We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichm\"uller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperelliptic automorphism comes from the 3-square-tiled translation surface itself. This makes the computation of an exact expression for period matrices of other Riemann surfaces in that family by the classical method impossible. We nevertheless give the solution to the Schottky problem for that family: we exhibit explicit necessary and sufficient conditions for a Riemann matrix to be a period matrix of a Riemann surface in the family, involving the vanishing of a genus 3 theta characteristic on a family of double covers.

The Real Analytic Theory of Teichmüller Space

On the Teichm\"uller geodesic generated by the L-shaped translation surface tiled by three squares

In this paper we consider the symplectic properties of the monodromy map of second order equations on a Riemann surface whose potential is meromorphic with second order poles. We show that the Poisson bracket defined by the periods of meromorphic quadratic differential implies the Goldman Poisson structure on character variety. These results generalize the previous works of the authors where the case of holomorphic potentials was considered and another paper of the second author where the symplectomorphism was proved for potentials with first order poles. We show that the leading term of the generating function of the monodromy symplectomorphism (the "Yang-Yang" function) is proportional to the Bergman tau-function on moduli spaces of meromorphic quadratic differentials with second order poles.

Yang-Yang generating function and Bergman tau-function

We prove a Gauss–Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the Gauss–Bonnet formula of Allendoerfer and Weil for Riemannian polyhedra.

A Gauss–Bonnet formula for moduli spaces of Riemann surfaces

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmuller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices. We show that the only Riemann surface in that family admitting a non-hyperelliptic automorphism comes from the 3-square-tiled translation surface itself. This makes the computation of an exact expression for period matrices of other Riemann surfaces in that family by the classical method impossible. We nevertheless give the solution to the Schottky problem for that family: we exhibit explicit necessary and sufficient conditions for a Riemann matrix to be a period matrix of a Riemann surface in the family, involving the vanishing of a genus 3 theta characteristic on a family of double covers.

On the Teichmüller geodesic generated by the L-shaped translation surface tiled by three squares

We prove that the Teichmuller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmuller metric) is almost isometric to the Teichmuller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmuller metric).

The Real Analytic Theory of Teichmüller Space

Citations

On the Teichm\"uller geodesic generated by the L-shaped translation surface tiled by three squares

Yang-Yang generating function and Bergman tau-function

A Gauss–Bonnet formula for moduli spaces of Riemann surfaces

On the Teichmüller geodesic generated by the L-shaped translation surface tiled by three squares

Almost isometries between Teichm\"uller spaces

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