Strebel Differentials on Riemann Surfaces

Updated 10 December 2025

Strebel differentials are meromorphic quadratic differentials on compact Riemann surfaces defined by double poles and closed horizontal trajectories that partition the surface into flat cylinders and critical graphs.
They induce a precise ribbon graph structure, providing cell decompositions of moduli spaces and bridging complex analysis with combinatorial geometry.
Applications of Strebel differentials span integrable systems, string field theory, and random matrix models, linking geometric decompositions to physical and statistical phenomena.

A Strebel differential, also known in the literature as a Jenkins–Strebel differential, is a meromorphic quadratic differential on a compact Riemann surface whose horizontal trajectories are almost everywhere closed, decomposing the surface into a finite union of flat cylinders joined along a finite metric ribbon (or critical) graph. The Strebel condition imposes strong constraints on the global foliation structure of the Riemann surface and leads to profound connections with the geometry of moduli spaces, combinatorial models of curves, integrable systems, and physical theories such as closed string field theory.

1. Definition and Existence–Uniqueness

A Strebel differential on a compact Riemann surface $(C; p_1,\dots,p_n)$ with $n$ marked points is a meromorphic quadratic differential $q(z)dz^2$ with the following properties:

$q$ is holomorphic outside the punctures and has only double poles at the points $p_j$ , with local expansion:

$q(z)\,dz^2 = -\left(\frac{L_j}{2\pi}\right)^2 \frac{dz^2}{z^2} \;(1+O(z)),$

where $L_j > 0$ are prescribed cylinder circumferences.

The horizontal trajectories of $q$ (locally $\Im \int^z \sqrt{q} = \text{const}$ ) are either closed loops around the punctures or connect zeroes of $q$ .
Outside a measure-zero subset (the critical graph), the surface is disjointly foliated by closed loops forming flat cylinders of the assigned circumferences.

The Strebel existence–uniqueness theorem establishes: for any compact Riemann surface of genus $g$ with $n$ marked points (with $2g - 2 + n > 0$) and any positive real vector $(L_1, \ldots, L_n)$ , there is a unique Strebel differential on $(C; p_1,\ldots,p_n)$ with cylinder circumferences $L_j$ at $p_j$ (Ishibashi, 15 Feb 2024, Bertola et al., 2018, Song et al., 2017).

2. Local and Global Structure: Critical Graphs and Ribbon Graphs

The horizontal foliation of a Strebel differential decomposes the surface into:

Open cylinders (maximal ring domains) foliated by closed trajectories around the marked points.
A finite critical (or ribbon) graph: the union of horizontal trajectories that connect zeros of $q$ . Each edge carries a flat length:

$\ell_e = \int_{e} \bigl|\sqrt{q(z)}\,dz\bigr|,$

and vertices correspond to zeroes of order $k$ (valence $k+2$ ).

The topological type of the ribbon graph encodes the combinatorics of the foliation, and its metric structure (all edge lengths) captures the shape of the Strebel differential (Bertola et al., 2018, Bogatyrev, 2010).

The complement of the critical graph yields a partition of the surface into $n$ open disks, each containing exactly one pole—a property unique to Strebel differentials (Bertola et al., 2018, Song et al., 2017).

3. Characterization via Extremal Problems and the Surface of Circumferences

Jenkins–Strebel differentials solve an extremal length problem for prescribed homotopy classes of curves: Given a collection $\Gamma = \{\gamma_1, \ldots, \gamma_k\}$ of nontrivial, pairwise disjoint, non-homotopic simple closed curves and positive circumferences $A_j > 0$ , the unique Jenkins–Strebel differential maximizes

$\sum_{j=1}^k A_j^2 \, \text{Mod}(R_j),$

with $R_j$ the maximal ring domains of type $\gamma_j$ (Amano, 2022).

This gives rise to the “surface of circumferences” $\mathcal{A}(\Sigma, \Gamma)$ , a smooth, convex hypersurface in $\mathbb{R}_{\geq 0}^k$ parameterizing Jenkins–Strebel differentials by their normalized squared circumferences. Degenerations of ring domains correspond to boundary points of this surface; under deformation of the Riemann surface, $\mathcal{A}$ varies continuously in Teichmüller space (Amano, 2022).

