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Important note: As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.

In these notes, random variables will be denoted in boldface.

Definition 1 A real random variable ${\mathbf{X}}$ is said to be normally distributed with mean ${x_0 \in {\bf R}}$ and variance ${\sigma^2 > 0}$ if one has

$\displaystyle \mathop{\bf E} F(\mathbf{X}) = \frac{1}{\sqrt{2\pi} \sigma} \int_{\bf R} e^{-(x-x_0)^2/2\sigma^2} F(x)\ dx$

for all test functions ${F \in C_c({\bf R})}$. Similarly, a complex random variable ${\mathbf{Z}}$ is said to be normally distributed with mean ${z_0 \in {\bf R}}$ and variance ${\sigma^2>0}$ if one has

$\displaystyle \mathop{\bf E} F(\mathbf{Z}) = \frac{1}{\pi \sigma^2} \int_{\bf C} e^{-|z-x_0|^2/\sigma^2} F(z)\ dx dy$

for all test functions ${F \in C_c({\bf C})}$, where ${dx dy}$ is the area element on ${{\bf C}}$.

A real Brownian motion with base point ${x_0 \in {\bf R}}$ is a random, almost surely continuous function ${\mathbf{B}^{x_0}: [0,+\infty) \rightarrow {\bf R}}$ (using the locally uniform topology on continuous functions) with the property that (almost surely) ${\mathbf{B}^{x_0}(0) = x_0}$, and for any sequence of times ${0 \leq t_0 < t_1 < t_2 < \dots < t_n}$, the increments ${\mathbf{B}^{x_0}(t_i) - \mathbf{B}^{x_0}(t_{i-1})}$ for ${i=1,\dots,n}$ are independent real random variables that are normally distributed with mean zero and variance ${t_i - t_{i-1}}$. Similarly, a complex Brownian motion with base point ${z_0 \in {\bf R}}$ is a random, almost surely continuous function ${\mathbf{B}^{z_0}: [0,+\infty) \rightarrow {\bf R}}$ with the property that ${\mathbf{B}^{z_0}(0) = z_0}$ and for any sequence of times ${0 \leq t_0 < t_1 < t_2 < \dots < t_n}$, the increments ${\mathbf{B}^{z_0}(t_i) - \mathbf{B}^{z_0}(t_{i-1})}$ for ${i=1,\dots,n}$ are independent complex random variables that are normally distributed with mean zero and variance ${t_i - t_{i-1}}$.

Remark 2 Thanks to the central limit theorem, the hypothesis that the increments ${\mathbf{B}^{x_0}(t_i) - \mathbf{B}^{x_0}(t_{i-1})}$ be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).

Real and complex Brownian motions exist from any base point ${x_0}$ or ${z_0}$; see e.g. this previous blog post for a construction. We have the following simple invariances:

Exercise 3

• (i) (Translation invariance) If ${\mathbf{B}^{x_0}}$ is a real Brownian motion with base point ${x_0 \in {\bf R}}$, and ${h \in {\bf R}}$, show that ${\mathbf{B}^{x_0}+h}$ is a real Brownian motion with base point ${x_0+h}$. Similarly, if ${\mathbf{B}^{z_0}}$ is a complex Brownian motion with base point ${z_0 \in {\bf R}}$, and ${h \in {\bf C}}$, show that ${\mathbf{B}^{z_0}+c}$ is a complex Brownian motion with base point ${z_0+h}$.
• (ii) (Dilation invariance) If ${\mathbf{B}^{0}}$ is a real Brownian motion with base point ${0}$, and ${\lambda \in {\bf R}}$ is non-zero, show that ${t \mapsto \lambda \mathbf{B}^0(t / |\lambda|^{1/2})}$ is also a real Brownian motion with base point ${0}$. Similarly, if ${\mathbf{B}^0}$ is a complex Brownian motion with base point ${0}$, and ${\lambda \in {\bf C}}$ is non-zero, show that ${t \mapsto \lambda \mathbf{B}^0(t / |\lambda|^{1/2})}$ is also a complex Brownian motion with base point ${0}$.
• (iii) (Real and imaginary parts) If ${\mathbf{B}^0}$ is a complex Brownian motion with base point ${0}$, show that ${\sqrt{2} \mathrm{Re} \mathbf{B}^0}$ and ${\sqrt{2} \mathrm{Im} \mathbf{B}^0}$ are independent real Brownian motions with base point ${0}$. Conversely, if ${\mathbf{B}^0_1, \mathbf{B}^0_2}$ are independent real Brownian motions of base point ${0}$, show that ${\frac{1}{\sqrt{2}} (\mathbf{B}^0_1 + i \mathbf{B}^0_2)}$ is a complex Brownian motion with base point ${0}$.

