A Zero Lyapunov Exponent in Genus $3$ Implies the Eierlegende Wollmilchsau

We prove that the closed orbit of the Eierlegende Wollmilchsau is the only $\text{SL}(2,\mathbb{R})$-orbit closure in genus three with a zero Lyapunov exponent in its Kontsevich-Zorich spectrum. The result recovers previous partial results in this direction by Bainbridge-Habegger-M\"oller and the first named author. The main new contribution is an understanding of the Forni subspace along a degeneration toward the boundary of the moduli space of curves. This results in a simple geometric criterion that excludes the existence of a Forni subspace. Another key ingredient is the solution to the jump problem from the work of Hu and the third named author.