Asymptotic distribution of period geodesics in strata of Abelian differentials, including geodesics transverse to the absolute period foliation

The moduli space of holomorphic 1-forms over the moduli space of genus-g Riemann surfaces carries period coordinates, and straight lines in these coordinates are the period geodesics; those contained in a single leaf of the absolute period foliation (the isoperiodic foliation, fixing absolute periods) are relative period geodesics. McMullen (J. Eur. Math. Soc. 15 (2013), 1223-1243) proved that a relative period geodesic is either complete or terminates via a collision of zeros or the creation of a node, and then posed the question, still open, of understanding the asymptotic distribution of the complete geodesic ray obtained as the connected sum of two flat tori with no positive real period, of other period geodesics, and of period geodesics transverse to the absolute period foliation. Later work established metric completeness of the leaves, classification of leaf closures, and almost-everywhere ergodicity, together with (conditional on a forthcoming measure classification) mixing of the horizontal (real Rel) flows and examples of dense and of divergent trajectories; but the behavior of the source's specific genus-2 ray and of any transverse period geodesic remains undetermined. The problem is to determine, with rigorous proof, the asymptotic behavior of these geodesics: recurrence, divergence, or neither; limit sets in the natural compactifications; and equidistribution of time averages with respect to the natural measures.

Background

Let $\mathcal{M}_g$ denote the moduli space of compact Riemann surfaces of genus $g$, and let $\Omega\mathcal{M}_g$ be the Hodge bundle over it, parameterizing pairs $(X,\omega)$ of a Riemann surface and a nonzero holomorphic 1-form $\omega$ (up to the finite group of automorphisms). A holomorphic 1-form defines a translation structure: a flat metric $|\omega|$ with conical singularities at the zeros of $\omega$. The space $\Omega\mathcal{M}_g$ is partitioned into strata $\Omega\mathcal{M}_g(m_1,\dots,m_s)$, indexed by the multiplicities of the zeros ($\sum_i m_i = 2g-2$), and each stratum carries a natural action of the group $SL_2(\mathbb{R})$ (by postcomposing the charts of the translation structure with linear maps, that is, by deforming the flat pictures linearly) and a Lebesgue-type measure, the Masur-Veech measure, defined by period coordinates. Period coordinates are the local coordinates on a stratum given by the periods (integrals) of $\omega$: absolute periods over closed cycles in $H_1(X)$ and relative periods over paths from a basepoint to the zeros; the tangent space to a stratum at $(X,\omega)$ is the relative cohomology group $H^1(X,Z(\omega))$, where $Z(\omega)$ is the zero set of $\omega$.

The absolute period foliation $\mathcal{A}$ of a stratum, also called the isoperiodic foliation or Rel foliation, is the foliation whose leaves consist of forms with fixed absolute periods; motion along a leaf changes only the relative periods, and is generated locally by Schiffer variations (slit the flat surface along a short segment through a zero and cross-reglue). A useful norm on these leafwise tangent vectors, written as twists $\sum_p a_p\,\mathrm{tw}(C_p)$ around small loops $C_p$ encircling the zeros, is $\max_{p,q}|a_p-a_q|$, the maximum of $|a_p-a_q|$ over all pairs of zeros; McMullen calls it the relative period metric and proves that paths of finite length in this metric converge in a suitable compactification by stable forms (limits on nodal stable curves, where the 1-form is allowed to vanish identically on whole components), so the leaves are complete metric spaces after stable forms are adjoined (the completeness question, raised by Hooper, Smillie and Weiss, was settled in McMullen's sequel).

In this framework a geodesic is a path $\gamma$ in a stratum whose velocity $\gamma'(t)\in H^1(X_t,Z(\omega_t))$ is locally constant, that is, a straight line in period coordinates; if $\gamma$ lies in a single leaf of $\mathcal{A}$ it is a relative period geodesic (in later literature such leafwise paths, for horizontal directions, are called real Rel trajectories, Rel denoting the relative-period direction), and if its velocity has a nonzero absolute-period part then the absolute periods drift linearly and the path crosses the leaves of $\mathcal{A}$, so it is transverse to $\mathcal{A}$. The source paper's Corollary 1.5 states that a relative period geodesic in a stratum is either complete or terminates via a collision of zeros or the creation of a node (a double point appearing on the stable-curve limit when the period across a saddle connection shrinks to zero). The prototype example takes two flat tori $(E_i,\omega_i)=(\mathbb{C}/\Lambda_i,dz)$, slits both along the horizontal segment $[0,t]$ from the origin, and cross-glues to form the connected sum $\gamma(t)=(E_1,\omega_1)\#(E_2,\omega_2)_{[0,t]}$, a relative period geodesic in $\Omega\mathcal{M}_2(1,1)$ defined for $t\in(0,T)$ with $T=\min(\lambda\in\Lambda_1\cup\Lambda_2 : \lambda>0)$; it is a complete geodesic ray exactly when $T=\infty$, i.e. when no nonzero element of either lattice is a positive real number (equivalently, both tori have non-periodic horizontal foliation). For $T<\infty$ the ray terminates by zero collision or at a stable form; the case $T=\infty$ produces a geodesic that runs forever, and the paper ends by asking what its distribution, and that of other period geodesics including transverse ones, asymptotically is.

