Riemann surfaces

A planar domain whose complement in the Riemann sphere has exactly two connected components. An annulus is the standard example. If neither complementary component is a single point, the domain is conformally equivalent to a circular annulus with finite positive conformal modulus of an annulus; punctured-disc and punctured-plane cases account for degenerate complementary components.

A Jordan domain together with four distinct boundary points in cyclic order, regarded up to a conformal map preserving those marked points. Such a domain maps to a rectangle with its marked points at the four corners. The opposite boundary-arc pairs specify two dual curve-joining problems whose extremal lengths are reciprocal.

Map a conformal quadrilateral to a rectangle of width $w$ and height $h$, with the chosen opposite boundary arcs mapped to its vertical sides. Their extremal length is $M=w/h$; the other pair has extremal length $1/M$. The constant conformal metric is extremal. For an arbitrary density, average horizontal joining lengths over height and apply the Cauchy-Schwarz inequality to obtain $L_{\rho }^{2}h\le w{A}_{\rho }$, proving the upper bound. Vertical joining segments prove the dual formula. Conformal invariance of extremal length transports both equalities back to the quadrilateral.

Glue two copies of $[0,w]\times [0,h]$ along their left, bottom and right sides. Their unglued top sides form the outer boundary of a disk; the two bottom corners are marked interior points. Let $\Gamma$ be the loops surrounding both points, and ${\Gamma }^{\ast }$ the boundary-to-boundary crosscuts separating them. Then $\lambda (\Gamma )\lambda ({\Gamma }^{\ast })=4$.

Each vertical seam joins a marked point to the outer boundary, so the Jordan curve theorem forces a loop surrounding both points to meet both seams. In the flat conformal metric, its total horizontal travel is therefore at least $2w$. Doubled horizontal segments attain this length. Each separating crosscut meets the bottom seam and starts and ends at height $h$, so its total vertical travel is at least $2h$; doubled vertical segments attain it. The total area is $2wh$.

For an arbitrary density, integrate the lengths of the doubled horizontal segments over their height. The Cauchy-Schwarz inequality yields $L_{\rho }(\Gamma {)}^{2}h\le 2w{A}_{\rho }$. Doubled vertical segments similarly give $L_{\rho }({\Gamma }^{\ast }{)}^{2}w\le 2h{A}_{\rho }$. These upper bounds equal the flat-metric ratios, proving both displayed formulas. Mapping the upper and lower halves of a disc to these rectangles gives the corresponding duality for a simply connected domain with two marked interior points.

A meromorphic differential is specified in each holomorphic chart by a meromorphic coefficient $f(z)$, with the coefficients transforming by the coordinate derivative so that the one-form agrees on overlaps. The derivative of a meromorphic function gives such a differential, and the ratio of two nonzero differentials is a meromorphic function. Orders of zeros and poles are independent of chart because coordinate derivatives are nonvanishing holomorphic factors. On a compact connected surface the sum of orders of a nonzero differential is $2g-2$, allowing its divisor to determine the genus.

A noncompact Riemann surface is hyperbolic in potential theory when its Green-function exhaustion has a finite limit, equivalently when it admits a positive Green function on a Riemann surface. This definition does not assume a disk uniformization. The unit disk is an example, since $-\log |z|$ is its positive Green function with pole zero. For simply connected surfaces this is equivalent to being conformally equivalent to the disk. For general surfaces, potential-theoretic hyperbolicity should not be conflated with the condition that the universal cover is the disk: some surfaces with disk universal cover have no positive Green function.

A Riemann surface which is compact as a topological space. For a connected one of genus $g$, the Riemann-Roch theorem gives $g$ independent holomorphic differential forms. The Riemann bilinear relations for a compact surface make their normalized periods a full lattice in ${\mathbb{C}}^g$, and the Abel-Jacobi map of a compact Riemann surface turns this lattice into a complex torus encoding divisor classes.

Integration of holomorphic differential forms along cycles embeds the first homology group as a full Euclidean lattice in $H^0(C,K_C{)}^{\vee }$. Fixing a base point, sum the path integrals to the points of an effective complex analytic divisor. Different path choices differ by that lattice. The Abel theorem for divisors identifies the fibres with linear equivalence of divisors.

A degree-zero complex analytic divisor $E$ on a compact Riemann surface is principal exactly when its Abel-Jacobi map of a compact Riemann surface is zero. Normalize a differential of the third kind with residues equal to the coefficients of $E$. The Riemann bilinear relations for a compact surface express its $b$-periods as $2\pi i$ times the Abel integrals. If those integrals are in the period lattice, subtract a suitable integral multiple of the normalized holomorphic differential forms; all periods then lie in $2\pi i\mathbb{Z}$, so exponentiation gives the required meromorphic function. Conversely, the logarithmic differential of a meromorphic function has integer residues and periods in $2\pi i\mathbb{Z}$, giving a zero Abel class by the same calculation.

