Small eigenvalues of the foliated Laplacian on moduli spaces of flat surfaces: existence below the $1/4$ barrier for Masur–Veech measures, accumulation at $1/4$, uniform spectral gaps, and the Ruelle zeta function of the Teichmüller flow

The $SL(2,\mathbb{R})$ action on a stratum of the moduli space of abelian or quadratic differentials on a Riemann surface, quotiented by the rotation group $SO(2,\mathbb{R})$, is foliated by copies of the hyperbolic plane; gluing the hyperbolic Laplacians of these leaves defines a foliated Laplacian $\Delta_\mu$ on $L^2$ of any invariant probability measure $\mu$. Avila and Gouëzel (Annals of Mathematics, 2013) proved that for every algebraic invariant measure the essential spectrum of $\Delta_\mu$ is contained in $[1/4,\infty)$, so that for each $\delta>0$ only finitely many eigenvalues lie in $(0,1/4-\delta)$ — but they left the fine structure of the small-eigenvalue region open. This problem records the surviving open cluster. (Q1) Do Masur–Veech measures on strata of genus $\ge 2$ have any eigenvalue in $(0,1/4)$ at all? Yoccoz conjectured that they do not — the analogue for moduli space of Selberg's eigenvalue conjecture — and this is verified only in genus one. (Q2) Can the eigenvalues of an algebraic measure in $(0,1/4)$ accumulate at $1/4$ from below, or is that set finite? (Q3) Does a fixed moduli space admit a spectral gap uniform over all algebraic measures it supports (known to fail when the genus is allowed to grow, by explicit Teichmüller curves with arbitrarily small eigenvalues)? (Q4) Does the Ruelle zeta function of the Teichmüller geodesic flow, restricted to the support of an algebraic measure, admit a meromorphic extension, and do the corresponding closed-geodesic asymptotics carry correction terms governed by the small eigenvalues?

Background

Flat surfaces and the Teichmüller flow. A translation surface is a Riemann surface equipped with a holomorphic $1$-form $\omega$; away from the zeros of $\omega$ the coordinate changes are translations, so the surface carries a flat metric with cone-type singularities at the zeros. More generally a quadratic differential defines a half-translation surface, where coordinate changes may also be half-turns. The moduli space of such surfaces decomposes into strata indexed by the orders of the zeros, and each connected stratum component $\mathcal{H}$ carries a canonical finite Lebesgue-class measure, ergodic for the natural $SL(2,\mathbb{R})$ action and known as the Masur–Veech measure $\nu_{\mathrm{MV}}$. Applying a matrix of $SL(2,\mathbb{R})$ to the translation charts of a surface defines an action on the stratum; the diagonal subgroup $g_t=\mathrm{diag}(e^t,e^{-t})$ is the Teichmüller geodesic flow, which renormalizes the straight-line flow on flat surfaces (the flow trajectory records the changing scale at which a straight-line trajectory on the surface is observed), and $SO(2,\mathbb{R})$ is the rotation subgroup. For the stratum of unit-area flat tori the space is $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$, the unit tangent bundle of the modular surface, where the classical theory of automorphic forms (spectral analysis of the Laplacian on quotients of a Lie group by a lattice) applies; in genus $\ge 2$ the strata are not homogeneous spaces, and how far homogeneous behavior persists is a central question of the field.

