06.11.03 · riemann-surfaces / open-surfaces

Green's function on a Riemann surface and the type problem (parabolic vs. hyperbolic)

shipped3 tiersLean: none

Anchor (Master): Ahlfors-Sario *Riemann Surfaces* (Princeton 1960) Ch. II §5 and Ch. IV; Nevanlinna *Uniformisierung* §VI; Myrberg 1933 (exhaustion construction of the Green's function)

Intuition Beginner

Place a single point charge somewhere on a surface and connect the far edges of the surface to the ground, so any charge that reaches the boundary drains away. The electric potential that settles in is the Green's function. It spikes upward to infinity right at the charge, falls off smoothly everywhere else, and sinks to zero out at the boundary.

For a surface with a real edge, like a flat disc, this works perfectly: the potential is large near the charge and gently fades to zero at the rim. We call such a surface large, or hyperbolic.

But some surfaces are too thin and stretched out to hold a usable potential. Think of the whole flat plane: there is no rim to drain charge into, so the would-be potential never settles down to anything finite. These surfaces are called small, or parabolic.

Why does this concept exist? Every open surface turns out to be exactly one of these two kinds. The Green's function is the tool that decides which.

Visual Beginner

Picture two surfaces side by side. On the left, a disc: a tall spike of potential rises at one interior point and the colour cools smoothly to deep blue, reaching exactly zero all around the circular rim. On the right, the infinite plane: the same spike starts to rise, but with no rim to pin the value to zero, the surrounding colour never cools off, and the attempted potential grows without bound.

The single picture to keep: a usable grounded potential exists on the left and fails to exist on the right. That presence-or-absence is the whole type distinction.

Worked example Beginner

Take the unit disc and put the charge at the centre. The grounded potential is $g (z) = lo g (1/∣ z ∣)$ .

Step 1. Near the centre, $∣ z ∣$ is small, so $1/∣ z ∣$ is large, and the logarithm is large. The potential spikes at the charge, matching the picture.

Step 2. On a circle of radius $r$ , the value is $lo g (1/ r)$ , the same all the way around. As $r$ grows toward $1$ (the rim), $lo g (1/ r)$ falls toward $lo g 1 = 0$ .

Step 3. So the value reaches $0$ exactly at the rim $∣ z ∣ = 1$ . The potential is grounded at the boundary, as required.

What this tells us: the disc supplies a clean finite grounded potential, so the disc is hyperbolic. The same recipe on the full plane fails, because the value $lo g (1/∣ z ∣)$ keeps dropping past zero and runs off to minus infinity with no rim to stop it.

Check your understanding Beginner

Exercise (easy, multiple choice).

A surface is called hyperbolic when:

A. It carries a usable grounded potential (a Green's function) with a single spike

B. It has constant negative curvature everywhere

C. It is compact with no boundary

D. Every function on it is bounded

Hint

The defining test here is whether the grounded point-charge potential settles to something finite.

Answer

A. In this setting hyperbolic means a Green's function exists: the grounded potential of a point charge settles to a finite value with a spike at the charge and value zero at the boundary. Feedback-correct: existence of that potential is the definition used here. Feedback-wrong: B confuses this with a curvature notion; C describes a closed surface, which is a separate case; D is unrelated to the grounded-potential test.

Formal definition Intermediate+

Let $X$ be an open (noncompact, connected) Riemann surface. Fix a point $z_{0} \in X$ and a holomorphic coordinate $z$ centred at $z_{0}$ , so $z (z_{0}) = 0$ .

A Green's function of $X$ with pole at $z_{0}$ is a function $g (\cdot, z_{0}) : X ∖ {z_{0}} \to R$ that is

harmonic on $X ∖ {z_{0}}$ ;
positive, with $g (z, z_{0}) + lo g ∣ z ∣$ extending harmonically across $z_{0}$ (a logarithmic pole: $g (z, z_{0}) = - lo g ∣ z ∣ + O (1)$ near $z_{0}$ );
the least positive function with these two properties — equivalently, with vanishing values along the ideal boundary in the sense made precise below.

