmonic on $X\setminus \{p_1\}\supseteq X\setminus Z_1$ [F3, step 5.1] and $\log \mathrm{\mid }{\varphi }_1\mathrm{\mid }$ is harmonic on $X\setminus Z_1$ [F10]. At $p_0\mathrm{\notin }{Z}_1$ one has ${\varphi }_1(p_0)=-\varphi (p_1)$, so $\log \mathrm{\mid }{\varphi }_1(p_0)\mathrm{\mid }=\log \mathrm{\mid }\varphi (p_1)\mathrm{\mid }=-g_0(p_1)=-g_X(p_1,p_0)$ by step 2.1; with the symmetry of step 5.1, $h(p_0)=g_X(p_0,p_1)-g_X(p_1,p_0)=0$.

$h$ vanishes identically. The function $h$ of step 6.1 is harmonic on the connected open set $X\setminus Z_1$ (step 4.2), satisfies $h\le 0$ there and $h(p_0)=0$; so $h$ attains its finite maximum at the interior point $p_0$, and the chartwise strong maximum principle [F7] gives $h\equiv 0$ on $X\setminus Z_1$.

The zero set of ${\varphi }_1$ is a single point. Suppose $z\in Z_1$ with $z\mathrm{\ne }{p}_1$. By step 3.1 there is an open neighbourhood $N$ of $z$ with $N\cap Z_1=\{z\}$, and $z$ is not a pole of $g_X(\cdot ,p_1)$, so $g_X(\cdot ,p_1)$ is continuous at $z$ [F3, F7]; shrinking $N$ within a chart we may assume $\mathrm{\mid }{g}_X(x,p_1)\mathrm{\mid }\le C$ on $N$ for some $C\ge 1$. Since ${\varphi }_1$ is continuous with ${\varphi }_1(z)=0$ [F1], after shrinking $N$ further we have $\mathrm{\mid }{\varphi }_1(x)\mathrm{\mid }<e^{-2C}$ for all $x\in N$, hence $h(x)\le C-2C=-C<0$ for every $x\in N\setminus \{z\}$, a set which is nonempty; but $N\setminus \{z\}\subseteq X\setminus Z_1$, where $h\equiv 0$ by step 7.1. This contradiction shows $Z_1=\{p_1\}$.

$\varphi$ is injective. Let $a,b\in X$ with $\varphi (a)=\varphi (b)$. If $\varphi (b)=0$ then $b=p_0$ and $\varphi (a)=0$ give $a=p_0$, so $a=b$ by step 2.1. Otherwise $c:=\varphi (b)\mathrm{\ne }0$, so $b\mathrm{\ne }{p}_0$; apply the construction of step 3.1 with $p_1:=b$, which gives ${\varphi }_1(a)=\frac{\varphi (a)-\varphi (b)}{1-\overline{\varphi (b)}\varphi (a)}=\frac{0}{1-\mathrm{\mid }c{\mathrm{\mid }}^2}=0$, that is, $a\in Z_1$; step 8.1 then gives $a=b$. Hence $\varphi$ is injective.

The image is a complex domain and $\varphi$ is a biholomorphism onto it. Let $\mathrm{\Omega }:=\varphi (X)$. For a point $a\in X$ choose a centred chart $z_a:U_a\to \mathbb{D}$ at $a$ [F1]. The chart expression $\varphi \circ z_a^{-1}:\mathbb{D}\to \mathbb{C}$ is holomorphic and injective (step 9.1), so by [F16] it is biholomorphic onto its open image $\varphi (U_a)$ and has nowhere-zero derivative; in particular $\varphi (U_a)$ is open and the inverse of $\varphi {\mathrm{\mid }}_{U_a}$ is holomorphic. The sets $U_a$ cover $X$, so $\mathrm{\Omega }={\bigcup }_a\varphi (U_a)$ is open; it is connected as a continuous image of the connected space $X$ [F12], and nonempty, while $\mathrm{\Omega }\subseteq \mathbb{D}⊊\mathbb{C}$ by step 2.1. Hence $\mathrm{\Omega }$ is a complex domain [F16], and the bijection $\varphi :X\to \mathrm{\Omega }$ (step 9.1) has a holomorphic inverse, since holomorphy is a local condition [F1]; thus $\varphi$ is a biholomorphism onto $\mathrm{\Omega }$.

Every based loop of $\mathrm{\Omega }$ is null as an element of the fundamental group. Let $\gamma$ be a loop in $\mathrm{\Omega }$ based at $y_0\in \mathrm{\Omega }$, and put $x_0:={\varphi }^{-1}(y_0)$ and $\tilde{\gamma }:={\varphi }^{-1}\circ \gamma$, continuous by step 10.1; then $\tilde{\gamma }$ is a loop in $X$ based at $x_0$, and $[\tilde{\gamma }]$ is the identity element of ${\pi }_1(X,x_0)$ because $X$ is simply connected [F13]. By the functoriality of the fundamental group [F13], $[\gamma ]=[\varphi \circ \tilde{\gamma }]={\varphi }_{\ast }[\tilde{\gamma }]={\varphi }_{\ast }(\text{identity})=\text{identity}$ in ${\pi }_1(\mathrm{\Omega },y_0)$; since $\gamma$ was an arbitrary based loop, every based loop of $\mathrm{\Omega }$ represents the identity class.

The image is homologically simply connected. By step 10.1, $\mathrm{\Omega }$ is a complex domain and by step 11.1 every based loop of $\mathrm{\Omega }$ represents the identity class in its fundamental group; hence $\mathrm{\Omega }$ is homologically simply connected by [F14].

Riemann mapping and conclusion. The domain $\mathrm{\Omega }⊊\mathbb{C}$ is proper and homologically simply connected by steps 10.1 and 12.1, so the Riemann mapping theorem [F15] provides a biholomorphic map $f:\mathrm{\Omega }\to \mathbb{D}$ normalised at $z_0:=\varphi (p_0)$. The composite $f\circ \varphi :X\to \mathbb{D}$ is then bijective and holomorphic with holomorphic inverse, being a composite of biholomorphisms [F1, F16]; in other words $X$ is biholomorphic to the unit disc. The Axiom of Choice [A1] is used exactly through [F15] and through the Countable Choice consumed by the envelope suppliers [F3] in steps 1.1, 4.1 and 5.1; the remaining selections are finite.