Canonical Green kernel on a Riemann surface

Let $X$ be a Riemann surface (Riemann surfaces and holomorphic atlases) and let $p\in X$ be a point. Harmonicity and subharmonicity on open subsets of $X$ are the chartwise notions of Chartwise harmonic and subharmonic functions on a Riemann surface.

Centred charts. A centred chart at $p$ is a chart $z:U\to \mathbb{D}$ of the complex structure with $z(p)=0$ and with closure $\bar{U}$ compact in $X$. Centred charts exist: a chart about $p$ may be post-composed with a Möbius automorphism of the disc carrying the image of $p$ to $0$, and the domain may then be shrunk so that its closure is compact; replacing $z$ by $z\mathrm{/}r$ for $0<r<1$ shrinks the domain around $p$, so one may also arrange $\bar{U}\mathrm{\ne }X$.

The Perron family. Let ${\mathcal{F}}_p={\mathcal{F}}_p(X)$ be the set of functions $v:X\setminus \{p\}\to [0,\mathrm{\infty })$ such that

1. $v$ is subharmonic on $X\setminus \{p\}$;
2. $v$ has compact support: $v=0$ on $X\setminus K$ for some compact set $K\subseteq X$;
3. $v$ has at most a unit logarithmic pole at $p$: for one, hence every, centred chart $z:U\to \mathbb{D}$ at $p$, ${\mathrm{lim\hspace{0.17em}sup}}_{q\to p}(v(q)+\log \mathrm{\mid }z(q)\mathrm{\mid })<\mathrm{\infty }\mathrm{.}$

Clause 3 does not depend on the centred chart. Indeed, if $w:V\to \mathbb{D}$ is a second centred chart, then $\tau :=w\circ z^{-1}$ is a biholomorphism between neighbourhoods of $0$ with $\tau (0)=0$ and ${\tau }^{\mathrm{\prime }}(0)\mathrm{\ne }0$, so $\mathrm{\mid }w(q)\mathrm{\mid }=\mathrm{\mid }{\tau }^{\mathrm{\prime }}(0)\mathrm{\mid }\mathrm{\mid }z(q)\mathrm{\mid }(1+o(1))$ and hence $\log \mathrm{\mid }w(q)\mathrm{\mid }=\log \mathrm{\mid }z(q)\mathrm{\mid }+\log \mathrm{\mid }{\tau }^{\mathrm{\prime }}(0)\mathrm{\mid }+o(1)$ as $q\to p$; therefore $v+\log \mathrm{\mid }w\mathrm{\mid }$ and $v+\log \mathrm{\mid }z\mathrm{\mid }$ differ by a quantity that is bounded above and below near $p$, and the two conditions are equivalent. The family consists of nonnegative functions by construction; The canonical Green kernel of a plane domain uses the same unit coefficient $\log \mathrm{\mid }z\mathrm{\mid }$ for its plane candidates, and the two normalizations are compared in the remarks below.

When $X$ is compact, this definition allows $K=X$. This is an explicit compact-surface extension of Marshall's Perron family, whose source definition requires $K\mathrm{\ne }X$; allowing the full compact support keeps the candidate family stable under finite maxima and makes the compact case identically infinite.

The envelope. For $q\in X\setminus \{p\}$ set $g_X(q,p):=\sup \{v(q):v\in {\mathcal{F}}_p\}\in [0,\mathrm{\infty }]\mathrm{.}$ The function $g_X(\cdot ,p)$ is the Perron envelope with pole $p$. It is well defined because clauses 1-3 are chart-independent, and it is nonnegative because every member of ${\mathcal{F}}_p$ is.

Canonical kernel and Greenian surfaces. Suppose $g_X(q,p)<\mathrm{\infty }$ for every $q\in X\setminus \{p\}$. Then the envelope is the canonical Green kernel of $X$ with pole $p$, written $g_X(\cdot ,p)$ as well, and one says that $X$ admits a finite canonical Green kernel at $p$. The surface $X$ is Greenian when it admits a finite canonical Green kernel at every point $p\in X$.

The family is nonempty. Choose a centred chart $\xi :V\to D(0,R)$ at $p$ and a radius $0<r<R$ whose closed disc lies in $D(0,R)$; set $U:={\xi }^{-1}(D(0,r))$ and $z:=\xi \mathrm{/}r$. Put $v_0(q)=\{\begin{array}{ll}-\log \mathrm{\mid }z(q)\mathrm{\mid },& q\in U\setminus \{p\},\\ 0,& q\in X\setminus (U\cup \{p\})\mathrm{.}\end{array}$ Then $v_0$ vanishes outside the compact set $\bar{U}$, and $v_0+\log \mathrm{\mid }z\mathrm{\mid }=0$ on $U\setminus \{p\}$, so clauses 2 and 3 hold. For clause 1, on $V\setminus \{p\}$ the chart expression of $v_0$ is $\max \{-\log \mathrm{\mid }\xi \mathrm{/}r\mathrm{\mid },0\}$, the maximum of two harmonic functions. Outside $\bar{U}$ it is locally zero, including near $\mathrm{\partial }V$. Thus every chart expression is subharmonic, by the finite maximum property (Positive linear combinations and finite maxima preserve subharmonicity) and conformal invariance of harmonicity (Plane harmonicity is preserved by holomorphic and antiholomorphic changes of coordinate). Hence $v_0$ is subharmonic on $X\setminus \{p\}$ and belongs to ${\mathcal{F}}_p$; in particular the envelope is at least $v_0\ge 0$ and strictly positive on $U\setminus \{p\}$.

