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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).

In the previous set of notes we introduced the notion of a complex diffeomorphism ${f: U \rightarrow V}$ between two open subsets ${U,V}$ of the complex plane ${{\bf C}}$ (or more generally, two Riemann surfaces): an invertible holomorphic map whose inverse was also holomorphic. (Actually, the last part is automatic, thanks to Exercise 40 of Notes 4.) Such maps are also known as biholomorphic maps or conformal maps (although in some literature the notion of “conformal map” is expanded to permit maps such as the complex conjugation map ${z \mapsto \overline{z}}$ that are angle-preserving but not orientation-preserving, as well as maps such as the exponential map ${z \mapsto \exp(z)}$ from ${{\bf C}}$ to ${{\bf C} \backslash \{0\}}$ that are only locally injective rather than globally injective). Such complex diffeomorphisms can be used in complex analysis (or in the analysis of harmonic functions) to change the underlying domain ${U}$ to a domain that may be more convenient for calculations, thanks to the following basic lemma:

Lemma 1 (Holomorphicity and harmonicity are conformal invariants) Let ${\phi: U \rightarrow V}$ be a complex diffeomorphism between two Riemann surfaces ${U,V}$.

• (i) If ${f: V \rightarrow W}$ is a function to another Riemann surface ${W}$, then ${f}$ is holomorphic if and only if ${f \circ \phi: U \rightarrow W}$ is holomorphic.
• (ii) If ${U,V}$ are open subsets of ${{\bf C}}$ and ${u: V \rightarrow {\bf R}}$ is a function, then ${u}$ is harmonic if and only if ${u \circ \phi: U \rightarrow {\bf R}}$ is harmonic.

Proof: Part (i) is immediate since the composition of two holomorphic functions is holomorphic. For part (ii), observe that if ${u: V \rightarrow {\bf R}}$ is harmonic then on any ball ${B(z_0,r)}$ in ${V}$, ${u}$ is the real part of some holomorphic function ${f: B(z_0,r) \rightarrow {\bf C}}$ thanks to Exercise 62 of Notes 3. By part (i), ${f \circ \phi: B(z_0,r) \rightarrow {\bf C}}$ is also holomorphic. Taking real parts we see that ${u \circ \phi}$ is harmonic on each ball ${B(z_0,r)}$ in ${V}$, and hence harmonic on all of ${V}$, giving one direction of (ii); the other direction is proven similarly. $\Box$

Exercise 2 Establish Lemma 1(ii) by direct calculation, avoiding the use of holomorphic functions. (Hint: the calculations are cleanest if one uses Wirtinger derivatives, as per Exercise 27 of Notes 1.)

Exercise 3 Let ${\phi: U \rightarrow V}$ be a complex diffeomorphism between two open subsets ${U,V}$ of ${{\bf C}}$, let ${z_0}$ be a point in ${U}$, let ${m}$ be a natural number, and let ${f: V \rightarrow {\bf C} \cup \{\infty\}}$ be holomorphic. Show that ${f: V \rightarrow {\bf C} \cup \{\infty\}}$ has a zero (resp. a pole) of order ${m}$ at ${\phi(z_0)}$ if and only if ${f \circ \phi: U \rightarrow {\bf C} \cup \{\infty\}}$ has a zero (resp. a pole) of order ${m}$ at ${z_0}$.

From Lemma 1(ii) we can now define the notion of a harmonic function ${u: M \rightarrow {\bf R}}$ on a Riemann surface ${M}$; such a function ${u}$ is harmonic if, for every coordinate chart ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ in some atlas, the map ${u \circ \phi_\alpha^{-1}: V_\alpha \rightarrow {\bf R}}$ is harmonic. Lemma 1(ii) ensures that this definition of harmonicity does not depend on the choice of atlas. Similarly, using Exercise 3 one can define what it means for a holomorphic map ${f: M \rightarrow {\bf C} \cup \{\infty\}}$ on a Riemann surface ${M}$ to have a pole or zero of a given order at a point ${p_0 \in M}$, with the definition being independent of the choice of atlas.

In view of Lemma 1, it is thus natural to ask which Riemann surfaces are complex diffeomorphic to each other, and more generally to understand the space of holomorphic maps from one given Riemann surface to another. We will initially focus attention on three important model Riemann surfaces:

• (i) (Elliptic model) The Riemann sphere ${{\bf C} \cup \{\infty\}}$;
• (ii) (Parabolic model) The complex plane ${{\bf C}}$; and
• (iii) (Hyperbolic model) The unit disk ${D(0,1)}$.

