Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

Abstract

We prove that if the Lyapunov spectrum of the Kontsevich–Zorich cocycle over an affine $\text{SL}_{2}(\mathbb{R})$-invariant submanifold is completely degenerate, i.e. if $\unicode[STIX]{x1D706}_{2}=\cdots =\unicode[STIX]{x1D706}_{g}=0$, then the submanifold must be an arithmetic Teichmüller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there is at most a finite number of such Teichmüller curves.