Several frameworks give meaning to 'asymptotic distribution' here: the Masur-Veech measure on the area-one locus of the stratum (the locus of forms normalized so that the flat area $(i/2)\int_X\omega\wedge\overline{\omega}$ equals one); the Lebesgue (translation-invariant) measure on a leaf, which in genus two is the flat structure of an explicit meromorphic quadratic differential $q(L)$ (a meromorphic section of the square of the cotangent bundle, defining a flat structure with poles) on the leaf $\mathcal{A}(L)\cong\mathbb{H}$ studied in McMullen's sequel; the Deligne-Mumford compactification $\overline{\mathcal{M}}_g$ by stable curves, which records limits of the underlying Riemann surfaces; and the WYSIWYG compactification of strata constructed by Chen and Wright (the acronym abbreviates 'what you see is what you get'), a partial compactification whose boundary points record flat-geometric limits of degenerating forms.

Problem Statement

Work in a stratum $\Omega\mathcal{M}_g(m_1,\dots,m_s)$ of the Hodge bundle, equipped with its period coordinates, and let $\mathcal{A}$ be its absolute period foliation. A period geodesic is a path $\gamma$ in the stratum whose velocity $\gamma'(t)\in H^1(X_t,Z(\omega_t))$ is locally constant; it is a relative period geodesic if it lies in a single leaf of $\mathcal{A}$, and transverse to $\mathcal{A}$ if its velocity has nonzero absolute-period part. McMullen proved that a relative period geodesic is either complete or terminates via a collision of zeros or the creation of a node, and for a pair of flat tori $(E_i,\omega_i)=(\mathbb{C}/\Lambda_i,dz)$ constructed the connected-sum path $\gamma(t)=(E_1,\omega_1)\#(E_2,\omega_2)_{[0,t]}$, a relative period geodesic in $\Omega\mathcal{M}_2(1,1)$ that is a complete geodesic ray exactly when $T=\min(\lambda\in\Lambda_1\cup\Lambda_2 : \lambda>0)=\infty$.

The problem, as posed at the end of the source, is to understand the asymptotic distribution of this complete ray and of other period geodesics, including those transverse to $\mathcal{A}$. Concretely, an answer should determine, with rigorous proof and for the three classes named in the source (the genus-two connected-sum rays $\gamma(t)=(E_1,\omega_1)\#(E_2,\omega_2)_{[0,t]}$ with $T=\infty$, parametrized by the pair of lattices $(\Lambda_1,\Lambda_2)$; the other complete relative period geodesics, meaning the complete period geodesics lying in a single leaf of $\mathcal{A}$ other than those connected-sum rays; and the complete period geodesics transverse to $\mathcal{A}$; non-complete geodesics are excluded, since they terminate at finite time and have no asymptotic distribution), the asymptotic behavior of the trajectory after renormalization to unit area: whether the renormalized trajectory is recurrent (returns infinitely often to every neighborhood of its initial form), divergent (eventually leaves every compact set), or neither; what its limit set, meaning the set of its accumulation points, is in the compactification specified for the class, namely the metric completion of the leaf for a relative period geodesic and the WYSIWYG compactification of the stratum for a transverse period geodesic; and whether its time averages equidistribute with respect to the measure specified for the class, namely the leafwise Lebesgue measure in period coordinates for a relative period geodesic, normalized on the leaf closure when that closure has finite total mass, and the Masur-Veech measure on the area-one locus for a transverse period geodesic, meaning that the empirical time averages of continuous observables along the renormalized trajectory converge to that measure. The recurrent/divergent verdict is determined by the limit set and the equidistribution where these settle it. For each class, the possible asymptotic behaviors and how they depend on the initial form and direction are to be determined for every member of the class, that is, the recurrence/divergence verdict, the limit set, and the equidistribution. The common objective is a rigorous description of the asymptotic distribution of period geodesics in strata of Abelian differentials; an answer should determine the asymptotic behavior, in this sense, for all three classes.

References

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