A meromorphic one-form on a compact Riemann surface with only simple poles is a differential of the third kind. Prescribed residues can be realized exactly when their sum is zero: in the residue exact sequence for $K_C(S)$, the obstruction in $H^1(C,K_C)\cong \mathbb{C}$ is the sum of residues. Subtracting a holomorphic differential form normalizes its $a$-periods. Integer residues and periods in $2\pi i\mathbb{Z}$ make the exponential of its integral a single-valued meromorphic function with those orders of zeros and poles.

For a compact Riemann surface with a symplectic basis $a_i,b_i$ of its first homology group, cutting along that basis and integrating a primitive givesfor closed smooth one-forms. For a holomorphic differential form $\omega$ and a differential of the third kind $\eta$, the corresponding boundary calculation givesThe paths and representatives of the cycles are chosen in the same cut surface. Normalized holomorphic differential forms consequently have a symmetric period matrix with positive definite imaginary part.

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 $K$.

If a conformal map or anticonformal map $T$ is an involution preserving a curve family, its extremal length can be computed using invariant conformal metrics. Replace any density by ${\rho }_s=(\rho +T^{\ast }\rho )/2$. Every curve and its image have length at least $L_{\rho }$, hence $L_{{\rho }_s}\ge L_{\rho }$. Convexity of the square and change of variables give $A_{{\rho }_s}\le A_{\rho }$, proving the assertion. For reflection in the real axis, invariant means $\rho (z)=\rho (\bar{z})$.^[ref]

For a conformal map $f$ and a conformal metric $\rho$ on its image, the pullback density is $\rho (f(z))|f^{\prime }(z)|$. The change of variables formula preserves its metric area, and the chain rule preserves the length of each corresponding curve. Applying this to $f$ and its inverse proves the title equality. This allows extremal length computations on rectangles, discs and annuli to be transported to other domains.

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 $M$.

For an inner circle of real center $c$ and radius $d$ lying strictly inside the unit circle, choose the real root $|a|<1$ of $c{a}^2-(1+c^2-d^2)a+c=0$. When $c=0$, take $a=0$. The automorphism of the unit disk $T_a$ sends the inner circle to a concentric circle. Indeed it sends the two real endpoints $c-d,c+d$ to opposite real numbers; preservation of circles and reflection symmetry then give an image centered at zero. Its radius is $r=T_a(c+d)\in (0,1)$.

The conformal invariance of harmonic measure and harmonic measure of an annulus give outer-boundary measure $\log (|T_a(z)|/r)/\log (1/r)$. For $c=d=2/5$, the quadratic gives $a=1/2$ and $r=1/2$, an explicit eccentric-annulus example.

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 $z$.

It is the holomorphic-function case of a germ.

A function element is a pair $(f,D)$ consisting of a nonempty disk and a holomorphic function on that disk. Its local value determines a germ of a holomorphic function. Elements agreeing on an overlapping neighborhood continue one another; chains of such continuations produce the complete analytic function generated by an element. An algebraic identity holding in one element persists along any continuation chain by the identity theorem.

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.

The level-two principal congruence subgroup $\Gamma (2)\subset {\mathrm{PSL}}_2(\mathbb{R})$ consists of integer determinant-one matrices congruent to the identity modulo two, modulo the scalar sign. It acts freely and properly discontinuously on the complex upper half-plane. An ideal quadrilateral with vertices $-1,0,1,\infty$ has its paired sides identified by $\tau \mapsto \tau +2$ and $\tau \mapsto \tau /(1-2\tau )$. The quotient has genus zero and three punctures, so it is conformally the sphere with three points removed. The invariant metric $|d\tau |/\mathrm{Im}\tau$ descends to its curvature-minus-one hyperbolic metric.