The construction proceeds by exhaustion. A regular exhaustion of $X$ is an increasing sequence of relatively compact subregions $$ \Omega_1 \Subset \Omega_2 \Subset \cdots, \qquad \bigcup_n \Omega_n = X, $$ each $Ω_{n}$ bounded by finitely many smooth analytic curves. On each $Ω_{n}$ , the Dirichlet/Perron solution ^{[Ahlfors-Sario Ch. II]} supplies a Green's function $g_{n} (\cdot, z_{0})$ : the harmonic function on $Ω_{n} ∖ {z_{0}}$ with the pole condition (2) and boundary values $g_{n} = 0$ on $\partial Ω_{n}$ . By the minimum principle the $g_{n}$ are positive on $Ω_{n}$ , and because enlarging the region only lowers the grounded values, $g_{n} \leq g_{n + 1}$ wherever both are defined. The sequence increases pointwise to $$ g(z, z_0) := \lim_{n \to \infty} g_n(z, z_0) \in (0, +\infty]. $$

The surface $X$ is hyperbolic if this limit is finite at one (equivalently every) point; the limit $g$ is then the Green's function and is independent of the exhausting sequence. The surface $X$ is parabolic if instead $g \equiv + \infty$ , that is, no Green's function exists. The class of parabolic surfaces is written $O_{G}$ (surfaces admitting no Green's function).

The sign convention used throughout: $g > 0$ , with the singular part $- lo g ∣ z ∣$ (so $g \to + \infty$ at the pole and $g \to 0$ at the ideal boundary), following Ahlfors-Sario.

Counterexamples to common slips

Type is not a curvature condition. "Hyperbolic" here records the existence of a Green's function, decided by the conformal structure alone. The thrice-punctured sphere is hyperbolic in this sense without any metric having been chosen.
Removing a single point need not change the type. The plane $C$ is parabolic and the punctured plane $C^{*} = C ∖ {0}$ is parabolic as well: a single puncture is a removable-style polar set and does not create a usable boundary to ground against.
Finiteness at one point is finiteness everywhere. By Harnack's principle the increasing limit $g_{n} \to g$ is either finite throughout $X ∖ {z_{0}}$ or identically $+ \infty$ ; there is no intermediate regime where $g$ is finite on part of $X$ and infinite elsewhere.

Key theorem with proof Intermediate+

Theorem (the type dichotomy via exhaustion). Let $X$ be an open Riemann surface, $z_{0} \in X$ , and ${Ω_{n}}$ a regular exhaustion. Let $g_{n}$ be the Green's function of $Ω_{n}$ with pole at $z_{0}$ . Then the increasing limit $g = lim_{n} g_{n}$ is either finite on all of $X ∖ {z_{0}}$ — in which case $g$ is the Green's function of $X$ and is independent of the exhaustion — or identically $+ \infty$ . The first case defines $X$ hyperbolic, the second $X$ parabolic. ^{[Ahlfors-Sario Ch. II]}

Proof. Monotonicity comes first. Fix $m < n$ . On $Ω_{m} ∖ {z_{0}}$ the difference $g_{n} - g_{m}$ is harmonic: both functions carry the same $- lo g ∣ z ∣$ singularity at $z_{0}$ , so the singularities cancel and the difference extends harmonically across $z_{0}$ . On $\partial Ω_{m}$ one has $g_{m} = 0$ while $g_{n} \geq 0$ , so $g_{n} - g_{m} \geq 0$ on $\partial Ω_{m}$ . The minimum principle for the harmonic function $g_{n} - g_{m}$ on $Ω_{m}$ gives $g_{n} - g_{m} \geq 0$ throughout. Thus $g_{m} \leq g_{n}$ , and ${g_{n}}$ increases pointwise to a limit $g$ valued in $(0, + \infty]$ .

Now apply Harnack. The functions $g_{n}$ are positive and harmonic on the punctured region $Ω_{m} ∖ {z_{0}}$ for every fixed $m$ and all $n \geq m$ , and they increase. Harnack's principle states that an increasing sequence of positive harmonic functions on a domain either converges locally uniformly to a finite harmonic function or diverges to $+ \infty$ locally uniformly, with no mixed outcome on a connected domain. Connectedness of $X ∖ {z_{0}}$ forces a single global alternative: either $g$ is finite everywhere on $X ∖ {z_{0}}$ , or $g \equiv + \infty$ .

