Riemann Surfaces

Riemann surfaces provide the natural geometric setting for multi-valued functions like $\sqrt{z}$ and $\log z$ . They are one-dimensional complex manifolds and serve as the bridge between complex analysis and algebraic geometry.

Motivation and Definition

Definition10.1Riemann surface

A Riemann surface is a connected, second-countable Hausdorff topological space $S$ equipped with a complex atlas: a collection of charts $\{(U_\alpha, \varphi_\alpha)\}$ where $U_\alpha$ are open sets covering $S$ , $\varphi_\alpha: U_\alpha \to \mathbb{C}$ are homeomorphisms onto open subsets of $\mathbb{C}$ , and the transition functions $\varphi_\beta \circ \varphi_\alpha^{-1}$ are holomorphic.

ExampleFundamental examples

The complex plane $\mathbb{C}$ with the identity chart.
The Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ with charts $z$ and $1/z$ .
The square root surface: The Riemann surface of $\sqrt{z}$ is a two-sheeted cover of $\mathbb{C}$ branched at $0$ and $\infty$ . Topologically, it is a sphere.
The logarithm surface: The Riemann surface of $\log z$ is an infinite-sheeted cover of $\mathbb{C}^*$ , topologically a helicoid.
Elliptic curves: $y^2 = 4x^3 - g_2 x - g_3$ defines a Riemann surface of genus $1$ (a torus).

Holomorphic Maps Between Riemann Surfaces

Definition10.2Holomorphic map

A continuous map $f: S_1 \to S_2$ between Riemann surfaces is holomorphic if for every pair of charts $(U_\alpha, \varphi_\alpha)$ on $S_1$ and $(V_\beta, \psi_\beta)$ on $S_2$ , the composition $\psi_\beta \circ f \circ \varphi_\alpha^{-1}$ is holomorphic wherever defined.

A bijective holomorphic map with holomorphic inverse is a biholomorphism (or conformal equivalence).

RemarkClassification by genus

Compact Riemann surfaces are classified topologically by their genus $g$ (the number of "handles"):

$g = 0$ : Riemann sphere (conformally unique).
$g = 1$ : Tori $\mathbb{C}/\Lambda$ (parametrized by the modular parameter $\tau$ ).
$g \geq 2$ : Moduli space is a complex variety of dimension $3g - 3$ .

Covering Spaces and Monodromy

Definition10.3Covering map

A holomorphic map $\pi: \tilde{S} \to S$ is a covering map if every point $p \in S$ has a neighborhood $U$ such that $\pi^{-1}(U)$ is a disjoint union of open sets, each mapped homeomorphically onto $U$ by $\pi$ . The number of sheets (possibly infinite) is the degree of the covering.

ExampleCovering maps

$z \mapsto z^n$ is an $n$ -sheeted covering $\mathbb{C}^* \to \mathbb{C}^*$ (branched covering of $\mathbb{C}$ ).
$z \mapsto e^z$ is a universal covering $\mathbb{C} \to \mathbb{C}^*$ with deck group $\mathbb{Z}$ (generated by $z \mapsto z + 2\pi i$ ).
The projection $\mathbb{C} \to \mathbb{C}/\Lambda$ is a universal covering of a torus.

The Uniformization Theorem

Theorem10.1Uniformization theorem

Every simply connected Riemann surface is biholomorphic to exactly one of:

The Riemann sphere $\hat{\mathbb{C}}$ ,
The complex plane $\mathbb{C}$ ,
The unit disk $\mathbb{D}$ .

Consequently, every Riemann surface is a quotient of one of these three by a group of deck transformations acting freely and properly discontinuously.

RemarkConsequences of uniformization

Compact surfaces of genus $0$ : universal cover is $\hat{\mathbb{C}}$ (the surface itself).
Compact surfaces of genus $1$ : universal cover is $\mathbb{C}$ (the surface is $\mathbb{C}/\Lambda$ ).
Compact surfaces of genus $\geq 2$ : universal cover is $\mathbb{D}$ (the surface is $\mathbb{D}/\Gamma$ for a Fuchsian group $\Gamma$ ).

This trichotomy mirrors the three geometries: spherical, Euclidean, and hyperbolic.