Symmetry of the canonical surface Green kernel

A compact bordered domain is connected after removing a closed disc. Let $S$ be a connected surface, that is, a nonempty connected open subset of a Riemann surface $X$ carrying the restricted complex structure and topology [F1], and let $C\subseteq S$ be a closed disc, that is, $C={\psi }^{-1}(\bar{B}(b,\sigma ))$ for a chart $\psi$ of $S$ and a radius $\sigma >0$ with $\bar{B}(b,\sigma )\subseteq \psi (\mathrm{dom}\psi )$, and suppose $S\setminus C\mathrm{\ne }\mathrm{\varnothing }$. Then $S\setminus C$ is connected. Indeed, put $\gamma :=\mathrm{\partial }C={\psi }^{-1}(\mathrm{\partial }B(b,\sigma ))$, a compact connected subset of $S$ [F12, F13]; if $S\setminus C=A\bigsqcup B$ with $A,B$ nonempty open in $S\setminus C$, then $S=C^{\circ }\bigsqcup \gamma \bigsqcup A\bigsqcup B$, every point of $\gamma$ lies in $\bar{A}\cup \bar{B}$ (a chart disc around it meets both the inside $C^{\circ }$ and the outside $A\cup B$), and these two traces are disjoint: at each boundary point a sufficiently small exterior half-disc is connected and lies entirely in one side of the separation. Since the traces are closed and cover the connected circle $\gamma$, one of them is all of $\gamma$; if $\gamma \subseteq \bar{A}$ then $\bar{B}$ meets neither $\gamma$ (the traces are disjoint) nor the open sets $A,C^{\circ }$. Hence $\bar{B}=B$, so $B$ is open and closed in $S$, and $B\mathrm{\ne }\mathrm{\varnothing }$ forces $B=S$, hence $A=\mathrm{\varnothing }$, a contradiction; the other case is symmetric. Hence $S\setminus C$ is connected.

Monotonicity in the exhaustion. Let $m\le n$ and let $v\in {\mathcal{F}}_p({\mathrm{\Omega }}_m)$. The function $v$ vanishes off a compact set $K\subseteq {\mathrm{\Omega }}_m$ with $K\cap \mathrm{\partial }{\mathrm{\Omega }}_m=\mathrm{\varnothing }$, so its extension $\tilde{v}$ by $0$ outside ${\mathrm{\Omega }}_m$ is subharmonic on $X\setminus \{p\}$ by locality [F7]; it is nonnegative, vanishes off the same compact $K\mathrm{\ne }X$, and keeps the unit logarithmic pole at $p$, so $\tilde{v}\in {\mathcal{F}}_p({\mathrm{\Omega }}_n)$ after restriction to ${\mathrm{\Omega }}_n$. Hence $v(x)\le g_{{\mathrm{\Omega }}_n}(x,p)$ for $x\in {\mathrm{\Omega }}_m\setminus \{p\}$, and taking the supremum over $v$ gives $g_{{\mathrm{\Omega }}_m}\le g_{{\mathrm{\Omega }}_n}$ on ${\mathrm{\Omega }}_m\setminus \{p\}$.