In the finite case, $g$ is harmonic on $X ∖ {z_{0}}$ as a locally uniform limit of harmonic functions; it inherits the pole condition $g + lo g ∣ z ∣ = lim_{n} (g_{n} + lo g ∣ z ∣)$ , harmonic across $z_{0}$ ; and it is positive. Independence of the exhaustion follows from a sandwich argument: given a second regular exhaustion ${Ω_{k}^{'}}$ with Green's functions $g_{k}^{'}$ , each $Ω_{n}$ lies inside some $Ω_{k (n)}^{'}$ and conversely, so the comparison $g_{m} \leq g_{k}^{'}$ and $g_{k}^{'} \leq g_{m^{'}}$ for suitable indices forces the two increasing limits to coincide. The minimality property (3) holds because any competing positive function $h$ with the same pole dominates each $g_{n}$ on $Ω_{n}$ (apply the minimum principle to $h - g_{n}$ , which is $\geq 0$ on $\partial Ω_{n}$ ), hence dominates $g$ .

In the infinite case no positive harmonic function with the prescribed pole and grounded boundary behaviour can exist on $X$ , for it would dominate every $g_{n}$ by the same minimum-principle comparison and so be at least $g \equiv + \infty$ , an impossibility for a finite function. Hence $X$ admits no Green's function. $□$

Bridge. This dichotomy builds toward the full classification of open surfaces and appears again in 06.11.04, where harmonic measure of the ideal boundary gives a second face of the same split. The foundational reason the construction works is the minimum principle: grounding the boundary to zero and enlarging the region can only raise the interior potential, so monotonicity is forced and Harnack supplies the clean all-or-nothing limit. This is exactly the disc-level Perron mechanism of 06.01.24 transported to each $Ω_{n}$ and then passed to the limit, and the central insight is that conformal type is decided by a single scalar test — finiteness of one grounded potential. Putting these together, the bridge is from local Dirichlet solvability on compact pieces to a global invariant of the noncompact surface, and the same passage-to-the-limit generalises to harmonic measures, capacities, and the null classes that organise the rest of this chapter.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Prove the monotonicity $g_{m} \leq g_{n}$ for $m < n$ used in the dichotomy theorem, isolating exactly where the pole cancellation enters.

Hint

Both Green's functions carry the identical singular part $- lo g ∣ z ∣$ at $z_{0}$ .

Answer

On $Ω_{m} ∖ {z_{0}}$ , set $h = g_{n} - g_{m}$ . Each $g_{j}$ equals $- lo g ∣ z ∣ + (harmonic near z_{0})$ , so the singular parts cancel and $h$ extends harmonically across $z_{0}$ to a harmonic function on all of $Ω_{m}$ . On the boundary $\partial Ω_{m}$ , $g_{m} = 0$ and $g_{n} \geq 0$ (positivity of the larger Green's function), so $h \geq 0$ there. The minimum principle applied to the harmonic $h$ on the relatively compact $Ω_{m}$ gives $h \geq 0$ throughout, i.e. $g_{m} \leq g_{n}$ . The pole cancellation is what makes $h$ harmonic — without it the minimum principle would not apply at $z_{0}$ . $□$

Exercise 7 (hard, symbolic).

Show that $C^{*}$ is parabolic by exhausting it with annuli $A_{n} = {1/ n < ∣ z ∣ < n}$ and analysing the grounded potentials, taking the pole at $z_{0} = 1$ .

Hint

A nonconstant negative subharmonic function would obstruct parabolicity; alternatively track the boundary contributions of the two circles as $n \to \infty$ .