The next lemma is a special case of the optional stopping theorem.

Lemma 4 (Optional stopping identities)

• (i) (Real case) Let ${\mathbf{B}^{x_0}}$ be a real Brownian motion with base point ${x_0 \in {\bf R}}$. Let ${\mathbf{t}}$ be a bounded stopping time – a bounded random variable with the property that for any time ${t \geq 0}$, the event that ${\mathbf{t} \leq t}$ is determined by the values of the trajectory ${\mathbf{B}^{x_0}}$ for times up to ${t}$ (or more precisely, this event is measurable with respect to the ${\sigma}$ algebra generated by this proprtion of the trajectory). Then

$\displaystyle \mathop{\bf E} \mathbf{B}^{x_0}(\mathbf{t}) = x_0$

and

$\displaystyle \mathop{\bf E} (\mathbf{B}^{x_0}(\mathbf{t})-x_0)^2 - \mathbf{t} = 0$

and

$\displaystyle \mathop{\bf E} (\mathbf{B}^{x_0}(\mathbf{t})-x_0)^4 = O( \mathop{\bf E} \mathbf{t}^2 ).$

• (ii) (Complex case) Let ${\mathbf{B}^{z_0}}$ be a real Brownian motion with base point ${z_0 \in {\bf R}}$. Let ${\mathbf{t}}$ be a bounded stopping time – a bounded random variable with the property that for any time ${t \geq 0}$, the event that ${\mathbf{t} \leq t}$ is determined by the values of the trajectory ${\mathbf{B}^{x_0}}$ for times up to ${t}$. Then

$\displaystyle \mathop{\bf E} \mathbf{B}^{z_0}(\mathbf{t}) = z_0$

$\displaystyle \mathop{\bf E} (\mathrm{Re}(\mathbf{B}^{z_0}(\mathbf{t})-z_0))^2 - \frac{1}{2} \mathbf{t} = 0$

$\displaystyle \mathop{\bf E} (\mathrm{Im}(\mathbf{B}^{z_0}(\mathbf{t})-z_0))^2 - \frac{1}{2} \mathbf{t} = 0$

$\displaystyle \mathop{\bf E} \mathrm{Re}(\mathbf{B}^{z_0}(\mathbf{t})-z_0) \mathrm{Im}(\mathbf{B}^{z_0}(\mathbf{t})-z_0) = 0$

$\displaystyle \mathop{\bf E} |\mathbf{B}^{x_0}(\mathbf{t})-z_0|^4 = O( \mathop{\bf E} \mathbf{t}^2 ).$

Proof: (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take ${x_0=0}$. Let ${T}$ be an upper bound for ${\mathbf{t}}$. Since ${\mathbf{B}^0(T)}$ is a real normally distributed variable with mean zero and variance ${T}$, we have

$\displaystyle \mathop{\bf E} \mathbf{B}^0( T ) = 0$

and

$\displaystyle \mathop{\bf E} \mathbf{B}^0( T )^2 = T$

and

$\displaystyle \mathop{\bf E} \mathbf{B}^0( T )^4 = 3T^2.$

By the law of total expectation, we thus have

$\displaystyle \mathop{\bf E} \mathop{\bf E}(\mathbf{B}^0( T ) | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = 0$

and

$\displaystyle \mathop{\bf E} \mathop{\bf E}((\mathbf{B}^0( T ))^2 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = T$

and

$\displaystyle \mathop{\bf E} \mathop{\bf E}((\mathbf{B}^0( T ))^4 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = 3T^2$

where the inner conditional expectations are with respect to the event that ${\mathbf{t}, \mathbf{B}^{0}(\mathbf{t})}$ attains a particular point in ${S}$. However, from the independent increment nature of Brownian motion, once one conditions ${(\mathbf{t}, \mathbf{B}^{0}(\mathbf{t}))}$ to a fixed point ${(t, x)}$, the random variable ${\mathbf{B}^0(T)}$ becomes a real normally distributed variable with mean ${x}$ and variance ${T-t}$. Thus we have

$\displaystyle \mathop{\bf E}(\mathbf{B}^0( T ) | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})$

and

$\displaystyle \mathop{\bf E}( (\mathbf{B}^0( T ))^2 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})^2 + T - \mathbf{t}$

and

$\displaystyle \mathop{\bf E}( (\mathbf{B}^0( T ))^4 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})^4 + 6(T - \mathbf{t}) \mathbf{B}^{z_0}(\mathbf{t})^2 + 3(T - \mathbf{t})^2$

which give the first two claims, and (after some algebra) the identity

$\displaystyle \mathop{\bf E} \mathbf{B}^{z_0}(\mathbf{t})^4 - 6 \mathbf{t} \mathbf{B}^{z_0}(\mathbf{t})^2 + 3 \mathbf{t}^2 = 0$

which then also gives the third claim. $\Box$

Exercise 5 Prove the second part of Lemma 4.