Compact surfaces have no finite canonical kernel. Suppose first that $X$ is compact and that $p_0\in X$ has a finite canonical Green kernel $g:=g_X(\cdot ,p_0)$. Fix a centred chart $z:U\to \mathbb{D}$ at $p_0$ and for $0<s<1$ put $D_s:=\{x\in U:\mathrm{\mid }z(x)\mathrm{\mid }<s\}$ and ${\mathrm{\Omega }}_s^{\mathrm{\prime }}:=X\setminus D_s$, a nonempty connected compact bordered domain with boundary $\mathrm{\partial }{\mathrm{\Omega }}_s^{\mathrm{\prime }}=\mathrm{\partial }{D}_s$ (connectedness is the complement-of-a-closed-disc argument of step 1.1 applied in the compact surface $X$). The functions $u:=g$ on $X\setminus \{p_0\}$ and $v:=1$ have $C^{\mathrm{\infty }}$ chart expressions near ${\mathrm{\Omega }}_s^{\mathrm{\prime }}$ [F16], $v$ is harmonic, and $g$ is harmonic on $X\setminus \{p_0\}\supseteq {\mathrm{\Omega }}_s^{\mathrm{\prime }}$ with $g+\log \mathrm{\mid }z\mathrm{\mid }$ harmonic on $U$ [F3]. Part 1 of [F6] applied to ${\mathrm{\Omega }}_s^{\mathrm{\prime }}$ gives $0={\int }_{{\mathrm{\Omega }}_s^{\mathrm{\prime }}}(g\mathrm{\Delta }1-1\mathrm{\Delta }g)dA={\int }_{\mathrm{\partial }{D}_s}(g{\mathrm{\partial }}_{\nu }1-1{\mathrm{\partial }}_{\nu }g)ds=-{\int }_{\mathrm{\partial }{D}_s}{\mathrm{\partial }}_{\nu }gds$, where $\nu$ is the outward conormal of ${\mathrm{\Omega }}_s^{\mathrm{\prime }}$ at $\mathrm{\partial }{\mathrm{\Omega }}_s^{\mathrm{\prime }}$, pointing into the removed disc $D_s$, that is, in the direction of decreasing $s=\mathrm{\mid }z\mathrm{\mid }$. Write $g=-m\log \mathrm{\mid }z\mathrm{\mid }+h$ with $m=1$ and $h$ harmonic on all of $U$ [F3], so that on the circle $\mathrm{\mid }z\mathrm{\mid }=s$ one has ${\mathrm{\partial }}_{\nu }g=-{\mathrm{\partial }}_{s}g=1\mathrm{/}s-{\mathrm{\partial }}_{s}h$. Therefore, with $ds=sd\theta$ on the circle, $0={\int }_{\mathrm{\partial }{D}_s}{\mathrm{\partial }}_{\nu }gds={\int }_0^{2\pi }(1\mathrm{/}s-{\mathrm{\partial }}_{s}h(s{e}^{i\theta }))sd\theta =2\pi -s{\int }_0^{2\pi }{\mathrm{\partial }}_{s}h(s{e}^{i\theta })d\theta$ for every $0<s<1$. Since $h$ is $C^{\mathrm{\infty }}$ near $0$ [F16], ${\mathrm{\partial }}_{s}h$ is bounded near $0$, so the last term tends to $0$ as $s\downarrow 0$, while the left-hand side is the constant $0$; taking the limit gives $0=2\pi$, a contradiction. Hence no compact $X$ admits a finite canonical Green kernel, and by [F3] the envelope of every compact $X$ is identically $+\mathrm{\infty }$.

The zero-boundary kernel of a compact bordered domain exists. Let $\mathrm{\Omega }\subseteq X$ be a connected relatively compact smooth-bordered domain with $\mathrm{\Omega }=\mathrm{int}\bar{\mathrm{\Omega }}$ and $\mathrm{\partial }\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, and let $p\in \mathrm{\Omega }$. We show that $g_{\mathrm{\Omega }}(\cdot ,p)$ is finite on $\mathrm{\Omega }\setminus \{p\}$, harmonic there with a unit logarithmic pole at $p$, and tends to $0$ at $\mathrm{\partial }\mathrm{\Omega }$. Choose a centred coordinate $z$ on a neighbourhood of a closed disc ${\bar{U}}_1\subseteq \mathrm{\Omega }$, scaled so that $U_1=\{\mathrm{\mid }z\mathrm{\mid }<1\}$, and fix $0<r<1$, put $r{U}_1:=\{x\in U_1:\mathrm{\mid }z(x)\mathrm{\mid }<r\}$ and $V:=\mathrm{\Omega }\setminus \overline{r{U}_1}$, a nonempty connected relatively compact smooth-bordered domain with boundary $\mathrm{\partial }V=\mathrm{\partial }\mathrm{\Omega }\bigsqcup \mathrm{\partial }(r{U}_1)$: connectedness follows from step 1.1 applied in the connected surface $\mathrm{\Omega }$ to the closed disc $\overline{r{U}_1}$, whose complement in $\mathrm{\Omega }$ is nonempty because $\overline{r{U}_1}\subseteq U_1\subseteq {\bar{U}}_1⫋\mathrm{\Omega }$. If $X$ is compact, choose $a_0\in X\setminus \bar{\mathrm{\Omega }}$; this set is nonempty since $\mathrm{\Omega }=\mathrm{int}\bar{\mathrm{\Omega }}$ has nonempty boundary. The punctured surface $X\setminus \{a_0\}$ is connected by the punctured-disc separation argument and noncompact since $a_0$ is not isolated; it contains $\bar{V}$ compactly. Apply [F4] in that ambient surface in the compact case, and in $X$ otherwise. There is a continuous $\omega :\bar{V}\to \mathbb{R}$, harmonic on $V$, with $\omega =1$ on $\mathrm{\partial }(r{U}_1)$ and $\omega =0$ on $\mathrm{\partial }\mathrm{\Omega }$.