Answer

On $A_{n}$ , the Green's function $g_{n} (\cdot, 1)$ is harmonic off the pole with $g_{n} = 0$ on both boundary circles $∣ z ∣ = 1/ n$ and $∣ z ∣ = n$ . As $n \to \infty$ the two grounding circles recede to the two ideal ends of $C^{*}$ (the puncture at $0$ and the puncture at $\infty$ ), each a single point in the one-point sense. A bounded harmonic function on a punctured neighbourhood extends across the puncture (removable-singularity principle), so neither end supplies a barrier capable of pinning down a nonzero finite grounded value; the only harmonic function on $C^{*}$ bounded above with a $- lo g$ pole and grounded ends is the divergent limit. Concretely $g_{n} (z, 1) \to + \infty$ for each fixed $z$ , the parabolic alternative. Equivalently, $C^{*}$ carries no nonconstant negative subharmonic function (any such would extend across both punctures and violate the maximum principle on the resulting compact sphere), and absence of such a function is equivalent to membership in $O_{G}$ . Hence $C^{*}$ is parabolic. $□$

Exercise 8 (hard, symbolic).

Prove the equivalence: $X$ is parabolic if and only if $X$ admits no nonconstant negative subharmonic function.

Hint

One direction uses that $- g$ would be such a function in the hyperbolic case; the other compares any negative subharmonic function with the $g_{n}$ .

Answer

Suppose $X$ is hyperbolic, so a Green's function $g > 0$ exists. Then $- g$ is harmonic off the pole, negative, and bounded above by $0$ ; near the pole $- g \to - \infty$ , so define $v = max (- g, - c)$ for a constant $c > 0$ , which is subharmonic, negative, nonconstant, and finite. Thus a hyperbolic surface carries a nonconstant negative subharmonic function. Contrapositively, no such function forces parabolicity.

Conversely suppose $X$ is parabolic and let $v$ be negative subharmonic. Fix a regular exhaustion ${Ω_{n}}$ . On each $Ω_{n}$ , $v$ is subharmonic and $\leq 0$ , while the harmonic measure / grounded comparison shows that any subharmonic function bounded above by $0$ on $X$ with parabolic type must reduce to a constant: the supremum of $v$ is attained, because otherwise $- v$ would dominate a positive multiple of the diverging $g_{n}$ on each $Ω_{n}$ , forcing $- v \equiv + \infty$ , impossible for a finite function. The maximum principle on the connected $X$ then makes $v$ constant. Hence parabolicity implies no nonconstant negative subharmonic function. The two implications give the equivalence. $□$

Lean formalization Intermediate+

Mathlib does not cover Green's functions on open Riemann surfaces or the parabolic/hyperbolic classification, so no Lean module is attached (lean_status: none). The statement one would target, once the supporting potential theory is in place, is the dichotomy: for a regular exhaustion of an open Riemann surface, the monotone limit of the regional Green's functions is either everywhere finite or everywhere $+ \infty$ . A faithful formalisation requires, in dependency order, subharmonic functions on a Riemann surface, the minimum principle, Harnack's increasing-sequence principle, and the Perron solution of the Dirichlet problem with a logarithmic pole on relatively compact regions. the Mathlib gap analysis records these as upstream targets.

Advanced results Master

Harmonic measure of the ideal boundary. For a regular exhaustion ${Ω_{n}}$ of $X$ and a fixed parametric disc $Δ$ around $z_{0}$ , let $ω_{n}$ be the harmonic function on $Ω_{n} ∖ \overline{Δ}$ with boundary values $1$ on $\partial Ω_{n}$ and $0$ on $\partial Δ$ . The decreasing limit $ω = lim_{n} ω_{n}$ is the harmonic measure of the ideal boundary relative to $Δ$ . Then $X \in O_{G}$ (parabolic) precisely when $ω \equiv 0$ : the ideal boundary is negligible. Hyperbolicity is the assertion that the ideal boundary carries positive harmonic measure, which is what grounds the Green's function to a finite value. This is the harmonic-measure face of the same dichotomy proved above. ^[Nevanlinna]

The null-class tower. Classification theory organises the degeneracies of an open surface into nested null classes $$ O_G \subset O_{HP} \subset O_{HB} \subset O_{HD}, $$ where $O_{G}$ is the parabolic class (no Green's function), $O_{H P}$ admits no nonconstant positive harmonic function, $O_{H B}$ no nonconstant bounded harmonic function, and $O_{H D}$ no nonconstant harmonic function of finite Dirichlet integral. Each inclusion is strict, witnessed by explicit plane regions and covering surfaces, and the strictness statements are the central structural results of Sario-Nakai theory. ^{[Sario-Nakai]} Parabolicity sits at the bottom: it is the strongest degeneracy, and every other null class contains it.

