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The total angle around the tip of a Euclidean cone. For a translation surface zero of order $m$ it equals $2\pi (m+1)$, while for a holomorphic quadratic differential zero of order $m$ it equals $(m+2)\pi$. An angle $2\pi$ is a regular point.

For a nonvanishing local $m$-differential $\varphi (z)d{z}^m$, choose $w=\int {\varphi }^{1/m}dz$. Then the differential is $d{w}^m$. Different root choices give $w\mapsto \zeta w+b$, ${\zeta }^m=1$. For $m=1,2$ the permitted linear parts are translations or signs, producing translation surfaces and half-translation surfaces.

First use the nonzero loci, as required by the standard SL2R action on differentials. A nonzero holomorphic one-form has, away from its zeros, flat coordinateswhose changes of coordinate are translations. A nonzero holomorphic quadratic differential similarly has local flat coordinates $w=\int \sqrt{q}$, with $q=d{w}^2$ and changes of coordinate $w_j=\pm w_i+c_{ij}$. These are respectively translation surfaces and half-translation surfaces.

Identify a flat coordinate with a vector in ${\mathbb{R}}^2$. For $A\in {\mathrm{SL}}_2(\mathbb{R})$, replace every flat coordinate by $W=Aw$. Since $A$ preserves orientation and commutes with multiplication by $-1$, the new changes of coordinate areThey are holomorphic in the new coordinates, and so define a new complex structure. Define ${\omega }_A=dW$ or $q_A=d{W}^2$ in that structure. The forms glue because translations preserve $dW$, and the extra signs preserve $d{W}^2$.

The zeros also extend. A zero of order $m$ of a holomorphic one-form has cone angle $2\pi (m+1)$; a zero of order $m$ of a holomorphic quadratic differential has cone angle $(m+2)\pi$. The real-linear deformation preserves the corresponding winding multiplicity. Filling the cone in a local coordinate $\zeta$ gives $W={\zeta }^{m+1}$ in the first case, or a local branch of $W={\zeta }^{(m+2)/2}$ in the second. Thus the resulting forms are constant multiples of ${\zeta }^{m}d\zeta$ or ${\zeta }^{m}d{\zeta }^2$ and have the same zero orders. This verifies extension across the missing points, rather than merely producing an atlas on the punctured surface.

An isomorphism preserving the original differential identifies its flat coordinates up to the permitted translations or signs; applying $A$ identifies the deformed atlases too. Hence the construction descends to the corresponding moduli spaces. Applying $B$ after $A$ replaces $w$ by $BAw$, soThe area of a quadratic differential, and the analogous area of a holomorphic one-form, are preserved because $\det A=1$.

For $q={\omega }^2$, the flat coordinates obtained from $\omega$ already give the required half-translation surface atlas for $q$. The same replacement $w\mapsto Aw$ therefore constructs both deformations, and

The printed sets include identically zero differentials. They have no flat coordinates, so the customary geometric group action is defined on the nonzero loci. One can obtain a set-theoretic action on the displayed entire sets by declaring $A\cdot (X,0)=(X,0)$; the same equivariance identity then holds at zero. This extension is generally not continuous: as $t\omega \to 0$, the deformed underlying surface is the same $X_A$ for every real $t>0$, and can differ from $X$. Thus a claim about the standard continuous geometric group action requires the nonzero convention.

The analogous flat-coordinate construction does not define a full ${\mathrm{SL}}_2(\mathbb{R})$ action on arbitrary cubic differentials. Away from zeros, a nonzero holomorphic cubic differential has flat coordinatesTheir changes of coordinate are $w_j=\zeta w_i+b_{ij}$ with ${\zeta }^3=1$. A branch change is therefore a rotation through $2\pi /3$, not merely a sign. Applying a real-linear map $A$ changes its linear part to $AR{A}^{-1}$, where $R$ is that rotation. In general this is not a complex-linear map, so the proposed changes of coordinate are not holomorphic and cannot define the required deformed complex structure.

For example, take $A=\mathrm{diag}(2,1/2)$ and $R=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ with $\theta =2\pi /3$. Thenwhose off-diagonal entries fail the condition for a complex-linear map. This dependence on the choice of cube-root coordinate is the obstruction even when considering descent from the locus $c={\omega }^3$: the three possible roots need not lead to the same deformation of the cubic pair.

The real matrices preserving orientation that normalize the order-three rotations are precisely the matrices of complex-linear maps; intersecting with ${\mathrm{SL}}_2(\mathbb{R})$ leaves $\mathrm{SO}(2)$. Indeed a nonreal rotation determines its complex structure, and conjugation to its inverse would reverse that structure's orientation. Thus there is a natural rotation action,The conclusion concerns the geometric group action analogous to that for translation surfaces and half-translation surfaces; it does not rule out artificial group actions unrelated to these atlases.