We now approach conformal maps from yet another perspective. Given an open subset ${U}$ of the complex numbers ${{\bf C}}$, define a univalent function on ${U}$ to be a holomorphic function ${f: U \rightarrow {\bf C}}$ that is also injective. We will primarily be studying this concept in the case when ${U}$ is the unit disk ${D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}$.

Clearly, a univalent function ${f: D(0,1) \rightarrow {\bf C}}$ on the unit disk is a conformal map from ${D(0,1)}$ to the image ${f(D(0,1))}$; in particular, ${f(D(0,1))}$ is simply connected, and not all of ${{\bf C}}$ (since otherwise the inverse map ${f^{-1}: {\bf C} \rightarrow D(0,1)}$ would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset ${V \subsetneq {\bf C}}$ of the complex numbers is the image of a univalent function on ${D(0,1)}$. Furthermore, if ${V}$ contains the origin, then the univalent function ${f: D(0,1) \rightarrow {\bf C}}$ with this image becomes unique once we normalise ${f(0) = 0}$ and ${f'(0) > 0}$. Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions ${f: D(0,1) \rightarrow {\bf C}}$ with ${f(0)=0}$ and ${f'(0)>0}$. We will focus particular attention on the univalent functions ${f: D(0,1) \rightarrow {\bf C}}$ with the normalisation ${f(0)=0}$ and ${f'(0)=1}$; such functions will be called schlicht functions.

One basic example of a univalent function on ${D(0,1)}$ is the Cayley transform ${z \mapsto \frac{1+z}{1-z}}$, which is a Möbius transformation from ${D(0,1)}$ to the right half-plane ${\{ \mathrm{Re}(z) > 0 \}}$. (The slight variant ${z \mapsto \frac{1-z}{1+z}}$ is also referred to as the Cayley transform, as is the closely related map ${z \mapsto \frac{z-i}{z+i}}$, which maps ${D(0,1)}$ to the upper half-plane.) One can square this map to obtain a further univalent function ${z \mapsto \left( \frac{1+z}{1-z} \right)^2}$, which now maps ${D(0,1)}$ to the complex numbers with the negative real axis ${(-\infty,0]}$ removed. One can normalise this function to be schlicht to obtain the Koebe function

$\displaystyle f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)$

which now maps ${D(0,1)}$ to the complex numbers with the half-line ${(-\infty,-1/4]}$ removed. A little more generally, for any ${\theta \in {\bf R}}$ we have the rotated Koebe function

$\displaystyle f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)$

that is a schlicht function that maps ${D(0,1)}$ to the complex numbers with the half-line ${\{ -re^{-i\theta}: r \geq 1/4\}}$ removed.

Every schlicht function ${f: D(0,1) \rightarrow {\bf C}}$ has a convergent Taylor expansion

$\displaystyle f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots$

for some complex coefficients ${a_1,a_2,\dots}$ with ${a_1=1}$. For instance, the Koebe function has the expansion

$\displaystyle f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n$

and similarly the rotated Koebe function has the expansion

$\displaystyle f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n.$

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients ${a_n}$ should obey the bound ${|a_n| \leq n}$ for all ${n}$. After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions ${g: D(0,1) \rightarrow {\bf C}}$ that are odd, thus ${g(-z)=-g(z)}$ for all ${z}$, and the Taylor expansion now reads

$\displaystyle g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots$

for some complex coefficients ${b_1,b_3,\dots}$ with ${b_1=1}$. One can transform a general schlicht function ${f: D(0,1) \rightarrow {\bf C}}$ to an odd schlicht function ${g: D(0,1) \rightarrow {\bf C}}$ by observing that the function ${f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}$, after removing the singularity at zero, is a non-zero function that equals ${1}$ at the origin, and thus (as ${D(0,1)}$ is simply connected) has a unique holomorphic square root ${(f(z^2)/z^2)^{1/2}}$ that also equals ${1}$ at the origin. If one then sets

$\displaystyle g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)$

it is not difficult to verify that ${g}$ is an odd schlicht function which additionally obeys the equation

$\displaystyle f(z^2) = g(z)^2. \ \ \ \ \ (4)$

Conversely, given an odd schlicht function ${g}$, the formula (4) uniquely determines a schlicht function ${f}$.