The foliated Laplacian and the $1/4$ barrier. Quotienting a stratum by $SO(2,\mathbb{R})$ collapses each $SL(2,\mathbb{R})$-orbit to a copy of the hyperbolic plane or a hyperbolic orbifold quotient. For an invariant probability measure $\mu$, the foliated Laplacian $\Delta_\mu$ is the unbounded self-adjoint operator on the $SO(2,\mathbb{R})$-invariant part of $L^2(\mu)$ obtained by letting the hyperbolic Laplacian act leaf by leaf on these copies. For the modular surface this is the classical Laplacian, whose spectrum consists of a discrete set of eigenvalues below $1/4$ and continuous spectrum $[1/4,\infty)$; the threshold $1/4$ is the bottom of the spectrum of the Laplacian on the hyperbolic plane itself. The meaning of the threshold is representation-theoretic: an eigenvalue $\lambda\in(0,1/4)$ of $\Delta_\mu$ corresponds to a complementary-series representation $\mathrm{Comp}_u$, with parameter $u=\sqrt{1-4\lambda}$, occurring in the decomposition of the regular representation of $SL(2,\mathbb{R})$ on $L^2(\mu)$, and the matrix coefficients of $\mathrm{Comp}_u$ decay only like $e^{-(1-u)t}$ (the remaining irreducible representations, the tempered or principal series, have coefficients decaying at the faster rate $e^{-t}$, up to subexponential factors). Small eigenvalues are therefore precisely the obstruction to fast mixing: a consequence of the main theorem of Avila and Gouëzel is that, for $SO(2,\mathbb{R})$-finite observables, correlations under $g_t$ decompose, for every $0<\delta<1$, as a finite sum of exponentials $e^{-a_i t}$ with $a_i=1-\sqrt{1-4\lambda_i}$ over the eigenvalues $\lambda_i$ of $\Delta_\mu$ in $[0,(1-\delta^2)/4]$, up to an error $o(e^{-(1-\delta)t})$; thus $\lambda$ close to $0$ means nearly no decay, while $\lambda$ approaching $1/4$ gives the optimal rate $e^{-t}$. A spectral gap — absence of spectrum of $\Delta_\mu$ on an interval $(0,\varepsilon)$ — is equivalent to exponential mixing at a rate proportional to $\varepsilon$. Selberg proved that every finite-volume hyperbolic surface has only finitely many eigenvalues in $(0,1/4)$ and conjectured that congruence covers of the modular surface have none at all; it is classical that $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ itself has none. This is the arithmetic prototype for the present questions. The obstruction is real in general ergodic situations: Kelmer and Sarnak constructed examples, for lattices $\Gamma$ in $G=SL(2,\mathbb{R})\times SU(2)$, where the foliated Laplacian on $SO(2,\mathbb{R})\backslash G/\Gamma$ has an accumulation point in $(0,1/4)$ — and whether a spectral gap always exists in that setting remains open.

Algebraic measures and the finiteness theorem. Strata also carry many invariant measures other than Masur–Veech measures, for instance the Liouville measures (normalized volume measures) on closed $SL(2,\mathbb{R})$-orbits: a Teichmüller curve is such a closed orbit — equivalently, the orbit of a surface whose group of affine automorphisms (its Veech group) is a lattice in $SL(2,\mathbb{R})$, whose image in the stratum is an algebraic curve; the simplest examples arise from square-tiled surfaces, that is, covers of a standard flat torus branched over a single point. The classification theorem of Eskin and Mirzakhani shows that every ergodic $SL(2,\mathbb{R})$-invariant probability measure on a stratum is algebraic (in the terminology of Avila and Gouëzel: admissible): it is supported on an affine invariant submanifold — a subset which, in period coordinates (coordinates recording the integrals of the $1$-form over relative homology classes), is locally defined by real-linear equations — and has Lebesgue-class conditional measures along orbit foliations. Building on earlier spectral-gap results for Masur–Veech measures by Avila–Gouëzel–Yoccoz and by Avila–Resende, and combining compact transfer operators (linear operators that encode one time-step of the dynamics on a space of functions, and whose compactness forces every non-zero spectral value to be an isolated eigenvalue of finite multiplicity) on anisotropic Banach spaces (spaces of distributions adapted to the hyperbolic directions of the flow) with a representation-theoretic measure-rigidity step, Avila and Gouëzel proved: for every algebraic invariant probability measure $\mu$ in a moduli space of quadratic differentials and every $\delta>0$, the spectrum of $\Delta_\mu$ in $[0,1/4-\delta]$ consists of finitely many eigenvalues of finite multiplicity; equivalently, the essential spectrum (the part of the spectrum stable under compact perturbations) is contained in $[1/4,\infty)$. The theorem is sharp in its formulation but loses the fixed fraction $\delta$ of the window $(0,1/4)$: by construction it says nothing on the full interval, nor at the endpoint.

What is known to exist, and what the paper left open. Some algebraic measures do have small eigenvalues: Ellenberg and McReynolds showed that for any finite-index subgroup $\Gamma$ of the congruence subgroup $\Gamma(2)$ containing $\{\pm 1\}$, the curve $SL(2,\mathbb{R})/\Gamma$ is realized as a Teichmüller curve, so quotients of the hyperbolic plane with complementary series sit inside moduli space; Matheus and Weitze-Schmithüsen constructed explicit square-tiled examples. Hence no spectral gap is uniform over all algebraic measures in all moduli spaces. Avila and Gouëzel explicitly record the remaining questions: whether Masur–Veech measures have any eigenvalue in $(0,1/4)$ at all (they write that it is tempting to conjecture that there is none, but refrain, for lack of evidence in either direction; Yoccoz had made precisely this conjecture in private, and it appeared in print later); whether the eigenvalues in $(0,1/4)$ can accumulate at $1/4$; whether there is a uniform spectral gap in each fixed moduli space; and they state the expectation that their techniques should give the meromorphic extension of the Ruelle zeta function $\zeta_{\mathrm{Ruelle}}(z)=\prod_\tau(1-e^{-z|\tau|})$ of the Teichmüller flow — the product over prime closed orbits $\tau$ of the flow with periods $|\tau|$ — and thereby precise asymptotic formulas for the number of closed geodesics in the support of any algebraic measure, including correction terms coming from small eigenvalues, in direct analogy with the Selberg trace formula and the prime geodesic theorem for hyperbolic surfaces.