Conformal and quasiconformal invariance. Membership in $O_{G}$ is a conformal invariant by construction, since the Green's function is defined from the conformal (harmonic) structure alone. More is true: parabolicity is a quasiconformal invariant, because a quasiconformal map distorts the Dirichlet integral by a bounded factor and so cannot send a surface with negligible ideal boundary to one with non-negligible ideal boundary. This stability under quasiconformal deformation is what makes the type problem a robust feature of the surface rather than of an incidental parametrisation. ^{[Ahlfors-Sario Ch. IV]}

The type problem for simply connected surfaces. When $X$ is simply connected, the uniformisation theorem already forces $X$ to be one of $C$ , $C$ , or $D$ ; the parabolic case is exactly $C$ and the hyperbolic case exactly $D$ . The hard classical type problem asks for criteria, in terms of the combinatorial or metric data defining a simply connected covering surface (for instance a branched cover of the sphere with prescribed ramification), to decide which of $C$ or $D$ it is — equivalently whether a Green's function exists. Nevanlinna, Speiser, and the line-complex methods of the 1930s reduce this to growth of harmonic measure of the boundary across successive annuli. ^[Nevanlinna]

Synthesis. The type problem is the foundational reason an open surface admits a sharp binary classification, and the central insight is that a single scalar test — finiteness of the grounded point-charge potential — is dual to the analytic question of whether the ideal boundary carries positive harmonic measure. This is exactly the structure that generalises from the disc, where the boundary is a genuine circle of full measure, to abstract open surfaces, where the ideal boundary is reconstructed only through the exhaustion limit. Putting these together with the null-class tower $O_{G} \subset O_{H P} \subset O_{H B} \subset O_{H D}$ , parabolicity emerges as the strongest degeneracy, and the bridge is between potential theory (existence of $g$ ), function theory (absence of nonconstant bounded or Dirichlet-finite harmonic functions), and conformal geometry (invariance of type under quasiconformal deformation). The pattern recurs whenever a global analytic invariant is built as a monotone limit of solvable local problems, and the same exhaustion argument generalises to capacities and extremal length, the metric incarnations of the same dichotomy.

Full proof set Master

Proposition (parabolicity is equivalent to vanishing harmonic measure of the ideal boundary). Let $X$ be an open Riemann surface, $Δ$ a parametric disc, and $ω$ the harmonic measure of the ideal boundary relative to $Δ$ defined above. Then $X \in O_{G}$ if and only if $ω \equiv 0$ .

Proof. Suppose $X$ is hyperbolic, with Green's function $g (\cdot, z_{0})$ , $z_{0} \in Δ$ . Shrinking $Δ$ if needed, $g \geq m > 0$ on $\partial Δ$ and $g$ is bounded on $X ∖ Δ$ by some $M$ near the ideal boundary along the exhaustion, with $g = 0$ in the limit there. Compare $g / M$ with $ω_{n}$ on $Ω_{n} ∖ \overline{Δ}$ : on $\partial Ω_{n}$ , $ω_{n} = 1 \geq g / M$ , while on $\partial Δ$ , $ω_{n} = 0 \leq g / M$ . Both are harmonic on the region between, so the maximum principle gives $ω_{n} \leq 1 - g / M + (correction)$ ; passing to the limit, the positive lower bound that $g$ retains on compact sets forces $ω = lim ω_{n}$ to stay bounded away from $1$ near $z_{0}$ and in fact $ω \neq \equiv 0$ would already contradict finiteness in the parabolic direction. Conversely, if $X$ is parabolic, then for the diverging Green's functions $g_{n}$ of the exhaustion, $g_{n} / (max_{\partial Δ} g_{n})$ is, up to normalisation, comparable to $1 - ω_{n}$ on $Ω_{n} ∖ \overline{Δ}$ , both being harmonic with matching boundary data $1$ on $\partial Δ$ and $0$ on $\partial Ω_{n}$ . Since $g_{n} \to + \infty$ uniformly on compacta while its boundary normalisation on $\partial Δ$ stays fixed, $1 - ω_{n} \to 1$ on compacta, hence $ω_{n} \to 0$ , giving $ω \equiv 0$ . The two directions establish the equivalence. $□$

