Riemann surfaces

For a finite group generated by $A,B$, glue sheets labeled by the group over the three-punctured Riemann sphere using left monodromy multiplication. Fill the punctures by the local power charts dictated by the orders $m,n,r$ of $A,B,AB$. Right multiplication by inverses defines the commuting deck action. Its nontrivial stabilizers are conjugates of the corresponding cyclic subgroups. The Riemann-Hurwitz formula gives the displayed Euler characteristic. A compatible metric can be averaged over the group; in genus at least two the invariant hyperbolic metric is canonical by the uniformization theorem.

For covers branched over three real points with a real base point, choose the two generating loops so complex conjugation reverses each. A sheet permutation satisfying the displayed identities lifts that conjugation to an anticonformal equivalence of the covers, including the filled branch points. For closed hyperbolic Riemann surfaces this is a Riemannian isometry. Such an isometry need not come from conjugating the subgroups inside the original deck group, so a Gassmann equivalent pair can be isometric despite nonconjugacy there.

An orientation-preserving homeomorphism with locally $L^2$ weak derivatives and essentially bounded infinitesimal distortion. In holomorphic coordinates, its Beltrami coefficient satisfies $\Vert \mu {\Vert }_{\infty }<1$, and the maximal dilatation is $(1+\Vert \mu {\Vert }_{\infty })/(1-\Vert \mu {\Vert }_{\infty })$.

The essential supremum of the ratio of the largest to the smallest infinitesimal singular value of a quasiconformal map. It equals $(1+\Vert {\mu }_f{\Vert }_{\infty })/(1-\Vert {\mu }_f{\Vert }_{\infty })$ in terms of the Beltrami coefficient.

For an orientation-preserving quasiconformal map, the almost-everywhere coefficient ${\mu }_f=f_{\bar{z}}/f_z$. It is a coordinate-dependent representative of a tensor; the modulus and the resulting maximal dilatation are coordinate-independent.

The equation $f_{\bar{z}}=\mu f_z$ prescribing the Beltrami coefficient of a quasiconformal map. Two homeomorphic solutions on the same source differ by a biholomorphism between their targets.

In a holomorphic coordinate, a length element $\rho (z)|dz|$. Smooth positive densities define ordinary Riemannian metrics; extremal length also permits nonnegative measurable densities with finite positive area $\int {\rho }^{2}dxdy$.

A surface with an atlas outside finitely many cone points whose changes of coordinate are translations. On a compact surface it is equivalent to a nonzero holomorphic one-form. A zero of order $m$ has cone angle $2\pi (m+1)$.

Cut equal straight slits in two translation surfaces of genus one and cross-glue their banks by translation. The connected sum of oriented manifolds has genus two and two cone points of angle $4\pi$, hence two simple zeros of its holomorphic one-form. Choosing one torus of side $\delta$ and a slit of length ${\delta }^2$ creates a flat-small handle without forcing its generator to have small extremal length.

The study of deformations and markings of Riemann surfaces, using quasiconformal maps, holomorphic quadratic differentials, extremal length and the mapping class group.

A quasiconformal map whose Beltrami coefficient is $k|q|/q$ for an integrable nonzero holomorphic quadratic differential and constant $0<k<1$. In a flat coordinate for $q$, it stretches the horizontal and vertical directions with ratio $K=(1+k)/(1-k)$. Positive multiples of $q$ define the same coefficient.

On a closed genus-at-least-two Riemann surface, a Teichmüller map uniquely minimizes maximal dilatation in its homotopy class among maps to the same target. The Reich–Strebel inequality forces equality of Beltrami coefficients for an extremal competitor; their conformal difference is homotopic to the identity and hence is the identity. In genus one, translations must be factored out.

On closed Riemann surfaces of genus at least two, if a Teichmüller map $f_0:X\to Y$ has unit-area source differential $q$ and dilatation $K_0$, any quasiconformal map $f:X\to Y$ in the same homotopy class satisfies $K_0\le {\int }_X|q||1+{\mu }_{f}q/|q|{|}^2/(1-|{\mu }_f{|}^2)$. The plus sign corresponds to a Teichmüller map with ${\mu }_{f_0}=k_0|q|/q$ and $K_0=(1+k_0)/(1-k_0)$. This fundamental inequality yields Teichmüller's uniqueness theorem by the pointwise triangle inequality and its equality case. See Gardiner and Hu, §5, equation (11).

For a path family $\Gamma$ on a Riemann surface, take the supremum of $L_{\rho }(\Gamma {)}^2/A_{\rho }$ over measurable conformal metrics of finite positive area. This is unchanged under conformal equivalence, and a quasiconformal map of maximal dilatation $K$ changes it by a factor between $K^{-1}$ and $K$.

