Fine structure of the Lyapunov spectrum of the Kontsevich-Zorich cocycle: individual exponents, zero exponents, and the rank filtration of the second fundamental form (open core of Forni's 2006 Handbook problem list)

Forni's 2006 Handbook of Dynamical Systems survey on the Kontsevich-Zorich (KZ) cocycle closed with five open problems; the surviving open core is stated below. Two of the five are settled: the Zorich/Kontsevich-Zorich simplicity conjecture for strata of abelian differentials was proved by Avila and Viana (2007), and the existence question for Veech surfaces with degenerate KZ spectrum in genus greater than 3 was answered affirmatively by Forni and Matheus' genus-4 example, with the containment question for the rank-one locus completely classified by subsequent work of Forni-Matheus-Zorich, Aulicino, Aulicino-Norton, and Aulicino-Benirschke-Norton. The remaining core, unified by Forni's variational approach through the second fundamental form of the Hodge bundle, comprises three linked questions: (i) closed formulas for individual Lyapunov exponents, or nontrivial partial sums, on strata of genus at least 3, beyond the Eskin-Kontsevich-Zorich total-sum formula; (ii) the dictionary between the rank filtration of the second fundamental form on the support of an invariant measure and the number of zero exponents, whose structural monodromy side was settled by Filip's semisimplicity and zero-exponent classification (2016-2017) but whose original rank-locus formulation remains open; and (iii) a description of the incidence of general Teichmuller discs with the determinant locus of the Hodge metric. These directly govern deviation spectra of typical translation flows and the rigidity theory of SL(2,R)-invariant submanifolds.

Background

A translation surface is a compact Riemann surface together with a holomorphic 1-form (an abelian differential) $\omega$; away from the zeros of $\omega$ the surface is locally isometric to the flat plane. The moduli space of unit-area abelian differentials of genus $g$ decomposes into strata $\mathcal{H}(m_1,\dots,m_k)$ labeled by the multiplicities of the zeros, $\sum_i m_i = 2g-2$. Each stratum carries an action of the group $SL(2,\mathbb{R})$ of area-preserving linear maps of the plane (the Teichmuller geodesic flow is its diagonal subgroup) and a finite invariant Masur-Veech measure, the natural volume class on the stratum. Over a stratum sits the Hodge bundle, whose fiber over a surface $(X,\omega)$ is the cohomology $H^1(X,\mathbb{C})$ with its decomposition into holomorphic and anti-holomorphic parts; the Kontsevich-Zorich cocycle is the monodromy action on flat cohomology along $SL(2,\mathbb{R})$ orbits. By Oseledets' multiplicative ergodic theorem the cocycle has Lyapunov exponents $1 = \lambda_1 \ge \dots \ge \lambda_g \ge 0$ (up to symmetry), which govern measurable dynamics on the flat surface, for example the power-law deviations of ergodic averages of linear flows and of saddle-connection holonomy vectors. Zorich conjectured, and Avila and Viana proved, that for the Masur-Veech measure on every stratum of abelian differentials the spectrum is simple and $\lambda_g > 0$; the analogous statement for strata of quadratic differentials (half-translation surfaces, where the local charts to the flat plane may be glued through the half-turn $z \mapsto -z$, so the vertical and horizontal directions need not be globally orientable) remains open, with only nonvanishing $\lambda_g > 0$ known (Treviño 2013). Forni's 2006 chapter developed the variational approach behind these problems: along the Teichmuller flow the infinitesimal movement of the Hodge decomposition is measured by the second fundamental form (Kodaira-Spencer map) $B_\omega$, an $Sp(2g,\mathbb{R})$-symmetric bilinear map on holomorphic differentials; equivalently one considers a Hermitian form $H_q$ on meromorphic functions whose pointwise eigenvalues $\Lambda_1 \ge \dots \ge \Lambda_g$ are the squared singular values of $B_\omega$. Forni's sum formula states $\lambda_1 + \dots + \lambda_g = \int (\Lambda_1 + \dots + \Lambda_g)\,d\mu$ for any ergodic $SL(2,\mathbb{R})$-invariant measure $\mu$. The loci where the last eigenvalues vanish pointwise, that is where $\operatorname{rank} B_\omega \le k$, are the higher vanishing loci of the chapter (denoted $R_g(k)$ there; later papers in the same school index these loci in opposite ways, with $k$ counting either the rank bound or the number of vanishing eigenvalues, so the conditions here are stated explicitly through ranks); the top case $k = g-1$, where $\det H_q = 0$, is the determinant locus, a real-analytic subvariety of the stratum. A Veech surface is a translation surface whose $SL(2,\mathbb{R})$ orbit is closed; such an orbit is an algebraic Teichmuller curve, and the $SL(2,\mathbb{R})$ orbit of a single surface is called a Teichmuller disc. The simplest Veech surfaces are square-tiled surfaces (finite covers of the standard torus branched over one point); among them, cyclic covers (Galois covers with cyclic deck group) admit explicit arithmetic computation of every individual Lyapunov exponent via isotypic components (the subspaces on which the deck group acts through a fixed irreducible representation). Forni's chapter exhibited a genus-3 square-tiled Veech surface (later nicknamed the Eierlegende Wollmilchsau) on which $\operatorname{rank} B = 1$ and $\lambda_2 = \lambda_3 = 0$, proving that Teichmuller discs can lie inside the determinant locus, and closed with five problems: simplicity of the spectrum on strata; Veech surfaces in higher vanishing loci for genus above 3; intersections of general Teichmuller discs with the determinant locus; closed formulas for individual exponents in genus at least 3; and the implication from support in the vanishing loci to vanishing exponents. The first two are now resolved, and the last three constitute the open core stated below.

