Related papers: Veech surfaces and complete periodicity in genus 2

We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus $g$ with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The…

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any…

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a…

We study infinite translation surfaces which are Z-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition…

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…

For a non-arithmetic Veech surface, it is known that the set points having finite orbit under the Veech group, called the set of periodic points, is finite. However, few examples of these periodic point sets have been computed. In what…

An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders…

We provide a complete classification of groups that can be realized as isometry groups of a translation surface $M$ with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if $S$ has no…

We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known…

We give a complete description of all classical Enriques surfaces with non-zero global vector fields. In particular we show that there are such surfaces. The obtained result also applies to supersingular Enriques surfaces fulfilling a…

In this note we are interested in the dynamics of the linear flow on infinite periodic $\mathbb{Z}^d$-covers of Veech surfaces. An elementary remark allows us to show that the kernel of some natural representations of the Veech group acting…

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any…

For fixed g and T we show that finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech groups contain a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we show that any…

We study how are permuted Weierstrass points of Veech surfaces in $\mathcal{H}(2)$, the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two. These surfaces were classified by McMullen relying on…

For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, \omega)$ in the…

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.

We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.

We obtain the first complete classification result for algebraically primitive Teichm\"{u}ller curves in a stratum component in genus greater than 2. Specifically, we show that the Veech 14-gon generates the unique algebraically primitive…

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS},…

In this paper we are interested in the stratum H^{hyp}(4) of translation surfaces, which consists of pairs (M,\omega), where M is a hyper-elliptic Riemann surface of genus 3, and \omega is a holopmorphic 1-form on M having only one zero. We…