Teichm\"uller 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 $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum, to a conjecture of M\"oller'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 $\text{SL}_2(\mathbb{R})$-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 Teichm\"uller discs contained in $\mathcal{D}_g (1)$. We show that if $(X,\omega)$ generates a Teichm\"uller disc in $\mathcal{D}_g (1)$, then $(X,\omega)$ is completely periodic. Furthermore, we show that there are no Teichm\"uller discs in $\mathcal{D}_g (1)$, for $g = 2$, and the two known examples of Teichm\"uller 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 Teichm\"uller discs in $\mathcal{D}_5(1)$, then there are no Teichm\"uller discs in $\mathcal{D}_g (1)$, for $g = 5,6$.