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

TL;DR

This paper proves that in genus three, the Eierlegende Wollmilchsau's orbit closure is uniquely characterized by having a zero Lyapunov exponent, using geometric and degeneration techniques in moduli space.

Contribution

It introduces a new geometric criterion to exclude Forni subspaces, advancing the understanding of Lyapunov spectra in genus three moduli spaces.

Findings

The Eierlegende Wollmilchsau is the unique orbit with zero Lyapunov exponent in genus three.

A geometric criterion is established to exclude Forni subspaces along degenerations.

The solution to the jump problem is applied to analyze boundary behavior in moduli space.

Abstract

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.