Problem Statement

The problem is to determine the fine structure of the spectrum of the foliated Laplacian below the $1/4$ barrier for $SL(2,\mathbb{R})$-invariant probability measures on moduli spaces of flat surfaces, as left open by Avila and Gouëzel (2013). Let $\mathcal{H}$ be a connected component of a stratum of abelian differentials of genus $\ge 2$ (the setting of quadratic differentials is equivalent for this problem: a quadratic differential lifts, on a canonical double cover of the surface, to the square of an abelian differential, and this is the reduction by which Avila and Gouëzel derive their theorem for quadratic differentials from the abelian case), let $\nu_{\mathrm{MV}}$ be its Masur–Veech measure, and let $\mu$ range over the algebraic $SL(2,\mathbb{R})$-invariant probability measures supported on affine invariant submanifolds of a fixed moduli space $\mathcal{M}^1_{g,n}$ of unit-area flat surfaces. The open questions are:

(Q1) Existence below the barrier (Yoccoz's conjecture). Does $\Delta_{\nu_{\mathrm{MV}}}$, acting on the $SO(2,\mathbb{R})$-invariant part of $L^2(\mathcal{H},\nu_{\mathrm{MV}})$, have any eigenvalue in $(0,1/4)$? Equivalently, does the regular representation of $SL(2,\mathbb{R})$ on $L^2(\mathcal{H},\nu_{\mathrm{MV}})$ contain complementary series representations? The conjectured answer is no (the analogue of Selberg's eigenvalue conjecture for congruence surfaces).

(Q2) Accumulation at the barrier. Is $\sigma(\Delta_\mu)\cap(0,1/4)$ finite for every algebraic measure $\mu$, or does some $\mu$ admit infinitely many eigenvalues in $(0,1/4)$ accumulating at $1/4$ from below? The finiteness theorem of Avila and Gouëzel leaves precisely this endpoint question open.

(Q3) Uniformity in a fixed moduli space. Does there exist, for each fixed $g,n$, a constant $\varepsilon(g,n)>0$ such that every algebraic measure $\mu$ supported in $\mathcal{M}^1_{g,n}$ satisfies $\sigma(\Delta_\mu)\cap(0,\varepsilon(g,n))=\varnothing$? It is known that no such constant exists uniformly over all genera and strata, since Matheus and Weitze-Schmithüsen constructed explicit Teichmüller curves whose smallest eigenvalues tend to $0$ as the genus grows; the fixed-moduli-space version is open.

(Q4) The Ruelle zeta function of the Teichmüller flow. For an algebraic measure $\mu$, let $\zeta_{\mathrm{Ruelle}}(z)=\prod_\tau(1-e^{-z|\tau|})$ be the Ruelle zeta function of the Teichmüller geodesic flow restricted to the support of $\mu$, the product over prime closed orbits $\tau$ with periods $|\tau|$. Does $\zeta_{\mathrm{Ruelle}}$ admit a meromorphic extension to a half-plane containing the imaginary axis (or to all of $\mathbb{C}$), and does the counting function of closed geodesics in the support of $\mu$ admit an asymptotic expansion with error terms in which each small eigenvalue $\lambda\in(0,1/4)$ of $\Delta_\mu$ contributes an explicit correction (in the hyperbolic-surface prototype, the prime geodesic theorem, a small eigenvalue $\lambda$ contributes a correction of order $e^{(1+\sqrt{1-4\lambda})R}$ up to polynomial factors, where $R$ is the period in the normalization $g_t=\mathrm{diag}(e^t,e^{-t})$, in which the leading term of the modular-surface count has exponent $2$)?

These four questions are facets of a single spectral object and its dynamical consequences, in the sense of the source: an eigenvalue $\lambda\in(0,1/4)$ simultaneously fixes the rate $e^{-(1-\sqrt{1-4\lambda})t}$ of decay of correlations of the Teichmüller flow, marks a complementary-series component of $L^2(\mu)$, and would enter the error term of closed-geodesic counting. Q1–Q2 determine the discrete spectrum below $1/4$, Q3 its uniformity over orbit closures, and Q4 its dynamical payoff. The problem is stated at the source's generality — all genera $\ge 2$, all stratum components, all algebraic measures — and no restriction of genus, stratum, or measure class is intended.