For instance, if ${f}$ is the Koebe function (1), ${g}$ becomes

$\displaystyle g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)$

which maps ${D(0,1)}$ to the complex numbers with two slits ${\{ \pm iy: y > 1/2 \}}$ removed, and if ${f}$ is the rotated Koebe function (2), ${g}$ becomes

$\displaystyle g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)$

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem) Let ${n \geq 1}$ be a natural number.

• (i) (Robertson conjecture) If ${g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}$ is an odd schlicht function, then

$\displaystyle \sum_{k=1}^n |b_{2k-1}|^2 \leq n.$

• (ii) (Bieberbach conjecture) If ${f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}$ is a schlicht function, then

$\displaystyle |a_n| \leq n.$

It is easy to see that the Robertson conjecture for a given value of ${n}$ implies the Bieberbach conjecture for the same value of ${n}$. Indeed, if ${f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}$ is schlicht, and ${g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}$ is the odd schlicht function given by (3), then from extracting the ${z^{2n}}$ coefficient of (4) we obtain a formula

$\displaystyle a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}$

for the coefficients of ${f}$ in terms of the coefficients of ${g}$. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of ${n}$ from the Robertson conjecture for the same value of ${n}$. We remark that Littlewood and Paley had conjectured a stronger form ${|b_{2k-1}| \leq 1}$ of Robertson’s conjecture, but this was disproved for ${k=3}$ by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of ${\log \frac{f(z)}{z}}$, known as the Milin conjecture. Next, one continuously enlarges the image ${f(D(0,1))}$ of the schlicht function to cover all of ${{\bf C}}$; done properly, this places the schlicht function ${f}$ as the initial function ${f = f_0}$ in a sequence ${(f_t)_{t \geq 0}}$ of univalent maps ${f_t: D(0,1) \rightarrow {\bf C}}$ known as a Loewner chain. The functions ${f_t}$ obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term ${\mu_t}$ (or ${\theta(t)}$, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of ${f(z)}$, ${g(z)}$, or ${\log \frac{f(z)}{z}}$. After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of ${n}$) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map ${f: U \rightarrow V}$ from an open subset ${U}$ of the complex plane to another open set ${V}$ is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of ${f}$ to be nowhere vanishing. We then say that the two open sets ${U,V}$ are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation ${f( z_0 + \Delta z) \approx f(z_0) + f'(z_0) \Delta z + O( |\Delta z|^2)}$ we see that they almost preserve small circles, indeed for ${\varepsilon}$ small the circle ${\{ z: |z-z_0| = \varepsilon\}}$ will approximately map to ${\{ w: |w - f(z_0)| = |f'(z_0)| \varepsilon \}}$.

Theorem 1 (Riemann mapping theorem) Let ${U}$ be a simply connected open subset of ${{\bf C}}$ that is not all of ${{\bf C}}$. Then ${U}$ is conformally equivalent to the unit disk ${D(0,1)}$.

This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps ${f: U \rightarrow D(0,1)}$ from ${U}$ to ${D(0,1)}$ that map some fixed point ${z_0 \in U}$ to ${0}$, pick one that maximises the magnitude ${|f'(z_0)|}$ of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If ${f(U)}$ avoids some point in ${D(0,1)}$, one can compose ${f}$ with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase ${|f'(z_0)|}$ without destroying injectivity; see the previous lecture notes for details. The conformal map ${\phi: U \rightarrow D(0,1)}$ is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points ${z_0,z_1}$ in ${U}$, and requiring ${\phi(z_0)}$ to be zero and ${\phi(z_1)}$ to be positive real.

It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).

To make the above discussion more precise we need some notation.

Definition 2 (Circle packing) A (finite) circle packing is a finite collection ${(C_j)_{j \in J}}$ of circles ${C_j = \{ z \in {\bf C}: |z-z_j| = r_j\}}$ in the complex numbers indexed by some finite set ${J}$, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices ${\{z_j: j \in J \}}$ are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)

It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.

The first basic theorem in the subject is the following converse statement:

Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.

Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:

Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.

Exercise 5 Let ${G}$ be a connected planar graph with ${n \geq 3}$ vertices. Show that the following are equivalent:

• (i) ${G}$ is a maximal planar graph.
• (ii) ${G}$ has ${3n-6}$ edges.
• (iii) Every drawing ${D}$ of ${G}$ divides the plane into faces that have three edges each. (This includes one unbounded face.)
• (iv) At least one drawing ${D}$ of ${G}$ divides the plane into faces that have three edges each.