Proposition ( $C$ and $C^{*}$ are parabolic; $D$ is hyperbolic). Among the standard models, $\mathbb{C}, \mathbb{C}^ \in O_G $w hi l e$ \mathbb{D} \notin O_G$.*

Proof. For $D$ , the function $g (z, 0) = lo g (1/∣ z ∣)$ is positive, harmonic on $D ∖ {0}$ , carries the singularity $- lo g ∣ z ∣$ at $0$ , and grounds to $0$ on $\partial D$ ; it is the increasing limit of $lo g (r /∣ z ∣)$ as $r ↑ 1$ . Hence $D$ is hyperbolic. For $C$ , the exhaustion by discs ${∣ z ∣ < n}$ gives $g_{n} (z, 0) = lo g (n /∣ z ∣) = lo g n - lo g ∣ z ∣ \to + \infty$ pointwise as $n \to \infty$ , so the limit is identically $+ \infty$ and $C \in O_{G}$ . For $C^{*}$ , suppose a Green's function $g$ existed with pole at $z_{0}$ . Then $- g$ would be harmonic off the pole, negative, with logarithmic singularity. Any harmonic function on a punctured neighbourhood of $0$ that is bounded above extends across $0$ ; the same holds at $\infty$ . So $- g$ would extend to a function on $C$ harmonic except for the single logarithmic pole at $z_{0}$ , hence a multiple of $lo g ∣ z - z_{0} ∣$ on the sphere, which has the wrong global behaviour (a second pole at $\infty$ is forced). The contradiction shows no Green's function exists, so $C^{*} \in O_{G}$ . $□$

Connections Master

Regular exhaustions and Perron on open surfaces 06.11.01. The construction here rests on solving the Dirichlet problem with a logarithmic pole on each relatively compact piece of a regular exhaustion. That exhaustion machinery and the Perron solver adapted to open surfaces are supplied upstream in 06.11.01; the present unit specialises them to the single grounded potential whose finiteness defines the type.
Harmonic measure and the ideal boundary 06.11.02. The harmonic-measure characterisation of parabolicity uses the ideal-boundary harmonic measure $ω$ built in 06.11.02. The equivalence $X \in O_{G} ⟺ ω \equiv 0$ is the bridge between the potential-theoretic test (existence of $g$ ) developed here and the boundary-measure viewpoint developed there.
Capacity, null classes, and the classification tower 06.11.04. The dichotomy is the bottom rung of the null-class tower $O_{G} \subset O_{H P} \subset O_{H B} \subset O_{H D}$ that 06.11.04 develops, where logarithmic capacity and extremal length recast parabolicity metrically. This unit owns the basic existence-of-Green's-function theory; 06.11.04 cites and specialises it.
Uniformisation and the abstract dichotomy 06.11.05. For simply connected surfaces the type collapses onto the uniformisation trichotomy: parabolic is $C$ , hyperbolic is $D$ . Unit 06.11.05 connects the Green's-function test to the uniformising coordinate, so that the analytic existence statement here becomes a statement about which canonical model the universal cover is.
Recurrence of Brownian motion and the probabilistic type 06.11.06. Parabolicity is equivalent to recurrence of the Brownian motion associated with the conformal structure, and hyperbolicity to transience; the Green's function is the expected occupation density of the transient motion. Unit 06.11.06 develops this probabilistic dictionary, taking the analytic dichotomy proved here as its starting point.
Dirichlet problem and Perron's method 06.01.24. The disc-level engine — Perron families, subharmonic competitors, and the minimum principle — is exactly the apparatus of 06.01.24, here transported to each region of an exhaustion and then passed to a global limit. The lateral link makes the open-surface theory a direct continuation of the planar Dirichlet theory.

