Veech surfaces and complete periodicity in genus two

We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair $(M, \omega)$ where $M$ is a Riemann surface and $\omega$ is an Abelian differential on $M$. Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of $\omega$.

A direction $v$ on a translation surface is completely periodic if any trajectory in the direction $v$ is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction $v$. Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group $SL(2, \mathbb{R})$ on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false.

In this paper, we use the $J$-invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space ${\mathcal H}(2)$ of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.