A Zero Lyapunov Exponent in Genus 3 Implies the Eierlegende Wollmilchsau

We prove that the closed orbit of the Eierlegende Wollmilchsau is the only SL 2 (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–Möller and the first named author. The main new contribution is the identification of the differentials in the Hodge bundle corresponding to the Forni subspace in terms of the degenerations of the surface. We use this description of the differentials in the Forni subspace to evaluate them on absolute homology curves and apply the jump problem from the work of Hu and the third named author to the differentials near the boundary of the orbit closure. This results in a simple geometric criterion that excludes the existence of a Forni subspace.