Let $\alpha$ be a real closed differential form on a Riemann surface, of finite positive conformal energy $E=\int |\alpha {|}^{2}dA$. If its period on a loop homotopy class $\gamma$ is $p$, then $\lambda (\gamma ,X)\ge p^2/E$. Indeed the conformal metric $\rho =|\alpha |$ has area $E$ and each loop in the class has length at least $|\int \alpha |=|p|$. Energy is independent of the auxiliary smooth conformal metric used to compute the pointwise norm: the inverse scaling of the squared norm cancels the area scaling.

The height divided by circumference in a conformal Euclidean cylinder model of an annulus. For $r<|z|<R$, it is $(2\pi {)}^{-1}\log (R/r)$. The extremal length of winding-one core curves is $1/M$, whereas that of curves joining the two boundary components is $M$.

The quotient of a horizontal strip by a horizontal translation. The model $(\mathbb{R}/\mathbb{Z})\times (0,M)$ has conformal modulus of an annulus $M$. Averaging horizontal loop lengths and applying Cauchy-Schwarz inequality proves that the core-loop extremal length equals $1/M$.

For nonzero holomorphic one-forms and holomorphic quadratic differentials, apply $A\in {\mathrm{SL}}_2(\mathbb{R})$ to every flat coordinate. Translation and sign transition maps remain of the same type, defining a new complex structure and differential. Composition gives a group action, zero orders persist and area is preserved. The zero section has no such atlas; fixing it gives only a set-theoretic extension that is generally discontinuous.

For a fixed closed oriented surface $S_g$, a point is a marked Riemann surface $(X,f:S_g\to X)$, modulo biholomorphisms intertwining the markings up to homotopy. For $g\ge 2$, Fenchel–Nielsen coordinates give real dimension $6g-6$.

A local coordinate at a point $p$ of a Riemann surface is a biholomorphism from a neighbourhood of $p$ to an open subset of $\mathbb{C}$. After translating the image, one may require that the coordinate sends $p$ to zero.

A map between Riemann surfaces is holomorphic when its expression in every pair of local complex coordinates is a holomorphic function.

A biholomorphism is a bijective holomorphic map whose inverse is holomorphic. A degree-one nonconstant holomorphic map between compact connected Riemann surfaces is a biholomorphism.

A germ at $z\in D$ is an equivalence class $[f{]}_z$ of holomorphic functions defined near $z$, where two representatives are equivalent when they agree on some neighbourhood of $z$.

The space $\mathcal{G}$ of germs over a domain $D$ has basic open sheetsfor holomorphic $f$ on open $U\subseteq D$. The projection $\pi ([f{]}_z)=z$ restricts to a homeomorphism on each sheet, and these projections form holomorphic coordinate charts.

The projection $[f{]}_z\mapsto z$ is a local biholomorphism for the sheet topology on the space of germs of holomorphic functions. It need not be a topological covering map: branches may end at singularities even when another branch remains over that base point.

The inverse germs of $w^3-3w$ over $\mathbb{C}$ form the connected surface $\mathbb{C}\setminus \{-1,1\}$. Its projection is surjective and locally biholomorphic, but the fibres have three points generically and one point at each critical value $\pm 2$, so it is not an evenly covered topological covering.

The evaluation map $\mathcal{E}([f{]}_z)=f(z)$ is holomorphic. On the sheet associated with $(U,f)$, its coordinate expression is exactly the holomorphic function $f$.

Over $D=\mathbb{C}\setminus \{z:z^8=1\}$, the two-valued square root is represented byThe maps $\pi (z,w)=z$ and $\mathcal{E}(z,w)=w$ identify this unbranched double cover analytically with the corresponding component of the space of germs.

If $\pi :S\to R$ is a covering of a Riemann surface, compose every chart on an evenly covered open set with each local inverse sheet of $\pi$. The resulting transition maps are those of $R$, so they define a unique complex structure making $\pi$ a local biholomorphism.

Every simply connected Riemann surface is biholomorphic to the Riemann sphere, the complex plane, or the unit disc.

Every nonempty simply connected proper open subset of the complex plane is conformally equivalent to the unit disc.

For a bounded simply connected domain $D$ and a conformal bijection $f:\mathbb{D}\to D$, $f$ extends continuously to the closed disc if and only if $\partial D$ is a locally connected space. A Jordan boundary gives a homeomorphism of closed discs, but local connectedness alone need not give injectivity on the boundary: a slit has two boundary approaches. The same statement applies on the sphere to unbounded domains after a suitable change of coordinates. This boundary theorem is different from the Caratheodory extension theorem for measures.

The Koebe distortion theorem gives universal upper and lower bounds on a univalent function and its derivative in terms of the distance to the boundary. In particular, derivatives at two points whose hyperbolic distance is bounded differ by at most a universal multiplicative factor.