// Destination: analysis.bigb

Exhaust a simply connected hyperbolic Riemann surface in potential theory by compact bordered Jordan domains $R_n$. Their Green functions on a Riemann surface $G_n(\cdot ,p)$ have logarithmic poles and zero boundary values. Exponentiating their harmonic conjugates produces proper degree-one maps $f_n:R_n\to \mathbb{D}$ with $f_n(p)=0$. If $G_n=-\log |\zeta |+c_n+o(1)$, normalize $f_n^{\prime }(p)=e^{-c_n}>0$. Domain monotonicity and a global positive Green function bound $c_n$ increasingly from above. Thus Montel theorem gives a nonconstant limit $F$, which is injective by the locally uniform limit of univalent functions theorem. The Schwarz lemma makes $F^{\prime }(p)$ extremal among normalized disk-valued maps. If $F$ omitted $a\ne 0$, taking a global holomorphic square root of a disk automorphism applied to $F$, then renormalizing at $p$, would multiply its derivative by $(1+|a|)/(2\sqrt{|a|})>1$. Thus $F$ is onto and gives the desired biholomorphism.

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

For a univalent function $f:\Omega \to \mathbb{D}$ with $f(z_0)=0$, suppose $a\ne 0$ is omitted. Apply the disk automorphism $T_a$ so the resulting image omits zero, take a holomorphic square root, then apply the disk automorphism taking the value at $z_0$ to zero. If $\Omega$ is a simply connected domain, the root exists. The resulting normalized univalent map has derivative magnitude multiplied by $(1+|a|)/(2\sqrt{|a|})>1$, since $(1-\sqrt{|a|}{)}^2>0$. Thus a map maximizing the derivative cannot omit an interior disk point.

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.

For a normalized univalent function $f$ on the unit disc, put $r=|z|<1$. The displayed sharp two-sided bounds hold. Rotations $z/(1-e^{i\theta }z{)}^2$ of the Koebe function attain the bounds along suitable radial directions. In particular, bounding the modulus controls the radius of a disk containing each smaller image $f(r\mathbb{D})$.

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 $w\in S$,

The degree counts sheets with multiplicity of the associated branched covering.

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.

For a nonconstant rational function of degree $n$, its derivative has degree between $n-1$ and $2n$. A finite pole of order $m$ becomes one of order $m+1$. If infinity is a pole of order $p-q>0$, it contributes $p-q-1$ to the derivative's pole count, where $p,q$ are the degrees of coprime numerator and denominator. This proves the lower bound; the quotient rule denominator $Q^2$ proves the upper bound. The functions $z^n$ and $1/(z^n-1)$ attain the bounds.

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, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

Let $f:X\to Y$ be a nonconstant holomorphic map between compact Riemann surfaces that are connected spaces, without boundary. Write $g_X,g_Y$ for their genera and $d\ge 1$ for the degree of a holomorphic map. With analytic ramification index $e_p$ at each $p\in X$, the formula isHere $\chi =2-2g$ is the Euler characteristic. Only finitely many ramification points contribute. The holomorphic map is automatically surjective; the valency theorem gives ${\sum }_{p\in f^{-1}(y)}{e}_p=d$ even at a branch value of a holomorphic map. Constant maps are excluded.^[ref]

The useful interpretation is that total ramification of a holomorphic map measures how much the source Euler characteristic falls short of $d$ copies of the target. The ramification of a power map of the Riemann sphere spends that entire correction at two points; the double cover of the sphere with four branch values spends it at four points and produces a genus-one surface.

Count ramification points upstairs in $X$, weighted by $e_p-1$, rather than counting distinct branch values of a holomorphic map downstairs in $Y$. Grouping by fibres givesThe cardinality counts distinct points; the degree of a holomorphic map counts them with their analytic ramification indices. Thus each summand is the number of missing distinct preimages in that fibre.^[ref]

The phrase branch point varies across conventions: to keep the two surfaces distinct, use ramification point for $p\in X$ and branch value of a holomorphic map for $f(p)\in Y$. The existing name branch point of a holomorphic map here denotes the latter. Several ramification points can share one branch value of a holomorphic map.

For an integer $d\ge 2$, extend $f(z)=z^d$ to a holomorphic map from the Riemann sphere to itself with $f(\infty )=\infty$. A nonzero finite regular value has $d$ distinct preimages, so the degree of a holomorphic map is $d$. At zero the analytic ramification index is $d$. At infinity, use local coordinates $u=1/z$ upstairs and $v=1/f(z)$ downstairs: the same holomorphic map becomes $v=u^d$, so $e_{\infty }=d$ too. Everywhere else the derivative $d{z}^{d-1}$ is nonzero.

There are two distinct ramification points and two distinct branch values of a holomorphic map, butFor $d=3$, the total ramification of a holomorphic map is four, not two. The local-circle panel shows three nearby preimages converging to the single ramification point. For $d=1$ the holomorphic map is the identity and neither zero nor infinity is ramified.

