19 Cutting into pants and varying twists

19.1 What data a cut curve retains

Fix an oriented hyperbolic surface of finite type, without cone points, and a pants decomposition with internal cuff lengths $a_1,\ldots,a_N>0$, where $N=3g-3+n$. Boundary lengths, including cusps, are fixed. Apart from components parallel to a cutting cuff, cutting a simple multicurve gives disjoint essential arcs in each pair of pants. Cutting a general immersed curve, even a filling one, need not give disjoint arcs: self-intersections may remain inside a pair of pants. Nor are the actual segments generally orthogonal to the cuffs.

Boundary intersection counts alone lose the relative homotopy classes of the arcs, their cyclic joining pattern, and winding information. For simple multicurves this information is organized by Dehn–Thurston coordinates. For a general immersed curve one must retain a fuller itinerary. Once this topological data is fixed, varying the metric twist is a well-defined and tractable problem.

There are two complementary exact descriptions. One uses arc lengths as generating functions in endpoint positions. The other uses products of holonomy matrices. Both keep the information that intersection counts alone discard.

19.2 An exact local distance function

Take two distinct ultraparallel geodesic lifts of positive-length cuffs in $\mathbb H^2$, at common perpendicular distance $d>0$. Let $u,v$ be signed distances on the two lines from the perpendicular feet, with compatible orientations. Then $$\boxed{ D_d(u,v)=\mathop{\mathrm{arcosh}}\bigl(\cosh d\cosh u\cosh v-\sinh u\sinh v\bigr).}$$(19.1) For example, in the hyperboloid model use $$p(u)=(\cosh u,\sinh u,0),\qquad q(v)=(\cosh d\cosh v,\sinh v,\sinh d\cosh v).$$ Then $\cosh D=-\langle p,q\rangle$ for the Lorentz form of signature $(-,+,+)$, and $\mathop{\mathrm{arcosh}}$ is its nonnegative real branch. Reversing one endpoint coordinate changes the sign of the mixed term.

For the ordinary seam between cuffs $a,b>0$ of pants with third cuff $c\geq0$, the right-angled-hexagon relation gives $$\cosh d=\frac{\cosh(c/2)+\cosh(a/2)\cosh(b/2)} {\sinh(a/2)\sinh(b/2)}.$$(19.2) It can also be checked from pants holonomy: translations of lengths $a,b$ on axes at distance $d$, with the opposed boundary directions, have product trace $2\cosh(a/2)\cosh(b/2)-2\cosh d\sinh(a/2)\sinh(b/2)$. Equating this to $-2\cosh(c/2)$ gives (19.2), with the same lift signs as (18.1). For a returning or winding arc, choose the boundary lifts appropriate to its relative homotopy class and use their common perpendicular instead. There is no single seam length that describes all relative arc classes.

For the reduced itinerary of a fixed nonperipheral closed curve, fix the boundary lifts for each segment and retain its integer winding data. Cuff gluing then imposes affine relations on the real endpoint positions, involving the twists $\tau_i$. Minimize the sum of the corresponding functions (19.1) over the crossing positions. Every competitor is a piecewise geodesic in the fixed free-homotopy class, so its length is at least the closed geodesic length. The geodesic representative realizes equality: geodesics have no bigons with the cutting cuffs and therefore realize the same reduced itinerary. At the minimum, first variation imposes smooth tangent matching at every crossing. A curve contained entirely in one pair of pants is handled directly by its holonomy; it has no endpoint variables.

The endpoint Hessian will also control variation in a twist. Put $z=\cosh D_d(u,v)$ and $S=\sqrt{z^2-1}>0$. Direct differentiation gives $$\begin{aligned} \nabla^2D_d&=\frac{\sinh^2d}{S^3} \begin{pmatrix} z\cosh^2v&-\cosh u\cosh v\\ -\cosh u\cosh v&z\cosh^2u \end{pmatrix},\\ \det\nabla^2D_d&= \frac{\sinh^4d\cosh^2u\cosh^2v}{S^4}>0. \end{aligned}$$(19.3) For example, $z_{uu}=z$ and $z^2-z_u^2-1=\sinh^2d\cosh^2v$, while $z_{uv}(z^2-1)-zz_uz_v=-\sinh^2d\cosh u\cosh v$. These identities give the displayed matrix. Its positive diagonal and determinant prove strict joint convexity in the two endpoint coordinates. Reversing an endpoint orientation conjugates the Hessian by a sign matrix and preserves positive definiteness.

19.3 Holonomy and first variation

In a basis adapted to a cuff axis, a twist displacement is represented by $$T_i(\tau_i)=\mathop{\mathrm{diag}}(e^{\tau_i/2},e^{-\tau_i/2}).$$ For a fixed nonperipheral curve word $\gamma$, insert these matrices and their inverses into the fixed pants holonomies. The resulting product $W_\gamma$ satisfies $$2\cosh(\ell_\gamma/2) =\left|\mathop{\mathrm{tr}}W_\gamma(\mathbf a,T_1(\tau_1),\ldots,T_N(\tau_N))\right|.$$(19.4) With the cuff lengths fixed, its trace is a finite Laurent polynomial in $e^{\tau_i/2}$. Its coefficients need not all be positive in an arbitrary choice of lifts and coordinates.