The designation of these model Riemann surfaces as elliptic, parabolic, and hyperbolic comes from Riemannian geometry, where it is natural to endow each of these surfaces with a constant curvature Riemannian metric which is positive, zero, or negative in the elliptic, parabolic, and hyperbolic cases respectively. However, we will not discuss Riemannian geometry further here.

All three model Riemann surfaces are simply connected, but none of them are complex diffeomorphic to any other; indeed, there are no non-constant holomorphic maps from the Riemann sphere to the plane or the disk, nor are there any non-constant holomorphic maps from the plane to the disk (although there are plenty of holomorphic maps going in the opposite directions). The complex automorphisms (that is, the complex diffeomorphisms from a surface to itself) of each of the three surfaces can be classified explicitly. The automorphisms of the Riemann sphere turn out to be the Möbius transformations ${z \mapsto \frac{az+b}{cz+d}}$ with ${ad-bc \neq 0}$, also known as fractional linear transformations. The automorphisms of the complex plane are the linear transformations ${z \mapsto az+b}$ with ${a \neq 0}$, and the automorphisms of the disk are the fractional linear transformations of the form ${z \mapsto e^{i\theta} \frac{\alpha - z}{1 - \overline{\alpha} z}}$ for ${\theta \in {\bf R}}$ and ${\alpha \in D(0,1)}$. Holomorphic maps ${f: D(0,1) \rightarrow D(0,1)}$ from the disk ${D(0,1)}$ to itself that fix the origin obey a basic but incredibly important estimate known as the Schwarz lemma: they are “dominated” by the identity function ${z \mapsto z}$ in the sense that ${|f(z)| \leq |z|}$ for all ${z \in D(0,1)}$. Among other things, this lemma gives guidance to determine when a given Riemann surface is complex diffeomorphic to a disk; we shall discuss this point further below.

It is a beautiful and fundamental fact in complex analysis that these three model Riemann surfaces are in fact an exhaustive list of the simply connected Riemann surfaces, up to complex diffeomorphism. More precisely, we have the Riemann mapping theorem and the uniformisation theorem:

Theorem 4 (Riemann mapping theorem) Let ${U}$ be a simply connected open subset of ${{\bf C}}$ that is not all of ${{\bf C}}$. Then ${U}$ is complex diffeomorphic to ${D(0,1)}$.

Theorem 5 (Uniformisation theorem) Let ${M}$ be a simply connected Riemann surface. Then ${M}$ is complex diffeomorphic to ${{\bf C} \cup \{\infty\}}$, ${{\bf C}}$, or ${D(0,1)}$.

As we shall see, every connected Riemann surface can be viewed as the quotient of its simply connected universal cover by a discrete group of automorphisms known as deck transformations. This in principle gives a complete classification of Riemann surfaces up to complex diffeomorphism, although the situation is still somewhat complicated in the hyperbolic case because of the wide variety of discrete groups of automorphisms available in that case.

We will prove the Riemann mapping theorem in these notes, using the elegant argument of Koebe that is based on the Schwarz lemma and Montel’s theorem (Exercise 57 of Notes 4). The uniformisation theorem is however more difficult to establish; we discuss some components of a proof (based on the Perron method of subharmonic functions) here, but stop short of providing a complete proof.

The above theorems show that it is in principle possible to conformally map various domains into model domains such as the unit disk, but the proofs of these theorems do not readily produce explicit conformal maps for this purpose. For some domains we can just write down a suitable such map. For instance:

Exercise 6 (Cayley transform) Let ${{\bf H} := \{ z \in {\bf C}: \mathrm{Im} z > 0 \}}$ be the upper half-plane. Show that the Cayley transform ${\phi: {\bf H} \rightarrow D(0,1)}$, defined by

$\displaystyle \phi(z) := \frac{z-i}{z+i},$

is a complex diffeomorphism from the upper half-plane ${{\bf H}}$ to the disk ${D(0,1)}$, with inverse map ${\phi^{-1}: D(0,1) \rightarrow {\bf H}}$ given by

$\displaystyle \phi^{-1}(w) := i \frac{1+w}{1-w}.$

Exercise 7 Show that for any real numbers ${a, the strip ${\{ z \in {\bf C}: a < \mathrm{Re}(z) < b \}}$ is complex diffeomorphic to the disk ${D(0,1)}$. (Hint: use the complex exponential and a linear transformation to map the strip onto the half-plane ${{\bf H}}$.)

We will discuss some other explicit conformal maps in this set of notes, such as the Schwarz-Christoffel maps that transform the upper half-plane ${{\bf H}}$ to polygonal regions. Further examples of conformal mapping can be found in the text of Stein-Shakarchi.