Label the polygon vertices $v_0,\dots ,v_{2n-1}$ cyclically. Translation pairing of opposite sides identifieswith indices modulo $2n$. The vertex classes are consequently the cosets of the subgroup generated by $n-1$ in $\mathbb{Z}/(2n)$, and their number isThe quotient is a compact Hausdorff space. An interior point has a disk neighbourhood, a paired-side point has two half-disks joined to a disk, and each vertex class has its incident sectors cyclically joined to a cone, topologically a disk. Thus it is a connected closed surface with an orientation. Its Euler characteristic is $V-n+1$, since there are $n$ paired edges and one face. Hence

The polygon interior and paired-side charts are translation surface charts with ${\omega }_n=dz$. Each corner angle is $(n-1)\pi /n$. For even $n$, all $2n$ corners meet, giving cone angle $2\pi (n-1)$; for odd $n$, each of the two classes contains $n$ corners, giving cone angle $(n-1)\pi$. Both are integral multiples of $2\pi$. A cone of angle $2\pi (k+1)$ has the local uniformizing coordinate $\zeta$ with $w={\zeta }^{k+1}$; the holomorphic one-form is $(k+1){\zeta }^{k}d\zeta$. Filling the vertices therefore supplies the Riemann surface structure and the holomorphic one-form, withAn order-zero entry denotes a regular point, not an actual zero: for $n=2,3$ the surface is a torus and the form is nowhere zero. The zero orders otherwise sum to $2g-2$, as a check against the degree of the canonical bundle.

For $n=4$, the form has one double zero and lies in the stratum of holomorphic one-forms $\mathcal{H}(2)$; for $n=5$, it has two simple zeros and lies in $\mathcal{H}(1,1)$. The SL2R action on differentials preserves zero multiplicities, as the local cone argument shows. ThereforeTheir equal genus and area do not distinguish the orbits; their different strata of holomorphic one-forms do.

The given $\gamma$ is a simple closed curve. Assume it is essential; for a contractible class both infimal lengths are already zero. Write ${\ell }_n={\ell }_{X_n}(\gamma )$. The collar lemma supplies an embedded annulus with coordinates $r\in (-w_n,w_n)$ and $t\in \mathbb{R}/\mathbb{Z}$, and metricSet $y(r)={\ell }_n^{-1}{\int }_0^r\mathrm{sech}sds$. This changes the metric to a positive scalar multiple of $d{t}^2+d{y}^2$, so the conformal modulus of an annulus isThe extremal length of its core curves is $1/M_n$. Allowing all curves homotopic to $\gamma$ in $X_n$ can only decrease infimal lengths, while the area of a metric on all of $X_n$ is at least its area on the collar. Thus $\lambda (\gamma ,X_n)\le 1/M_n$. Use the particular conformal metric $|q_n{|}^{1/2}$, whose area of a quadratic differential is one:This proves the required implication uniformly over all the area-one holomorphic quadratic differentials on these surfaces.

The converse is false. Here is an explicit slit connected sum of translation tori. Start with square flat copies of a torus $T_1=\mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$ and $T_{\delta }=\mathbb{C}/(\delta \mathbb{Z}+i\delta \mathbb{Z})$, where $\delta \to 0$. Cut a horizontal slit of physical length ${\delta }^2$ in each, centred in an interior coordinate disk, and cross-glue the banks by translation. The resulting surface has genus two. Its two slit endpoints have cone angle $4\pi$, so the locally defined $dz$ extends to a holomorphic one-form ${\omega }_{\delta }$ with two simple zeros. Its flat area is $1+{\delta }^2$. Setwhich is a holomorphic quadratic differential of area one. Let $\gamma$ be a horizontal generator in the small torus, taken away from the slit and fixed by the marking of this small handle. Then

To verify that its hyperbolic length does not tend to zero, construct a uniform lower bound on extremal length. On the unit square torus choose a disk $D$ about the eventual slit centre and a smooth cutoff function $\chi$ equal to one on a smaller disk and supported in $D$. Let $x$ denote a local real coordinate on $D$; in the first term below, $dx$ is the globally defined torus one-form, while $\chi x$ is extended by zero outside $D$. The real closed differential formis globally defined, vanishes on the smaller disk, and has period one on the horizontal generator. Pull it to $T_{\delta }$ by the rescaling map $z\mapsto z/\delta$, and extend it by zero across the slit and over the other torus. For sufficiently small $\delta$, the slit is inside the region where the form vanishes. Hence this extension ${\alpha }_{\delta }$ is a smooth closed differential form on the connected sum, with ${\int }_{\gamma }{\alpha }_{\delta }=1$.

Define a nonnegative conformal metric density by the pointwise norm of ${\alpha }_{\delta }$ relative to the flat metric. Two-dimensional scale invariance givesindependently of $\delta$. For every representative $\tilde{\gamma }$ homotopic to $\gamma$,because the period of a closed differential form is unchanged by homotopy. This is the extremal length lower bound from a closed one-form; therefore $\lambda (\gamma ,X_{\delta })\ge 1/C>0$. If ${\ell }_{X_{\delta }}(\gamma )$ tended to zero along any subsequence, the collar estimate would force $\lambda (\gamma ,X_{\delta })\to 0$, a contradiction. In fact its hyperbolic lengths are uniformly bounded away from zero. ThusFor every fixed $g>2$, replace $T_1$ by the area-one translation surface $X_{2g-2}$ constructed in the polygon argument; its genus is $g-1$. Cut its slit inside a nonsingular flat coordinate disk. The new connected sum has genus $g$, and the area normalization, small-handle length bound and closed-one-form energy argument are unchanged. Thus the converse fails at every fixed genus $g\ge 2$.

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.

On an affine space modeled on a vector space, the map $T_v(x)=x+v$ for a fixed vector $v$. Its derivative is the identity, and $T_v{T}_w=T_{v+w}$. Changes between flat coordinates on a translation surface have this form.