The exhaustion kernels increase to the canonical kernel. Assume now that $X$ admits a finite canonical Green kernel at $p$, so that $g_X(\cdot ,p)<\mathrm{\infty }$ on $X\setminus \{p\}$ and, by [F3], $g_X(\cdot ,p)$ is harmonic there with a unit logarithmic pole at $p$. Each $g_{{\mathrm{\Omega }}_n}(\cdot ,p)$ is dominated by $g_X(\cdot ,p)$ on ${\mathrm{\Omega }}_n\setminus \{p\}$: by step 1.2 the extension by zero of any $v\in {\mathcal{F}}_p({\mathrm{\Omega }}_n)$ lies in ${\mathcal{F}}_p(X)$, so $v\le g_X(\cdot ,p)$ and hence $g_{{\mathrm{\Omega }}_n}(\cdot ,p)\le g_X(\cdot ,p)$. Conversely, let $x\in X\setminus \{p\}$ and let $v\in {\mathcal{F}}_p(X)$; its support $K$ is compact, so $K\subseteq {\mathrm{\Omega }}_N$ for some $N$ because the exhaustion is increasing with union $X$, and then $v$ restricts to a member of ${\mathcal{F}}_p({\mathrm{\Omega }}_N)$ (it is subharmonic on ${\mathrm{\Omega }}_N\setminus \{p\}$, nonnegative, vanishes off $K\subseteq {\mathrm{\Omega }}_N$ with $K\mathrm{\ne }{\mathrm{\Omega }}_N$, and has the unit pole), so $v(x)\le g_{{\mathrm{\Omega }}_N}(x,p)\le {\sup }_n{g}_{{\mathrm{\Omega }}_n}(x,p)$. Taking the supremum over $v\in {\mathcal{F}}_p(X)$ gives $g_X(x,p)\le {\sup }_n{g}_{{\mathrm{\Omega }}_n}(x,p)\le g_X(x,p)$, so the increasing sequence $g_{{\mathrm{\Omega }}_n}(x,p)$ converges to $g_X(x,p)$ for every $x\in X\setminus \{p\}$.

The hypothesis of part 1 forces noncompactness and produces an exhaustion. If $X$ admits finite canonical Green kernels at the distinct points $p$ and $q$, then $X$ is not compact by step 2.1, so ${\mathrm{A}\mathrm{C}}_{\omega }$ and [F4] provide a regular exhaustion ${\mathrm{\Omega }}_1\subseteq {\mathrm{\Omega }}_2\subseteq \cdots$ with $\overline{{\mathrm{\Omega }}_n}\subseteq {\mathrm{\Omega }}_{n+1}$, $X={\bigcup }_n{\mathrm{\Omega }}_n$, and each ${\mathrm{\Omega }}_n$ connected, relatively compact, smooth-bordered with ${\mathrm{\Omega }}_n=\mathrm{int}\overline{{\mathrm{\Omega }}_n}$ and $\mathrm{\partial }{\mathrm{\Omega }}_n\mathrm{\ne }\mathrm{\varnothing }$. Since $\{p,q\}$ is compact and the ${\mathrm{\Omega }}_n$ increase to $X$, some index $N$ has $p,q\in {\mathrm{\Omega }}_N$; discarding the first $N-1$ domains and relabelling gives an exhaustion with $p,q\in {\mathrm{\Omega }}_1$. Fix such an exhaustion for the rest of the proof; it exists in part 1 whenever the symmetry hypothesis holds, and in part 2 it is assumed.

