Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum

Abstract

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 $g=5,6$.