Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum

We reduce a question of Eskin–Kontsevich–Zorich and Forni–Matheus–Zorich, which asks for a classification of all ${SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich–Zorich spectrum, to a conjecture of Möller's. Let $\mathcal{D}_g (1)$ be the subset of the moduli space of Abelian differentials $\mathcal{M}_g$ whose elements have period matrix derivative of rank one. There is an ${SL}_2(\mathbb{R})$-invariant ergodic probability measure $\nu$ with completely degenerate Kontsevich–Zorich spectrum, i.e. $\lambda_1 = 1 > \lambda_2 = \cdots = \lambda_g = 0$, if and only if $\nu$ has support contained in $\mathcal{D}_g (1)$. We approach this problem by studying Teichmüller discs contained in $\mathcal{D}_g (1)$. We show that if $(X,\omega)$ generates a Teichmüller disc in $\mathcal{D}_g (1)$, then $(X,\omega)$ is completely periodic. Furthermore, we show that there are no Teichmüller discs in $\mathcal{D}_g (1)$, for $g = 2$, and the two known examples of Teichmüller discs in $\mathcal{D}_g (1)$, for $g = 3, 4$, are the only two such discs in those genera. Finally, we prove that if there are no genus five Veech surfaces generating Teichmüller discs in $\mathcal{D}_5(1)$, then there are no Teichmüller discs in $\mathcal{D}_g (1)$, for $g = 5,6$.