Basic properties of the barrier and of the candidates. With $\omega$ as in step 2.2, the functions $\omega$ and $1-\omega$ are subharmonic on $V$ [F5, F10], so the boundary maximum principle (proved as in the companion argument of this batch: a subharmonic function on a nonempty proper connected open subset of $X$ with compact closure and boundary limsup at most $0$ is at most $0$, by the chartwise strong maximum principle [F8] applied to a maximising sequence) gives $0\le \omega \le 1$ on $V$. Moreover $\omega$ is harmonic on the connected $V$ and satisfies $0\le \omega \le 1$. If $1-\omega (x)=0$ for some $x\in \mathrm{\partial }{U}_1$, then $\omega$ attains its finite maximum $1$ at the interior point $x\in V$ (because $\mathrm{\partial }{U}_1\subseteq V$: indeed $\mathrm{\partial }{U}_1\cap \overline{r{U}_1}=\mathrm{\varnothing }$ and $\mathrm{\partial }{U}_1\subseteq {\bar{U}}_1\subseteq \mathrm{\Omega }$), so $\omega \equiv 1$ on $V$ by [F8]; by continuity $\omega =1$ on $\mathrm{\partial }\mathrm{\Omega }$, contradicting $\omega =0$ there. Hence ${\max }_{\mathrm{\partial }{U}_1}\omega \le 1-\delta$ for $\delta :={\min }_{\mathrm{\partial }{U}_1}(1-\omega )>0$. Second, for every $v\in {\mathcal{F}}_p(\mathrm{\Omega })$ and every $\epsilon >0$ the modified function $x\mapsto v(x)+(1+\epsilon )\log \mathrm{\mid }z(x)\mathrm{\mid }$ on $U_1\setminus \{p\}$ extends to an upper semicontinuous subharmonic function on $U_1$ with the value $-\mathrm{\infty }$ at $p$: subharmonicity on $U_1\setminus \{p\}$ follows from [F5], [F9], [F10] and [F15], and upper semicontinuity at $p$ from the unit pole condition of [F2], which gives $v_z(\zeta )\le -\log \mathrm{\mid }\zeta \mathrm{\mid }+C$, hence $v_z(\zeta )+(1+\epsilon )\log \mathrm{\mid }\zeta \mathrm{\mid }\le \epsilon \log \mathrm{\mid }\zeta \mathrm{\mid }+C\to -\mathrm{\infty }$. Subharmonicity across the centre follows by the decreasing finite-max truncation argument in the proof of [F3].

Symmetry on a compact bordered domain. Fix $n$ and put $G_p:=g_{{\mathrm{\Omega }}_n}(\cdot ,p)$ and $G_q:=g_{{\mathrm{\Omega }}_n}(\cdot ,q)$. By step 2.2 both extend continuously with value zero to the boundary, are harmonic off their poles and zero on $\mathrm{\partial }{\mathrm{\Omega }}_n$. Smooth-boundary Dirichlet regularity [F17] gives $C^2$ chart expressions up to $\mathrm{\partial }{\mathrm{\Omega }}_n$; away from that boundary, harmonicity gives interior $C^{\mathrm{\infty }}$ regularity [F16]. Choose small disjoint closed coordinate discs $D_p,D_q$ about $p,q$. The functions therefore meet the $C^2$-near-closure hypothesis in the punctured second Green identity [F6] on ${\mathrm{\Omega }}_K:={\mathrm{\Omega }}_n\setminus (D_p^{\circ }\cup D_q^{\circ })$, and both vanish on $\mathrm{\partial }{\mathrm{\Omega }}_n$. Thus [F6] gives ${\int }_{\mathrm{\partial }{D}_p}(G_p{\mathrm{\partial }}_{\nu }{G}_q-G_q{\mathrm{\partial }}_{\nu }{G}_p)ds+{\int }_{\mathrm{\partial }{D}_q}(G_p{\mathrm{\partial }}_{\nu }{G}_q-G_q{\mathrm{\partial }}_{\nu }{G}_p)ds=0,$ with the conormal outward from ${\mathrm{\Omega }}_K$. Near $p$, write $G_p=-\log \mathrm{\mid }{z}_p\mathrm{\mid }+h_p$ with $h_p$ harmonic and bounded, while $G_q=G_q(p)+O(s)$ and ${\mathrm{\partial }}_{\nu }{G}_q=O(1)$ on $\mathrm{\mid }{z}_p\mathrm{\mid }=s$. Since ${\mathrm{\partial }}_{\nu }{G}_p=1\mathrm{/}s-{\mathrm{\partial }}_s{h}_p$, the first integral is $-2\pi G_q(p)+O(s\log (1\mathrm{/}s))$. The same computation with $p,q$ interchanged shows that the second integral is $2\pi G_p(q)+O(s\log (1\mathrm{/}s))$. Letting $s\downarrow 0$ yields $g_{{\mathrm{\Omega }}_n}(p,q)=g_{{\mathrm{\Omega }}_n}(q,p)$.