References

1. Artur Avila and Sébastien Gouëzel, "Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow", Annals of Mathematics 178, no. 2 (2013), 385–442. DOI: 10.4007/annals.2013.178.2.1. https://doi.org/10.4007/annals.2013.178.2.1
2. Artur Avila, Sébastien Gouëzel, and Jean-Christophe Yoccoz, "Exponential mixing for the Teichmüller flow", Publications Mathématiques de l'IHÉS 104 (2006), 143–211. DOI: 10.1007/s10240-006-0001-5. https://doi.org/10.1007/s10240-006-0001-5
3. Artur Avila and Maria João Resende, "Exponential mixing for the Teichmüller flow in the space of quadratic differentials", Commentarii Mathematici Helvetici 87, no. 3 (2012), 589–638. DOI: 10.4171/CMH/263. https://doi.org/10.4171/CMH/263
4. Alex Eskin and Maryam Mirzakhani, "Invariant and stationary measures for the SL(2,R) action on Moduli space", Publications Mathématiques de l'IHÉS 127 (2018), 95–324. DOI: 10.1007/s10240-018-0099-2. https://doi.org/10.1007/s10240-018-0099-2
5. Jordan S. Ellenberg and D. B. McReynolds, "Arithmetic Veech sublattices of SL(2,Z)", Duke Mathematical Journal 161, no. 3 (2012), 415–429. DOI: 10.1215/00127094-1507412. https://doi.org/10.1215/00127094-1507412
6. Carlos Matheus and Gabriela Weitze-Schmithüsen, "Explicit Teichmüller curves with complementary series", Bulletin de la Société Mathématique de France 141, no. 4 (2013), 557–602. DOI: 10.24033/bsmf.2656. https://doi.org/10.24033/bsmf.2656
7. Dubi Kelmer and Peter Sarnak, "Strong spectral gaps for compact quotients of products of PSL(2,R)", Journal of the European Mathematical Society 11, no. 2 (2009), 283–313. DOI: 10.4171/JEMS/151. https://doi.org/10.4171/JEMS/151
8. Michael Magee, "On Selberg's eigenvalue conjecture for moduli spaces of abelian differentials", Compositio Mathematica 155, no. 12 (2019), 2354–2398. DOI: 10.1112/S0010437X1900767X. https://doi.org/10.1112/S0010437X1900767X
9. Michael Magee and Rene Rühr, with an appendix by Rodolfo Gutiérrez-Romo, "Counting saddle connections in a homology class modulo q", Journal of Modern Dynamics 14 (2019), 237–262. DOI: 10.3934/jmd.2019020. https://doi.org/10.3934/jmd.2019020
10. Jayadev S. Athreya, Jean Lagacé, Martin Möller, and Martin Raum, "Spectral decomposition and Siegel–Veech transforms for strata: the case of marked tori", Journal of Spectral Theory 15, no. 2 (2025), 895–959. DOI: 10.4171/JST/563. https://doi.org/10.4171/JST/563
11. Alex Eskin, Maryam Mirzakhani, and Kasra Rafi, "Counting closed geodesics in strata", Inventiones mathematicae 215, no. 2 (2019), 535–607. DOI: 10.1007/s00222-018-0832-y. https://doi.org/10.1007/s00222-018-0832-y
12. Frédéric Faure, Sébastien Gouëzel, and Erwan Lanneau, "Ruelle spectrum of linear pseudo-Anosov maps", Journal de l'École polytechnique — Mathématiques 6 (2019), 811–877. DOI: 10.5802/jep.107. https://doi.org/10.5802/jep.107
13. François Ledrappier, "Three problems solved by Sébastien Gouëzel", Journal of Modern Dynamics 16 (2020), 373–387. DOI: 10.3934/jmd.2020015. https://doi.org/10.3934/jmd.2020015
14. Carlos Matheus, "Quelques contributions à la théorie de l'action de SL(2,R) sur les espaces de modules de surfaces plates", mémoire d'habilitation à diriger des recherches (2016). arXiv:1612.02457. https://arxiv.org/abs/1612.02457
15. Osama Khalil, "Exponential mixing via additive combinatorics", Journal of the American Mathematical Society 39, no. 3 (2026), 781–856. DOI: 10.1090/jams/1076. https://doi.org/10.1090/jams/1076
16. Atle Selberg, "On the estimation of Fourier coefficients of modular forms", Proceedings of Symposia in Pure Mathematics, vol. VIII (1965), 1–15, American Mathematical Society. DOI: 10.1090/pspum/008/0182610. https://doi.org/10.1090/pspum/008/0182610