(Hint: use Euler’s formula ${V-E+F=2}$, where ${F}$ is the number of faces including the unbounded face.)

Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:

Conjecture 6 (Informal Thurston conjecture) Let ${U}$ be a simply connected domain, with two distinct points ${z_0,z_1}$. Let ${\phi: U \rightarrow D(0,1)}$ be the conformal map from ${U}$ to ${D(0,1)}$ that maps ${z_0}$ to the origin and ${z_1}$ to a positive real. For any small ${\varepsilon>0}$, let ${{\mathcal C}_\varepsilon}$ be the portion of the regular hexagonal circle packing by circles of radius ${\varepsilon}$ that are contained in ${U}$, and let ${{\mathcal C}'_\varepsilon}$ be an circle packing of ${D(0,1)}$ with all “boundary circles” tangent to ${D(0,1)}$, giving rise to an “approximate map” ${\phi_\varepsilon: U_\varepsilon \rightarrow D(0,1)}$ defined on the subset ${U_\varepsilon}$ of ${U}$ consisting of the circles of ${{\mathcal C}_\varepsilon}$, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that ${\phi_\varepsilon(z_0)}$ is zero and ${\phi_\varepsilon(z_1)}$ is a positive real. Then ${\phi_\varepsilon}$ converges to ${\phi}$ as ${\varepsilon \rightarrow 0}$.

A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation ${\frac{\partial \phi}{\partial \overline{z}} = 0}$, while (sufficiently smooth) quasiconformal maps only obey an inequality ${|\frac{\partial \phi}{\partial \overline{z}}| \leq \frac{K-1}{K+1} |\frac{\partial \phi}{\partial z}|}$. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map ${\phi_\varepsilon}$ alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).

The fundamental object of study in real differential geometry are the real manifolds: Hausdorff topological spaces ${M = M^n}$ that locally look like open subsets of a Euclidean space ${{\bf R}^n}$, and which can be equipped with an atlas ${(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}$ of coordinate charts ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ from open subsets ${U_\alpha}$ covering ${M}$ to open subsets ${V_\alpha}$ in ${{\bf R}^n}$, which are homeomorphisms; in particular, the transition maps ${\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )}$ defined by ${\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}}$ are all continuous. (It is also common to impose the requirement that the manifold ${M}$ be second countable, though this will not be important for the current discussion.) A smooth real manifold is a real manifold in which the transition maps are all smooth.

In a similar fashion, the fundamental object of study in complex differential geometry are the complex manifolds, in which the model space is ${{\bf C}^n}$ rather than ${{\bf R}^n}$, and the transition maps ${\tau_{\alpha\beta}}$ are required to be holomorphic (and not merely smooth or continuous). In the real case, the one-dimensional manifolds (curves) are quite simple to understand, particularly if one requires the manifold to be connected; for instance, all compact connected one-dimensional real manifolds are homeomorphic to the unit circle (why?). However, in the complex case, the connected one-dimensional manifolds – the ones that look locally like subsets of ${{\bf C}}$ – are much richer, and are known as Riemann surfaces. For sake of completeness we give the (somewhat lengthy) formal definition:

Definition 1 (Riemann surface) If ${M}$ is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection ${(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}$ of homeomorphisms from open subsets ${(U_\alpha)_{\alpha \in A}}$ of ${M}$ that cover ${M}$ to open subsets ${V_\alpha}$ of the complex numbers ${{\bf C}}$, such that the transition maps ${\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )}$ defined by ${\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}}$ are all holomorphic. Here ${A}$ is an arbitrary index set. Two atlases ${(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}$, ${(\phi'_\beta: U'_\beta \rightarrow V'_\beta)_{\beta \in B}}$ on ${M}$ are said to be equivalent if their union is also an atlas, thus the transition maps ${\phi'_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U'_\beta) \rightarrow \phi'_\beta(U_\alpha \cap U'_\beta)}$ and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space ${M}$ equipped with an equivalence class of one-dimensional complex atlases.

A map ${f: M \rightarrow M'}$ from one Riemann surface ${M}$ to another ${M'}$ is holomorphic if the maps ${\phi'_\beta \circ f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap f^{-1}(U'_\beta)) \rightarrow {\bf C}}$ are holomorphic for any charts ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$, ${\phi'_\beta: U'_\beta \rightarrow V'_\beta}$ of an atlas of ${M}$ and ${M'}$ respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category).