Nonnegativity does not change the envelope. Marshall's family imposes clauses 1-3 on functions $v:X\setminus \{p\}\to [-\mathrm{\infty },\mathrm{\infty })$ without requiring $v\ge 0$. If such a $v$ satisfies clauses 1-3, then $v^{+}:=\max (v,0)$ is subharmonic on $X\setminus \{p\}$ (in each chart this is the maximum of the two subharmonic functions $v_{\phi }$ and $0$, Positive linear combinations and finite maxima preserve subharmonicity), vanishes on $X\setminus K$, and satisfies $v^{+}\le \max (-\log \mathrm{\mid }z\mathrm{\mid }+C,0)\le -\log \mathrm{\mid }z\mathrm{\mid }+\max (C,0)$ near $p$ whenever $v\le -\log \mathrm{\mid }z\mathrm{\mid }+C$ there; hence $v^{+}\in {\mathcal{F}}_p$ and $v^{+}\ge v$. Taking suprema, the envelope over the family above equals the envelope over the family without the nonnegativity requirement. Replacing a candidate by its positive part is therefore harmless, and by the same chartwise argument finite maxima of members of ${\mathcal{F}}_p$ are again members of ${\mathcal{F}}_p$.

Promised properties. The definition above fixes the envelope only. Its basic properties are not assumed here; they are proved in Green envelope dichotomy, logarithmic pole and leastness on a Riemann surface (which assumes Countable Choice). For a fixed pole $p$:

1. either $g_X(q,p)=+\mathrm{\infty }$ for every $q\in X\setminus \{p\}$, or $g_X(\cdot ,p)$ is finite and strictly positive on $X\setminus \{p\}$;
2. in the finite case $g_X(\cdot ,p)$ is harmonic on $X\setminus \{p\}$, and $g_X(\cdot ,p)+\log \mathrm{\mid }z\mathrm{\mid }$ extends from $U\setminus \{p\}$ to a harmonic function on all of $U$;
3. in the finite case $g_X(\cdot ,p)$ is least among the positive harmonic unit-pole functions: $g_X(\cdot ,p)\le H$ for every $H>0$ harmonic on $X\setminus \{p\}$ such that $H+\log \mathrm{\mid }z\mathrm{\mid }$ extends harmonically across $p$.

Normalization and the interface with the plane-domain kernel. By clause 2 of the promise above, in a centred chart the kernel has the local form $g_X(q,p)=-\log \mathrm{\mid }z(q)\mathrm{\mid }+h(q)$ with $h$ harmonic on $U$, so the coefficient of the logarithmic singularity is exactly $1$ and the corrector is finite at $p$. This is precisely the normalization fixed by The canonical Green kernel of a plane domain for plane domains, whose candidates require $u+\log \mathrm{\mid }z-a\mathrm{\mid }$ to extend harmonically across the pole $a$; restricted to the chart disc, $g_X(\cdot ,p)$ is a logarithmic-pole candidate at $0$ for the plane domain $z(U)$ in the sense of that definition. The flux form of the normalization is the one used by the arguments on this page: with $\nu$ the unit normal of the circle $\{\mathrm{\mid }z\mathrm{\mid }=r\}$ pointing toward $p$, one has ${\mathrm{\partial }}_{\nu }(-\log \mathrm{\mid }z\mathrm{\mid })=1\mathrm{/}r$, while a harmonic function has zero flux through a circle (its mean value over the circle is constant in the radius), so ${\int }_{\mathrm{\mid }z\mathrm{\mid }=r}{\mathrm{\partial }}_{\nu }{g}_X(\cdot ,p)ds=2\pi$ for all $0<r<1$. That constant is the unit point-charge normalization recorded on the plane-domain page in the distributional form $-\mathrm{\Delta }g=2\pi {\delta }_a$. The arguments below use this flux form of the normalization, and none of them identifies the surface kernel with a plane-domain Green function.

Continuous candidates. The candidate $v_0$ above is continuous, finite maxima of members of ${\mathcal{F}}_p$ are again in ${\mathcal{F}}_p$ and are continuous when their entries are, and the maximum principle applies to such candidates directly. Marshall records (Comment 3) that the proofs may be carried out with continuous subharmonic candidates alone, applying the maximum principle on a punctured region wherever an isolated value $-\mathrm{\infty }$ would otherwise appear; the explicit candidate $v_0$ above is continuous.