Problem Statement

The open core of Forni's five closing problems (Forni, Handbook of Dynamical Systems 1B, 2006; problems 1 and 2 are resolved) is the fine structure of the KZ Lyapunov spectrum beyond the total sum and beyond arithmetic families, unified by one objective: determine the individual exponents, and the position and count of zero exponents, from the geometry of the second fundamental form $B$ along $SL(2,\mathbb{R})$ orbits. Precisely, with $\mathcal{H}$ a stratum of unit-area abelian differentials of genus $g$, $\mu$ an ergodic $SL(2,\mathbb{R})$-invariant probability measure, $B_\omega$ the second fundamental form of the Hodge bundle at $\omega$, and $D_g = \{\omega : \operatorname{rank} B_\omega \le g-1\}$ the determinant locus: (i) Closed formulas: obtain closed-form expressions — explicit finite formulas of the same character as the Eskin-Kontsevich-Zorich total-sum formula — for individual Lyapunov exponents $\lambda_k$, or for nontrivial partial sums $\lambda_1 + \dots + \lambda_k$ with $1 < k < g$, of the KZ cocycle for the Masur-Veech measure on strata of genus $g \ge 3$; beyond the Eskin-Kontsevich-Zorich formula for the total sum, beyond the genus-2 spectrum, and beyond square-tiled cyclic covers. (ii) Zero exponents from the rank filtration: for an arbitrary ergodic $SL(2,\mathbb{R})$-invariant measure $\mu$ whose support is contained in the locus $\{\operatorname{rank} B_\omega \le k\}$ for some $k$ with $2 \le k < g$, must the cocycle have at least $g-k$ zero exponents, that is, at most $k$ positive exponents? (The case $k = 1$ is excluded because it is a theorem of Forni: an ergodic invariant measure is supported in the rank-one locus $\{\operatorname{rank} B_\omega \le 1\}$ if and only if $\lambda_2 = \dots = \lambda_g = 0$.) The related but strictly stronger Problem 1 of Forni-Matheus-Zorich asks whether the number of strictly positive exponents can ever differ from the maximal rank of $B$ on the support, and precisely when the neutral Oseledets subspace (the subbundle of vectors whose Lyapunov exponent is zero) coincides with the kernel of $B$ on the support, in the sense of Forni-Matheus-Zorich the annihilator of the image of the real second fundamental form; this stronger problem is background for (ii) and is not an additional required deliverable. (iii) Orbits and the determinant locus: describe the intersection of a Teichmuller disc with $D_g$. For the rank-one locus $D_g(1) = \{\omega : \operatorname{rank} B_\omega \le 1\}$, which by Forni's theorem supports exactly the measures with completely degenerate spectrum $\lambda_2 = \dots = \lambda_g = 0$, the containment question is completely settled: the only Teichmuller discs in $D_g(1)$ are the genus-3 Eierlegende Wollmilchsau and the genus-4 Ornithorynque, and none exist in genus 2 or genus at least 5 (Aulicino, Comment. Math. Helv. 90 (2015) and ETDS 38 (2018); Aulicino-Norton, arXiv 1904.01625). What remains open is the incidence question for the determinant locus $D_g$ and for general discs: whether a Teichmuller disc can be entirely contained in $D_g$ without being contained in $D_g(1)$ (a case of the open implication in part (ii)), and, for a general Teichmuller disc, a description of the intersection with $D_g$. For (i), the target is a closed-form expression valid in general, not merely on an isolated stratum.

