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 SL$(2,\mathbb{R})$-invariant submanifold is completely degenerate, i.e. ${\lambda }_2=\cdots ={\lambda }_g=0$, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there are at most finitely many such Teichmueller curves.

arXiv comment26 pages, 4 figures: The proof of the main theorem was edited so that it is independent of arXiv:1205.2359