Here are some basic examples of Riemann surfaces.

Example 2 (Quotients of ${{\bf C}}$) The complex numbers ${{\bf C}}$ clearly form a Riemann surface (using the identity map ${\phi: {\bf C} \rightarrow {\bf C}}$ as the single chart for an atlas). Of course, maps ${f: {\bf C} \rightarrow {\bf C}}$ that are holomorphic in the usual sense will also be holomorphic in the sense of the above definition, and vice versa, so the notion of holomorphicity for Riemann surfaces is compatible with that of holomorphicity for complex maps. More generally, given any discrete additive subgroup ${\Lambda}$ of ${{\bf C}}$, the quotient ${{\bf C}/\Lambda}$ is a Riemann surface. There are an infinite number of possible atlases to use here; one such is to pick a sufficiently small neighbourhood ${U}$ of the origin in ${{\bf C}}$ and take the atlas ${(\phi_\alpha: U_\alpha \rightarrow U)_{\alpha \in {\bf C}/\Lambda}}$ where ${U_\alpha := \alpha+U}$ and ${\phi_\alpha(\alpha+z) := z}$ for all ${z \in U}$. In particular, given any non-real complex number ${\omega}$, the complex torus ${{\bf C} / \langle 1, \omega \rangle}$ formed by quotienting ${{\bf C}}$ by the lattice ${\langle 1, \omega \rangle := \{ n + m \omega: n,m \in {\bf Z}\}}$ is a Riemann surface.

Example 3 Any open connected subset ${U}$ of ${{\bf C}}$ is a Riemann surface. By the Riemann mapping theorem, all simply connected open ${U \subset {\bf C}}$, other than ${{\bf C}}$ itself, are isomorphic (as Riemann surfaces) to the unit disk (or, equivalently, to the upper half-plane).

Example 4 (Riemann sphere) The Riemann sphere ${{\bf C} \cup \{\infty\}}$, as a topological manifold, is the one-point compactification of ${{\bf C}}$. Topologically, this is a sphere and is in particular connected. One can cover the Riemann sphere by the two open sets ${U_1 := {\bf C}}$ and ${U_2 := {\bf C} \cup \{\infty\} \backslash \{0\}}$, and give these two open sets the charts ${\phi_1: U_1 \rightarrow {\bf C}}$ and ${\phi_2: U_2 \rightarrow {\bf C}}$ defined by ${\phi_1(z) := z}$ for ${z \in {\bf C}}$, ${\phi_2(z) := 1/z}$ for ${z \in {\bf C} \backslash \{0\}}$, and ${\phi_2(\infty) := 0}$. This is a complex atlas since the ${1/z}$ is holomorphic on ${{\bf C} \backslash \{0\}}$.

An alternate way of viewing the Riemann sphere is as the projective line ${\mathbf{CP}^1}$. Topologically, this is the punctured complex plane ${{\bf C}^2 \backslash \{(0,0)\}}$ quotiented out by non-zero complex dilations, thus elements of this space are equivalence classes ${[z,w] := \{ (\lambda z, \lambda w): \lambda \in {\bf C} \backslash \{0\}\}}$ with the usual quotient topology. One can cover this space by two open sets ${U_1 := \{ [z,1]: z \in {\bf C} \}}$ and ${U_2: \{ [1,w]: w \in {\bf C} \}}$ and give these two open sets the charts ${\phi: U_1 \rightarrow {\bf C}}$ and ${\phi_2: U_2 \rightarrow {\bf C}}$ defined by ${\phi_1([z,1]) := z}$ for ${z \in {\bf C}}$, ${\phi_2([1,w]) := w}$. This is a complex atlas, basically because ${[z,1] = [1,1/z]}$ for ${z \in {\bf C} \backslash \{0\}}$ and ${1/z}$ is holomorphic on ${{\bf C} \backslash \{0\}}$.

Exercise 5 Verify that the Riemann sphere is isomorphic (as a Riemann surface) to the projective line.