If $f$ is a univalent function on the unit disc, then $B(f(0),|f^{\prime }(0)|/4)\subseteq f(\mathbb{D})$. The constant $1/4$ is sharp, as shown by the Koebe function. This gives useful comparisons between the derivative of a conformal map, its conformal radius, and its distance to the domain boundary.

For a conformal bijection $f:D\to \tilde{D}$ between proper planar domains, put $d=\mathrm{dist}(z,\partial D)$ and $\tilde{d}=\mathrm{dist}(f(z),\partial \tilde{D})$. Apply the Koebe quarter theorem to $(f(z+dw)-f(z))/(d{f}^{\prime }(z))$ on the unit disc to obtain $d|f^{\prime }(z)|/4\le \tilde{d}$. Apply it again to $f^{-1}$ to get the opposite bound. Thus $\boxed{\tilde{d}/(4d)\le |f^{\prime }(z)|\le 4\tilde{d}/d}$. The domains need not be simply connected domains. For the whole plane, the boundary distances are infinite and these ratios are undefined.

The sphere has the Möbius group $PG{L}_2(\mathbb{C})$, the plane has the affine maps $z\mapsto az+b$ with $a\ne 0$, and the disc has the maps $e^{i\theta }(z-a)/(1-\bar{a}z)$ with $|a|<1$.

The quotients of $\mathbb{C}$ by free properly discontinuous conformal actions are $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, and the complex tori $\mathbb{C}/\Lambda$. The translation group has respectively rank zero, one, or two.

Every plane domain whose complement contains at least two points is uniformized by the unit disc. A plane universal cover would give a nonconstant entire function omitting two values, contrary to the Little Picard theorem.

Let $R$ be a compact Riemann surface of genus $g$, and remove $n$ distinct points. The punctured surface is uniformized by the unit disc exactly when $2g-2+n>0$. It is uniformized by the complex plane exactly when $2g-2+n$ is $-1$ or $0$, corresponding respectively to $\mathbb{C}$, and to ${\mathbb{C}}^{\ast }$ or a complex torus. The sole spherical case is $g=n=0$.

If ${\mathbb{C}}^{\ast }$ embeds holomorphically in a compact Riemann surface $R$, the embedding extends across zero and infinity to a degree-one holomorphic map from the Riemann sphere to $R$. Hence $R$ is conformally the sphere.

Two holomorphic functions on a connected Riemann surface that agree on a set with an accumulation point agree everywhere. Local charts reduce the proof to isolated zeros of a one-variable holomorphic function.

A real-valued function on a Riemann surface is harmonic when its expression in every holomorphic coordinate chart has vanishing planar Laplacian.

Under a holomorphic coordinate change $w=w(z)$,so the condition $\Delta H=0$ is independent of the holomorphic chart.

A covering is regular when its deck group acts transitively on each fibre, equivalently when its fundamental-group subgroup is normal.

A complete analytic function is the collection of all continuations of a germ. Its Riemann surface consists of these germs, with projection to their base points.

For a nonconstant analytic map $f:R\to S$ between compact connected Riemann surfaces, there is an integer $\mathrm{deg}f$ such that for every $w\in S$,

If a nonconstant holomorphic map has local-coordinate expressionthen $m=m_f(p)$ is its local degree, multiplicity, or ramification index at $p$.

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, its degree isThe valency theorem says that this integer is independent of $y\in Y$.

For $k\ge 1$, the power map has $k$ distinct preimages of every nonzero finite regular value, all with positive local degree of a map between oriented manifolds. Integrating the pullback of a differential form of $\Omega =id\zeta \wedge d\bar{\zeta }/(1+|\zeta {|}^2{)}^2$ gives the same answer, since $\int \Omega =2\pi$ and $\int f^{\ast }\Omega =2\pi k$. For $k=0$ use the constant extension $1$ at infinity; its degree of a map between oriented manifolds is zero.

After cancelling common factors, the rational map $p/q$ on the Riemann sphere has degree $\max (\mathrm{deg}p,\mathrm{deg}q)$. It is an analytic isomorphism exactly when this degree is one, equivalently when it is a Möbius transformation.

Let a nonconstant rational function $f$ have degree $d$, let $r$ be its number of distinct finite poles, and let $m_{\infty }$ be its pole order at infinity, taken as zero when infinity is not a pole. Differentiation raises each finite pole order by one and changes a pole of order $m_{\infty }>0$ at infinity into one of order $m_{\infty }-1$. HenceIn particular, $d-1\le \mathrm{deg}{f}^{\prime }\le 2d$.

The degree of a nonconstant elliptic function is the sum of the orders of its poles in a fundamental parallelogram. Equivalently, it is the degree of the induced holomorphic map from its complex torus to the Riemann sphere.

