Teichmüller theory

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 $6g-6$.