Moduli spaces of isoperiodic forms on Riemann surfaces

@article{McMullen2014ModuliSO,
  title={Moduli spaces of isoperiodic forms on Riemann surfaces},
  author={Curtis T. McMullen},
  journal={Duke Mathematical Journal},
  year={2014},
  volume={163},
  pages={2271-2323}
}
  • C. McMullen
  • Published 11 March 2014
  • Mathematics
  • Duke Mathematical Journal
This paper describes the intrinsic geometry of a leaf A(L) of the absolute period foliation of the Hodge bundle ΩMg: its singular Euclidean structure, its natural foliations and its discretized Teichmüller dynamics. We establish metric completeness of A(L) for general g, and then turn to a study of the case g = 2. In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on A(L) ∼= H, whose zeros, poles and exotic trajectories are analyzed in detail. 
Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms
Let Sg be a closed oriented surface of genus g, and let ΩMg(κ) be a stratum of the moduli space of holomorphic 1-forms of genus g. We show that the absolute period foliation of ΩMg(κ) is ergodic on
The Riemann-hilbert Mapping for Sl 2 -systems over Genus Two Curves
We prove in two different ways that the monodromy map from the space of irreducible sl2-differential-systems on genus two Riemann surfaces, towards the character variety of SL2-representations of the
The boundary of an affine invariant submanifold
We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann
A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds
We define on any affine invariant orbifold M a foliation F^M that generalises the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in
Isoperiodic meromorphic forms: two simple poles
In this paper we prove that isoperiodic moduli spaces of meromorphic differentials with two simple poles on homologically marked smooth curves are non empty and connected, unless they correspond to
Isoperiodic dynamics in rank 1 affine invariant orbifolds
Let M be a rank 1 affine invariant orbifold in a stratum of the moduli space of flat surfaces. We show that the leaves of the M-isoperiodic foliation are either all closed or all dense. In the second
Schiffer variations and Abelian differentials
Real-Normalized Differentials and the Elliptic Calogero-Moser System
In our recent works (Grushevsky and Krichever, The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces. In: Surveys in differential geometry. Vol. XIV.
DENSE REAL REL FLOW ORBITS AND ABSOLUTE PERIOD LEAVES
We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, we
Dynamical properties of the absolute period foliation
We show that the absolute period foliation of the principal stratum of abelian differentials on a surface of genus at least 3 is ergodic. We also investigate the absolute period foliation of affine
...
1
2
3
4
...

References

SHOWING 1-10 OF 38 REFERENCES
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe
Foliations of Hilbert modular surfaces
The Hilbert modular surface XD is the moduli space of Abelian varieties A with real multiplication by a quadratic order of discriminant D > 1. The locus where A is a product of elliptic curves
Billiards and Teichmüller curves on Hilbert modular surfaces
This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmuller curves lie on Hilbert modular surfaces
Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities
Abstract Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component.
Dynamics of SL 2 ( R ) over moduli space in genus two
This paper classifies orbit closures and invariant measures for the natural action of SL2(R) on ΩM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.
Dynamics of SL2(ℝ) Over Moduli Space in Genus Two
This paper classifies orbit closures and invariant measures for the natural action of SL2(R) on UM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.
Spaces of elliptic differentials
We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers
Flat Surfaces
Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a
On unipotent flows in ℋ(1,1)
Abstract We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures
Euler characteristics of Teichmüller curves in genus two
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These
...
1
2
3
4
...