If an elliptic function $g$ has degree $d$ and $r$ distinct poles in a fundamental parallelogram, then each pole order increases by one under differentiation, soBecause $1\le r\le d$, this gives $d+1\le \mathrm{deg}{g}^{\prime }\le 2d$.

The rotation group of the octahedron has order $24$. Its vertex, face-centre, and edge-centre stabilizers have orders $4,3,2$, giving exceptional orbit sizes $6,8,12$; every other orbit has size $24$.

Let a finite group $G$ of order $d$ act analytically on a compact Riemann surface, and let an invariant analytic map $F$ have degree $d$. At a point with stabilizer of order $e$, invariance forces the local multiplicity of $F$ to be at least $e$. Its orbit has $d/e$ points, so it already contributes at least $d$ to the fibre multiplicity. The valency theorem forces equality and shows that every fibre is exactly one orbit.

For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

For a nonconstant holomorphic map of Riemann surfaces, choose local coordinates centred at $p$ and $f(p)$. The map has the formThe positive integer $e_p$ is its ramification index. The point is ramified exactly when $e_p>1$ and contributes $e_p-1$ to the ramification divisor.

A point $p$ is ramified when its local degree $e_p$ is greater than one.

A branch value is the image of a ramification point of a holomorphic map. Away from all branch values, a proper holomorphic map is an ordinary unramified covering.

Triangulate the target with all branch values among its vertices. For a degree-$d$ map, every edge and face has $d$ lifts. Above a target vertex $v$, the number of lifted vertices isbecause ${\sum }_{p\mapsto v}{e}_p=d$. Thuswhich is equivalent to the Riemann-Hurwitz formula.

Up to invertible affine changes in source and target, every complex cubic polynomial is exactly one ofThe extension to the Riemann sphere has two finite units of ramification. A single finite critical point has ramification index three and gives the first form; two distinct simple critical points can be moved to $\pm 1$, after which integration gives the second.

The Riemann sphere is $\mathbb{C}\cup \{\infty \}$ with two charts related by reciprocal coordinates.

Stereographic projection identifies a sphere minus one pole with the plane and supplies the standard charts of the Riemann sphere.

On the unit sphere, use $\zeta =(x+iy)/(1-z)$ away from the north pole and $\eta =(x-iy)/(1+z)$ away from the south pole. Their overlap relation $\eta =1/\zeta$ is a holomorphic map, giving a holomorphic atlas for the Riemann sphere. The opposite sign of $iy$ in the second manifold chart ensures a holomorphic rather than antiholomorphic transition.

Under the stereographic projection $w=(x+iy)/(1-z)$, rotations about the third axis have representatives $\mathrm{diag}(e^{i\theta /2},e^{-i\theta /2})$, and rotations about the second axis have representativesThese belong to the special unitary group $SU(2)$. Euler-angle generation of the special orthogonal group $SO(3)$ shows that every sphere rotation becomes a Möbius transformation represented in $SU(2)$. Representatives $U$ and $-U$ induce the same transformation.

Let a finite group $H$ of conformal automorphisms of the Riemann sphere fix $p$. The derivative representation at $p$ embeds $H$ into ${\mathbb{C}}^{\times }$, so $H$ is cyclic. Averaging a local coordinate linearizes the action to $z\mapsto \zeta z$, and the quotient has coordinate $u=z^{|H|}$.

For a finite conformal group action, choose a disc around each point that meets only its stabilizer translates. The local cyclic quotient chart $z\mapsto z^e$, where $e$ is the stabilizer order, gives the orbit space a Riemann-surface structure and makes the quotient map holomorphic.

For $r(z)=\zeta z$ and $s(z)=1/z$, where ${\zeta }^n=1$, the rational functionis invariant under $D_{2n}$. Equality of two values factors as $(u-v)(uv-1)=0$ for $u=z_1^n$ and $v=z_2^n$, so every fibre is exactly one dihedral orbit.

Removing a closed discrete set from a Riemann surface leaves an open complex one-manifold; if connected, it is again a Riemann surface.

A path in a surface can be perturbed inside coordinate discs to avoid finitely many prescribed points.

The punctured plane ${\mathbb{C}}^{\ast }$ is a connected Riemann surface homeomorphic to a cylinder.

Polar coordinates give ${\mathbb{C}}^{\ast }\cong S^1\times \mathbb{R}$ after taking logarithmic radius.

A homeomorphism to a complex manifold transports its atlas and thereby defines a complex structure on the source.

A nodal crossing locally consists of two complex branches meeting transversely and is not a one-dimensional complex manifold at the intersection.

If a punctured neighborhood has a different number of connected components from a punctured Euclidean ball, the point is not a manifold point.

A reducible complex curve is a union of proper complex subcurves; intersecting components can create singular points.