  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.

Dynamical Systems;Geometric Topology