Example 6 (Smooth algebraic plane curves) Let ${P(z_1,z_2,z_3)}$ be a complex polynomial in three variables which is homogeneous of some degree ${d \geq 1}$, thus

$\displaystyle P( \lambda z_1, \lambda z_2, \lambda z_3) = \lambda^d P( z_1, z_2, z_3). \ \ \ \ \ (1)$

Define the complex projective plane ${\mathbf{CP}^2}$ to be the punctured space ${{\bf C}^3 \backslash \{0\}}$ quotiented out by non-zero complex dilations, with the usual quotient topology. (There is another important topology to place here of fundamental importance in algebraic geometry, namely the Zariski topology, but we will ignore this topology here.) This is a compact space, whose elements are equivalence classes ${[z_1,z_2,z_3] := \{ (\lambda z_1, \lambda z_2, \lambda z_3)\}}$. Inside this plane we can define the (projective, degree ${d}$) algebraic curve

$\displaystyle Z(P) := \{ [z_1,z_2,z_3] \in \mathbf{CP}^2: P(z_1,z_2,z_3) = 0 \};$

this is well defined thanks to (1). It is easy to verify that ${Z(P)}$ is a closed subset of ${\mathbf{CP}^2}$ and hence compact; it is non-empty thanks to the fundamental theorem of algebra.

Suppose that ${P}$ is irreducible, which means that it is not the product of polynomials of smaller degree. As we shall show in the appendix, this makes the algebraic curve connected. (Actually, algebraic curves remain connected even in the reducible case, thanks to Bezout’s theorem, but we will not prove that theorem here.) We will in fact make the stronger nonsingularity hypothesis: there is no triple ${(z_1,z_2,z_3) \in {\bf C}^3 \backslash \{(0,0,0)\}}$ such that the four numbers ${P(z_1,z_2,z_3), \frac{\partial}{\partial z_j} P(z_1,z_2,z_3)}$ simultaneously vanish for ${j=1,2,3}$. (This looks like four constraints, but is in fact essentially just three, due to the Euler identity

$\displaystyle \sum_{j=1}^3 z_j \frac{\partial}{\partial z_j} P(z_1,z_2,z_3) = d P(z_1,z_2,z_3)$

that arises from differentiating (1) in ${\lambda}$. The fact that nonsingularity implies irreducibility is another consequence of Bezout’s theorem, which is not proven here.) For instance, the polynomial ${z_1^2 z_3 - z_2^3}$ is irreducible but singular (there is a “cusp” singularity at ${[0,0,1]}$). With this hypothesis, we call the curve ${Z(P)}$ smooth.

Now suppose ${[z_1,z_2,z_3]}$ is a point in ${Z(P)}$; without loss of generality we may take ${z_3}$ non-zero, and then we can normalise ${z_3=1}$. Now one can think of ${P(z_1,z_2,1)}$ as an inhomogeneous polynomial in just two variables ${z_1,z_2}$, and by nondegeneracy we see that the gradient ${(\frac{\partial}{\partial z_1} P(z_1,z_2,1), \frac{\partial}{\partial z_2} P(z_1,z_2,1))}$ is non-zero whenever ${P(z_1,z_2,1)=0}$. By the (complexified) implicit function theorem, this ensures that the affine algebraic curve

$\displaystyle Z(P)_{aff} := \{ (z_1,z_2) \in {\bf C}^2: P(z_1,z_2,1) = 0 \}$

is a Riemann surface in a neighbourhood of ${(z_1,z_2,1)}$; we leave this as an exercise. This can be used to give a coordinate chart for ${Z(P)}$ in a neighbourhood of ${[z_1,z_2,z_3]}$ when ${z_3 \neq 0}$. Similarly when ${z_1,z_2}$ is non-zero. This can be shown to give an atlas on ${Z(P)}$, which (assuming the connectedness claim that we will prove later) gives ${Z(P)}$ the structure of a Riemann surface.

Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface.

Exercise 8

• (i) Show that all (irreducible plane projective) algebraic curves of degree ${1}$ are isomorphic to the Riemann sphere. (Hint: reduce to an explicit linear polynomial such as ${z_3}$.)
• (ii) Show that all (irreducible plane projective) algebraic curves of degree ${2}$ are isomorphic to the Riemann sphere. (Hint: to reduce computation, first use some linear algebra to reduce the homogeneous quadratic polynomial to a standard form, such as ${z_1^2+z_2^2+z_3^2}$ or ${z_2 z_3 - z_1^2}$.)

Exercise 9 If ${a,b}$ are complex numbers, show that the projective cubic curve

$\displaystyle \{ [z_1, z_2, z_3]: z_2^2 z_3 = z_1^3 + a z_1 z_3^2 + b z_3^3 \}$

is nonsingular if and only if the discriminant ${-16 (4a^3 + 27b^2)}$ is non-zero. (When this occurs, the curve is called an elliptic curve (in Weierstrass form), which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.)

