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 ${\text{SL}}_2(\mathbb{R})$-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 ${\text{SL}}_2(\mathbb{R})$-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 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 $g=5,6$.

arXiv comment68 pages, 7 figures: A gap was found in the previous version and corrected here in low genus (g \leq 6). A stronger version of the original result is still true by arXiv:1302.0913. To appear in Comm. Math. Helv