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 , ω ) (M, \omega ) where M M is a Riemann surface and ω \omega is an Abelian differential on M 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 ω \omega . A direction v v on a translation surface is completely periodic if any trajectory in the direction v v is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction v 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 S L ( 2 , R ) 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 J -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space H ( 2 ) {\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.