The two elementary inequalities. Let $v\in {\mathcal{F}}_p(\mathrm{\Omega })$, put $M_v:={\max }_{\mathrm{\partial }(r{U}_1)}v$ and define $w:=v+(1+\epsilon )\log \mathrm{\mid }z\mathrm{\mid }$ on $U_1$ as in step 3.2. Applying the boundary-value maximum principle (a subharmonic function on a compactly contained nonempty proper domain, extended upper semicontinuously to the closure, has ${\sup }_{{\bar{U}}_1}w={\max }_{\mathrm{\partial }{U}_1}w$; this is step 3.2's maximum principle applied to $w$ and to sequences maximising $w$ on ${\bar{U}}_1$) to $w$ on the disc $U_1$, and using $w=v$ on $\mathrm{\partial }{U}_1$ and $w=v+(1+\epsilon )\log r$ on $\mathrm{\partial }(r{U}_1)$, gives ${\max }_{\mathrm{\partial }(r{U}_1)}v+(1+\epsilon )\log r\le {\max }_{\mathrm{\partial }{U}_1}v$; letting $\epsilon \downarrow 0$, ${\max }_{\mathrm{\partial }(r{U}_1)}v\le {\max }_{\mathrm{\partial }{U}_1}v+\log (1\mathrm{/}r)$. On the other hand the subharmonic function $u:=v-M_v\omega$ on $V$ satisfies ${\mathrm{lim\hspace{0.17em}sup}}_{x\to \zeta }u(x)\le 0$ at every $\zeta \in \mathrm{\partial }V$ (the limsup being that of the upper semicontinuity definition [F11]): at $\zeta \in \mathrm{\partial }(r{U}_1)$ one has $\omega (\zeta )=1$ and $v$ is upper semicontinuous at $\zeta$ with $v(\zeta )\le M_v$, while at $\zeta \in \mathrm{\partial }\mathrm{\Omega }$ the candidate $v$ vanishes on a neighbourhood of $\zeta$ by its compact support [F2] and $\omega (\zeta )=0$; hence the boundary maximum principle of step 3.2 gives $v\le M_v\omega$ on $V$, and evaluating on $\mathrm{\partial }{U}_1\subseteq V$ gives ${\max }_{\mathrm{\partial }{U}_1}v\le (1-\delta ){\max }_{\mathrm{\partial }(r{U}_1)}v$. Adding the two inequalities gives $\delta {\max }_{\mathrm{\partial }(r{U}_1)}v\le \log (1\mathrm{/}r)$, so that $M_v\le C:={\delta }^{-1}\log (1\mathrm{/}r)$ for every $v\in {\mathcal{F}}_p(\mathrm{\Omega })$.

Conclusion of step 2.2. Fix $q_1\in \mathrm{\partial }{U}_1\subseteq V$. For $v\in {\mathcal{F}}_p(\mathrm{\Omega })$, step 4.1 gives $v(q_1)\le M_v\omega (q_1)\le C$, hence $g_{\mathrm{\Omega }}(q_1,p)\le C<+\mathrm{\infty }$; the dichotomy [F3] now shows that the envelope is finite everywhere on $\mathrm{\Omega }\setminus \{p\}$, harmonic and positive there, with a unit logarithmic pole at $p$. Moreover for $x\in V$ the comparison $v(x)\le M_v\omega (x)\le C\omega (x)$ gives $g_{\mathrm{\Omega }}(x,p)\le C\omega (x)$, and $\omega$ is continuous on $\bar{V}$ with $\omega =0$ on $\mathrm{\partial }\mathrm{\Omega }$, so ${\lim }_{x\to \zeta ,x\in \mathrm{\Omega }}{g}_{\mathrm{\Omega }}(x,p)=0$ for every $\zeta \in \mathrm{\partial }\mathrm{\Omega }$ because the envelope is nonnegative and bounded above by a function tending to $0$.

Symmetry on $X$. Under the hypothesis of part 1, steps 3.1 and 2.3 give $g_X(p,q)={\lim }_n{g}_{{\mathrm{\Omega }}_n}(p,q)={\lim }_n{g}_{{\mathrm{\Omega }}_n}(q,p)=g_X(q,p)$, the middle equality by step 3.3. This proves the symmetry assertion for any Riemann surface with finite canonical kernels at two distinct poles, and step 2.3 proves the exhaustion approximation.