Simple closed geodesics on hyperelliptic translation surfaces and a classification theorem for translation surfaces in ${\mathcal{H}^{\mathrm{hyp}}}(4)$

In this paper, we give the maximum of the numbers $n$ such that we can take $n$ simple closed geodesics without singularities that are disjoint to each other for translation surfaces in the hyperelliptic components ${\mathcal{H}^{\mathrm{hyp}}}(2g-2)$ and ${\mathcal{H}^{\mathrm{hyp}}}(g-1, g-1)$. The maximum is different from the maximum number of pairwise disjoint simple closed geodesics on a hyperbolic surface. We also give a classification theorem for translation surfaces in ${\mathcal{H}^{\mathrm{hyp}}}(4)$ with respect to their Euclidean structures.