The twist derivative has the cosine formula $$\frac{\partial\ell_\gamma}{\partial\tau_i} =\sum_{p\in\gamma\cap\alpha_i}\cos\vartheta_p, \qquad \left|\frac{\partial\ell_\gamma}{\partial\tau_i}\right| \le i(\gamma,\alpha_i).$$(19.5) Angles and twist orientation are chosen consistently, and intersections are counted by passages, including repeated passages for an iterate. Cutting at each crossing proves the cosine formula: endpoint variations inside a segment cancel by stationarity, while the relative displacement across the cuff contributes the tangential projection $\cos\vartheta_p$ [24]. The same argument applies to immersed geodesics. If $\gamma$ is a power of a cuff, the intersection sum is empty and the derivative is zero.

Length is strictly convex in one twist $\tau_i$ whenever $i(\gamma,\alpha_i)>0$. Here is a proof that includes nonsimple curves. Hold all pants lengths and the other twists fixed, and use the fixed reduced itinerary above. The segment distances $d$ remain positive and fixed. For two twist values choose the minimizing endpoint configurations. Their affine interpolation is admissible for the interpolated twist. The two full endpoint vectors cannot coincide: at a crossing of $\alpha_i$, the gluing equation contains $\tau_i$ with coefficient $1$ or $-1$, with fixed winding data. Thus a change of this twist cannot be absorbed while leaving all segment endpoints fixed. Strict convexity of the sum in (19.3), followed by minimization, gives the strict convexity inequality for the geodesic length. This also explains why the conclusion excludes a disjoint cuff, whose twist leaves the length constant. The simple-curve earthquake theorem provides a broader context [65, Theorem 1]; no extension of that cited statement is needed for the local argument here.

These ideas overlap with several parts of Mirzakhani’s work, but serve different roles there. Dehn–Thurston coordinates and the definition of combinatorial length for simple multicurves are given in [77, Theorem 3.4 and equation (3.6)]. The comparison with geometric length for an $L$-bounded pants decomposition is [77, Proposition 3.5 and equation (3.7), p. 107]. The work on general curve orbits studies asymptotically piecewise-linear length, including filling curves and length sublevel volumes [78, Theorem 4.1 and Section 8]. In the basic volume recursion, the cutting curves are simple and the kernel depends on their lengths, so the twist integral itself is just the period $a_i$.

19.4 Unfolding a twist lattice

Let $D_i$ be the Dehn twist about the $i$th cuff. With a consistent sign convention, $$\ell_{D_1^{k_1}\cdots D_N^{k_N}\gamma}(\mathbf a,\boldsymbol\tau) =\ell_\gamma(\mathbf a,\boldsymbol\tau+\mathbf k\mathbf a).$$ For a nonnegative function $F$, Tonelli’s theorem gives $$\boxed{ \int_{\prod_i[0,a_i)}\sum_{\mathbf k\in\mathbb Z^N} F\bigl(\ell_{D_1^{k_1}\cdots D_N^{k_N}\gamma} (\mathbf a,\boldsymbol\tau)\bigr)\,\mathop{}\!\mathrm d\boldsymbol\tau =\int_{\mathbb R^N}F\bigl(\ell_\gamma(\mathbf a,\mathbf u)\bigr)\,\mathop{}\!\mathrm d\mathbf u.}$$(19.6) As written, the sum is over lattice elements, so the nonnegative equality is valid even if both sides are infinite. For signed functions, impose absolute integrability on $\mathbb R^N$. To interpret the sum as one term per curve, first quotient any stabilizer and unfold the corresponding nonredundant twist directions; twisting about a disjoint cuff must not produce infinitely many copies of one curve. This quotient may change the lattice and its fundamental domain, so one cannot simply divide an infinite multiplicity out of the displayed formula.

A fixed curve’s length is not periodic in a twist coordinate. The sum over its twist orbit is periodic, which is exactly what makes this unfolding work. The accompanying WP measure is $\prod_i\mathop{}\!\mathrm da_i\,\mathop{}\!\mathrm d\tau_i$. However, the positive cuff orthant times one twist box is the quotient by this particular twist subgroup, not a fundamental domain for the full mapping class group. A further full-moduli integration needs the appropriate orbit unfolding.

For example, with weight $(2\cosh(\ell_\gamma/2))^{-s}$ and $x_i=e^{u_i/2}$, the right side becomes the Mellin-type integral $$2^N\int_{(0,\infty)^N} \left|\mathop{\mathrm{tr}}W_\gamma(\mathbf a,\mathbf x)\right|^{-s} \prod_i\frac{\mathop{}\!\mathrm dx_i}{x_i},$$(19.7) whenever it converges. This is an explicit analytic object attached to the curve itinerary and the fixed pants lengths.