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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<b}, 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}}.)

Exercise 8 Show that for any real numbers {a<b<a+2\pi}, the strip {\{ re^{i\theta}: r>0, a < \theta < b \}} is complex diffeomorphic to the disk {D(0,1)}. (Hint: use a branch of either the complex logarithm, or of a complex power {z \mapsto z^\alpha}.)

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.

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