A recurring theme in mathematics is that an object ${X}$ is often best studied by understanding spaces of “good” functions on ${X}$. In complex analysis, there are two basic types of good functions:

Definition 10 Let ${X}$ be a Riemann surface. A holomorphic function on ${X}$ is a holomorphic map from ${X}$ to ${{\bf C}}$; the space of all such functions will be denoted ${{\mathcal O}(X)}$. A meromorphic function on ${X}$ is a holomorphic map from ${X}$ to the Riemann sphere ${{\bf C} \cup \{\infty\}}$, that is not identically equal to ${\infty}$; the space of all such functions will be denoted ${M(X)}$.

One can also define holomorphicity and meromorphicity in terms of charts: a function ${f: X \rightarrow {\bf C}}$ is holomorphic if and only if, for any chart ${\phi_\alpha: U_\alpha \rightarrow {\bf C}}$, the map ${f \circ \phi^{-1}_\alpha: \phi_\alpha(U_\alpha) \rightarrow {\bf C}}$ is holomorphic in the usual complex analysis sense; similarly, a function ${f: X \rightarrow {\bf C} \cup \{\infty\}}$ is meromorphic if and only if the preimage ${f^{-1}(\{\infty\})}$ is discrete (otherwise, by analytic continuation and the connectedness of ${X}$, ${f}$ will be identically equal to ${\infty}$) and for any chart ${\phi_\alpha: U_\alpha \rightarrow X}$, the map ${f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha) \rightarrow {\bf C} \cup \{\infty\}}$ becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite. One consequence of this alternate definition is that the space ${{\mathcal O}(X)}$ of holomorphic functions is a commutative complex algebra (a complex vector space closed under pointwise multiplication), while the space ${M(X)}$ of meromorphic functions is a complex field (a commutative complex algebra where every non-zero element has an inverse). Another consequence is that one can define the notion of a zero of given order ${k}$, or a pole of order ${k}$, for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting (from the holomorphicity of transition maps and their inverses) that this does not depend on the choice of chart. (However, one cannot similarly define the residue of a meromorphic function on ${X}$ this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic ${1}$-forms, a concept we will introduce later.) A third consequence is analytic continuation: if two holomorphic or meromorphic functions on ${X}$ agree on a non-empty open set, then they agree everywhere.

On the complex numbers ${{\bf C}}$, there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions (with non-zero denominator) will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem (covered in previous quarters).

It turns out, however, that the situation changes dramatically when the Riemann surface ${X}$ is compact, with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:

Lemma 11 Let ${f \in \mathcal{O}(X)}$ be a holomorphic function on a compact Riemann surface ${X}$. Then ${f}$ is constant.

This result should be seen as a close sibling of Liouville’s theorem that all bounded entire functions are constant. (Indeed, in the case of a complex torus, this lemma is a corollary of Liouville’s theorem.)

Proof: As ${f}$ is continuous and ${X}$ is compact, ${|f(z_0)|}$ must attain a maximum at some point ${z_0 \in X}$. Working in a chart around ${z_0}$ and applying the maximum principle, we conclude that ${f}$ is constant in a neighbourhood of ${z_0}$, and hence is constant everywhere by analytic continuation. $\Box$

This dramatically cuts down the number of possible meromorphic functions – indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional. The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic ${1}$-forms (also known as Abelian differentials).

A more detailed study of Riemann surface (and more generally, complex manifolds) can be found for instance in Griffiths and Harris’s “Principles of Algebraic Geometry“.

Next quarter (starting Monday, April 2) I will be teaching Math 246C (complex analysis) here at UCLA.  This is the third in a three-series graduate course on complex analysis; a few years ago I taught the first course in this series (246A), so this course can be thought of in some sense as a sequel to that one (and would certainly assume knowledge of the material in that course as a prerequisite), although it also assumes knowledge of material from the second course 246B (which covers such topics as Weierstrass factorization and the theory of harmonic functions).

246C is primarily a topics course, and tends to be a somewhat miscellaneous collection of complex analysis subjects that were not covered in the previous two installments of the series.  The initial topics I have in mind to cover are

• The Riemann-Roch theorem;
• Circle packings;
• The Bieberbach conjecture (proven by de Branges); and
• the Schramm-Loewner equation (SLE).
• This list is however subject to change (it is the first time I will have taught on any of these topics, and I am not yet certain on the most logical way to arrange them; also I am not completely certain that I will be able to cover all the above topics in ten weeks).  I welcome reference recommendations and other suggestions from readers who have taught on one or more of these topics.

As usual, I will be posting lecture notes on this blog as the course progresses.

[Update: Mar 13: removed elliptic functions, as I have just learned that this was already covered in the prior 246B course.]