Suppose a nonconstant holomorphic map $f:X\to {\mathbb{P}}^1$ from a connected compact Riemann surface has degree of a holomorphic map two and exactly four distinct branch values of a holomorphic map. The valency theorem forces each of their fibres to contain one ramification point with $e_p=2$, a simple ramification point of a holomorphic map. HenceThis determines the genus without recognizing a torus in a drawing.

A concrete realization is the compact Riemann surface obtained by adding the two points at infinity to the even-degree hyperelliptic model given by the algebraic curveProjection to $z$ has degree of a holomorphic map two. At a simple zero $a$ of $P$, $w$ is a local coordinate and $z-a=w^2/P^{\prime }(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm 1$, each unramified since $u$ is a local coordinate. The four finite roots are therefore all the branch values of a holomorphic map.^[ref]

Choose a triangulation of a surface on the Riemann sphere with these four branch values of a holomorphic map as vertices, combinatorially the boundary of a tetrahedron. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving Euler characteristic $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The Triangulation proof of the Riemann-Hurwitz formula applies the same counting idea after refining a general triangulation of a surface around the branch values of a holomorphic map.^[ref]

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 ramification point is simple when its analytic ramification index is two. Its contribution to total ramification of a holomorphic map is one. In suitable local coordinates, the holomorphic map is $t\mapsto t^2$; two nearby preimages merge there. A ramification point with $e_p=3$ contributes two even though it is just one point.

For a nonconstant rational map, choose local source and target coordinates centred at $p$ and $R(p)$. If the local expression is $a{u}^e+O(u^{e+1})$ with $a\ne 0$, its local degree is $e$, and $p$ is critical exactly when $e\ge 2$. This definition covers poles and infinity; a simple pole is not critical merely because the usual finite-coordinate expression is unbounded. Coordinate changes have nonzero first derivatives and leave $e$ unchanged.

A critical orbit is the forward sequence $p,R(p),R^2(p),\dots$ starting at a critical point of a rational map. The postcritical set records the forward images, usually from the first iterate onward, of all critical points; a single critical orbit and the union of all such images are distinct notions. For quadratic polynomials there is one finite critical point, zero, and its orbit controls connectedness of the filled Julia set and membership in the Mandelbrot set. A bounded orbit can be preperiodic to a repelling cycle, as $0\mapsto i\mapsto -1+i\mapsto -i\mapsto -1+i$ for $z^2+i$; boundedness then does not imply Fatou membership.

Local degrees multiply under composition: $e_p(S\circ R)=e_p(R)e_{R(p)}(S)$, directly by substituting the two local power-series expansions. ConsequentlyIn particular every critical point of $R$ remains critical for each positive iterate, and every critical point of an iterate reaches a critical point of $R$ in fewer than $n$ steps. This covers infinity as well as finite points. For polynomials at finite points the same conclusion follows from $(P^n{)}^{\prime }(z)={\prod }_{j=0}^{n-1}{P}^{\prime }(P^j(z))$.

At a point of local degree $e$, the critical multiplicity is $e-1$. At an ordinary finite point this is the order of vanishing of the derivative. The invariant local-coordinate definition also works at poles and infinity. For $R(z)=z^d$, zero and infinity both have multiplicity $d-1$, so a critical-point count must count multiplicities: there are two distinct critical points but total multiplicity $2d-2$.

A degree-$d$ rational map has total critical multiplicity $2d-2$. Here is an algebraic proof. Critical points are isolated zeros of local derivatives and are finite in number by compactness. Change domain and range Möbius coordinates so that the poles are $d$ simple finite poles and infinity is a regular point with a finite image. Write $R=P/Q$, with $\mathrm{deg}Q=d$, and $R(z)=a+b/z+O(z^{-2})$, $b\ne 0$. Then $W=P^{\prime }Q-P{Q}^{\prime }$ has degree exactly $2d-2$, since $R^{\prime }=W/Q^2\sim -b/z^2$. At a pole, $W=-P{Q}^{\prime }\ne 0$. Thus the roots of $W$ are precisely the critical points and their zero orders are the critical multiplicities. The fundamental theorem of algebra gives the count; the coordinate changes preserve it.

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.

In the compactified Riemann surface of $w^r=z^n-1$, a positive circuit enclosing the $n$ simple finite zeros sends the sheet label $j$ to $j+n$ modulo $r$. This permutation has $d=\gcd (n,r)$ cycles of length $r/d$. Each cycle fills in one point over infinity, with ramification index $r/d$. Hence infinity is unramified exactly when $r$ divides $n$. Each finite zero has ramification index $r$, so the base sphere has exactly $n$ branch values in that case and $n+1$ otherwise.

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.