4. Moduli Space, Ribbon Graphs, and Homological Structure

Strebel differentials induce a cell decomposition of the moduli space $\mathcal{M}_{g,n}$ of punctured Riemann surfaces via their critical graphs (ribbon graphs). Given a topological type, the metric ribbon graph is uniquely realized (up to isomorphism) by a Strebel differential with assigned perimeters—the contour lengths around each puncture (Bertola et al., 2018, Zorich, 2010).

On the double cover branched at zeros and double poles (canonical cover), the periods of $\sqrt{q}$ on anti-invariant homology cycles provide Darboux coordinates for the real symplectic form (Kontsevich's form) on the combinatorial moduli cell—as explicit period integrals: $z_{i} = \int_{\gamma_i^-} \sqrt{q},\;\; i=1,\ldots,2g_{-} + n.$ These homological (period) coordinates enable explicit computations in flat geometry and intersection theory on moduli space (Bertola et al., 2018).

Tau-functions (Hodge and Prym) constructed via the geometry of these ribbon graphs yield explicit sections of determinant line bundles and encode Chern classes in terms of their arguments winding around codimension-two loci in the ribbon-graph complex, providing combinatorial analogues of Mumford's relations (Bertola et al., 2018).

5. Explicit Models and Applications: Scattering, Random Matrices, and Physical Theories

Matrix Models, Scattering, and Strebel Graphs

In certain $n$ -particle scattering configurations, the saddle-point equations for a Penner-type matrix model yield, in the large- $N$ limit, a spectral curve whose analytic structure is the Strebel differential on a three-punctured Riemann sphere. The residues match kinematic variables, and the critical graph encodes the localization of solutions (with edge lengths as linear functions of the kinematics) (Maity, 2021).

Random Matrix Theory and Boutroux Differentials

Numerical algorithms for constructing rational Jenkins–Strebel differentials with prescribed polar behavior utilize Boutroux conditions—purely imaginary periods of the square root differential—to produce minimal-capacity sets (“S-curves”) in potential theory and serve as global models for steepest descent in Riemann–Hilbert problems for random matrices (Bertola, 27 Aug 2024). The critical trajectories give extremal networks, and the metric geometry matches that of the Strebel ribbon graphs.

String Theory and Moduli Parametrization

In closed string field theory, contact interactions (joining of semi-infinite cylinders) are encoded by the Strebel differential on punctured surfaces. The combinatorial Fenchel–Nielsen coordinates extracted from the metric ribbon graph serve as the moduli for off-shell string amplitudes; recursion relations governing amplitudes correspond to combinatorial gluing of Strebel ribbon graphs and are mirrored in the Fokker–Planck Hamiltonian structure (Ishibashi, 15 Feb 2024).

6. Combinatorics, Volume Asymptotics, and Statistical Geometry

Counting Jenkins–Strebel differentials with given topological types, such as "lattice" JS differentials on $\mathbb{CP}^1$ with prescribed simple zeros and poles and integer period data, is equivalent to counting integer points in certain moduli strata and, via pillowcase covers, computing explicit volumes of moduli space components. Asymptotic enumeration yields deep combinatorial identities expressing moduli space volumes as weighted sums over decorated trees (corresponding to cylinder decompositions of surfaces) (Athreya et al., 2012).

As extremal length increases, the critical graphs of Jenkins–Strebel differentials equidistribute in the metric ribbon graph moduli space according to the Kontsevich measure, revealing fundamental connections between the probabilistic geometry of large surfaces, intersection theory, and geometric representation of random surfaces (Arana-Herrera et al., 2023).

7. Special Constructions and Generalizations

Explicit models on the sphere and tori: Concrete formulas for Strebel differentials with simple or double poles are obtained via elliptic functions (Weierstrass $\wp$ ), Belyi maps, or symmetrization; examples include the sphere with four simple poles, three double poles, and hyperelliptic or real curves (Bogatyrev, 2010, Chen et al., 2016, Song et al., 2019).
Half-plane differentials: Strebel’s theory for double poles generalizes to higher-order poles, leading to "half-plane" structures whose metric completion yields surfaces built from Euclidean half-planes glued by interval exchange (Gupta, 2013).
Real algebraic curves: On real curves with reflection, the square of a real holomorphic 1-form satisfying certain period vanishing conditions is a Jenkins–Strebel differential; this yields explicit parameterizations for such differentials in terms of real cycles (Bogatyrev, 2010).