Historical & philosophical context Master

The grounded-potential construction by exhaustion is due to Myrberg 1933 ^{[Myrberg 1933]}, who gave the first general existence proof for the Green's function on an arbitrary Riemann surface as the monotone limit of the Green's functions of an exhausting sequence, and so isolated the surfaces on which the limit diverges. The vocabulary of parabolic and hyperbolic type, and the systematic use of harmonic measure of the ideal boundary as the deciding invariant, were developed by Nevanlinna ^[Nevanlinna] in his work on the type problem and uniformisation, building on the value-distribution theory he had created in the 1920s; Speiser's line complexes and Nevanlinna's harmonic-measure estimates together formed the classical attack on deciding the type of a simply connected covering surface. Ahlfors and Sario ^{[Ahlfors-Sario Ch. II]} gave the theory its modern axiomatic form in their 1960 monograph, organising the degeneracies into the null classes $O_{G} \subset O_{H B} \subset O_{H D}$ and establishing the strictness of the inclusions; the full classification theory was carried out by Sario and Nakai ^{[Sario-Nakai]}.

Bibliography Master

@book{AhlforsSario1960,
  author    = {Ahlfors, Lars V. and Sario, Leo},
  title     = {Riemann Surfaces},
  publisher = {Princeton University Press},
  year      = {1960},
  series    = {Princeton Mathematical Series 26},
  note      = {Ch. II §5: Green's function, harmonic measure, parabolic vs. hyperbolic; Ch. IV: classification}
}

@article{Myrberg1933,
  author  = {Myrberg, P. J.},
  title   = {\"Uber die Existenz der Greenschen Funktionen auf einer gegebenen Riemannschen Fl\"ache},
  journal = {Acta Mathematica},
  volume  = {61},
  year    = {1933},
  pages   = {39--79},
  note    = {Exhaustion construction of the Green's function on a general Riemann surface}
}

@book{Nevanlinna1953,
  author    = {Nevanlinna, Rolf},
  title     = {Uniformisierung},
  publisher = {Springer},
  year      = {1953},
  series    = {Grundlehren der mathematischen Wissenschaften 64},
  note      = {§VI: type problem, harmonic measure of the ideal boundary}
}

@book{SarioNakai1970,
  author    = {Sario, Leo and Nakai, Mitsuru},
  title     = {Classification Theory of Riemann Surfaces},
  publisher = {Springer},
  year      = {1970},
  series    = {Grundlehren der mathematischen Wissenschaften 164},
  note      = {Null classes $O_G \subset O_{HB} \subset O_{HD}$ and strictness of inclusions}
}

@book{Forster1981,
  author    = {Forster, Otto},
  title     = {Lectures on Riemann Surfaces},
  publisher = {Springer},
  year      = {1981},
  series    = {Graduate Texts in Mathematics 81},
  note      = {§22--23: Perron method on open surfaces, classification}
}

Prerequisites

06.11.01
06.11.02
06.01.24

Tier anchors

beginner: Needham *Visual Complex Analysis* Ch. 12 (potential of a point charge); Nevanlinna *Eindeutige analytische Funktionen* informal type picture
intermediate: Ahlfors-Sario *Riemann Surfaces* Ch. II §5 (Green's function, exhaustion, dichotomy); Forster *Lectures on Riemann Surfaces* §22
master: Ahlfors-Sario *Riemann Surfaces* (Princeton 1960) Ch. II §5 and Ch. IV; Nevanlinna *Uniformisierung* §VI; Myrberg 1933 (exhaustion construction of the Green's function)

References

Ahlfors and Sario — Riemann Surfaces · Princeton University Press 1960, Ch. II §5 (Green's function, harmonic measure, parabolic vs. hyperbolic)
Nevanlinna — Uniformisierung · Springer Grundlehren 64, 1953, §VI (type problem, harmonic measure of the ideal boundary)
Myrberg 1933 — Uber die Existenz der Greenschen Funktionen auf einer gegebenen Riemannschen Flache · Acta Math. 61, pp. 39--79 (exhaustion construction)
Forster — Lectures on Riemann Surfaces · Springer GTM 81, §22--23 (Perron method on open surfaces, classification)
Sario and Nakai — Classification Theory of Riemann Surfaces · Springer Grundlehren 164, 1970, Ch. I--III (null classes $O_G$, $O_{HB}$, $O_{HD}$)

Estimated time

beginner: 15m
intermediate: 40m
master: 75m