References

1. Giovanni Forni, 'On the Lyapunov exponents of the Kontsevich-Zorich cocycle', Handbook of Dynamical Systems, vol. 1B, Elsevier, 2006, pp. 549-580. DOI 10.1016/S1874-575x(06)80033-0, https://doi.org/10.1016/S1874-575x(06)80033-0
2. Artur Avila and Marcelo Viana, 'Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture', Acta Mathematica 198 (2007), no. 1, 1-56. DOI 10.1007/s11511-007-0012-1, arXiv math/0508508, https://doi.org/10.1007/s11511-007-0012-1
3. Matt Bainbridge, 'Euler characteristics of Teichmüller curves in genus two', Geometry and Topology 11 (2007) 1887-2073. DOI 10.2140/gt.2007.11.1887, arXiv math/0611409, https://doi.org/10.2140/gt.2007.11.1887
4. Giovanni Forni and Carlos Matheus, 'An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum', preprint, 2008. arXiv 0810.0023, https://arxiv.org/abs/0810.0023
5. Frank Herrlich and Gabriela Schmithüsen, 'An extraordinary origami curve', Mathematische Nachrichten 281 (2008), no. 2, 219-237. DOI 10.1002/mana.200510597, arXiv math/0509195, https://doi.org/10.1002/mana.200510597
6. Giovanni Forni, Carlos Matheus and Anton Zorich, 'Square-tiled cyclic covers', Journal of Modern Dynamics 5 (2011), no. 2, 285-318. DOI 10.3934/jmd.2011.5.285, arXiv 1007.4275, https://doi.org/10.3934/jmd.2011.5.285
7. Alex Eskin, Maxim Kontsevich and Anton Zorich, 'Lyapunov spectrum of square-tiled cyclic covers', Journal of Modern Dynamics 5 (2011), no. 2, 319-353. DOI 10.3934/jmd.2011.5.319, arXiv 1007.5330, https://doi.org/10.3934/jmd.2011.5.319
8. Giovanni Forni, 'A geometric criterion for the nonuniform hyperbolicity of the Kontsevich–Zorich cocycle', Journal of Modern Dynamics 5 (2011), no. 2, 355-395. DOI 10.3934/jmd.2011.5.355, arXiv 1009.4655, https://doi.org/10.3934/jmd.2011.5.355
9. Martin Möller, 'Shimura and Teichmüller curves', Journal of Modern Dynamics 5 (2011), no. 1, 1-32. DOI 10.3934/jmd.2011.5.1, arXiv math/0501333, https://doi.org/10.3934/jmd.2011.5.1
10. David Aulicino, 'Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum', Commentarii Mathematici Helvetici 90 (2015), no. 3, 573-643. DOI 10.4171/CMH/365, arXiv 1205.2359, https://doi.org/10.4171/CMH/365
11. Giovanni Forni, Carlos Matheus and Anton Zorich, 'Lyapunov spectrum of invariant subbundles of the Hodge bundle', Ergodic Theory and Dynamical Systems 34 (2014), no. 2, 353-408. DOI 10.1017/etds.2012.148, arXiv 1112.0370, https://doi.org/10.1017/etds.2012.148
12. Giovanni Forni, Carlos Matheus and Anton Zorich, 'Zero Lyapunov exponents of the Hodge bundle', Commentarii Mathematici Helvetici 89 (2014), no. 2, 489-535. DOI 10.4171/CMH/325, arXiv 1201.6075, https://doi.org/10.4171/CMH/325
13. Alex Eskin, Maxim Kontsevich and Anton Zorich, 'Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow', Publications Mathematiques de l'IHES 120 (2014) 207-333. DOI 10.1007/s10240-013-0060-3, arXiv 1112.5872, https://doi.org/10.1007/s10240-013-0060-3
14. Simion Filip, 'Semisimplicity and rigidity of the Kontsevich-Zorich cocycle', Inventiones Mathematicae 205 (2016), no. 3, 617-670. DOI 10.1007/s00222-015-0643-3, arXiv 1307.7314, https://doi.org/10.1007/s00222-015-0643-3
15. Simion Filip, 'Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle', Duke Mathematical Journal 166 (2017), no. 4, 657-706. DOI 10.1215/00127094-3715806, arXiv 1410.2129, https://doi.org/10.1215/00127094-3715806
16. David Aulicino, 'Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum', Ergodic Theory and Dynamical Systems 38 (2018), no. 1, 10-33. DOI 10.1017/etds.2016.26, arXiv 1302.0913, https://doi.org/10.1017/etds.2016.26
17. Rodolfo Gutiérrez-Romo, 'Classification of Rauzy–Veech groups: proof of the Zorich conjecture', Inventiones Mathematicae 215 (2019), no. 3, 741-778. DOI 10.1007/s00222-018-0836-7, arXiv 1706.04923, https://doi.org/10.1007/s00222-018-0836-7
18. David Aulicino, Frederik Benirschke and Chaya Norton, 'A zero Lyapunov exponent in genus 3 implies the Eierlegende Wollmilchsau', Annales Henri Lebesgue 7 (2024) 207-237. DOI 10.5802/ahl.198, arXiv 2203.00841, https://doi.org/10.5802/ahl.198
19. Rodrigo Treviño, 'On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials', Geometriae Dedicata 163 (2013), no. 1, 311-338. DOI 10.1007/s10711-012-9751-z, arXiv 1010.1038, https://doi.org/10.1007/s10711-012-9751-z
20. David Aulicino and Chaya Norton, 'Shimura-Teichmüller curves in genus 5', preprint, 2019. arXiv 1904.01625, https://arxiv.org/abs/1904.01625