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

Aulicino, David; Benirschke, Frederik; Norton, Chaya

Annales Henri Lebesgue, Tome 7 (2024), pp. 207-237

We prove that the closed orbit of the Eierlegende Wollmilchsau is the only ${\text{SL}}_2\left(ℝ\right)$-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.

Nous démontrons que l’adhérence de l’orbite du Eierelegende Wollmilchsau est la seule adhérence d’orbite de ${\text{SL}}_2\left(ℝ\right)$ en genre trois avec un exposant de Lyapunov nul dans son spectre de Kontsevich-Zorich. Ce résultat étend des résultats partiels de Bainbridge–Habegger–Möller et du premier auteur. La contribution principale de notre article est l’identification des différentielles dans le fibré de Hodge correspondant au sous-espace de Forni en termes de dégénérescences de la surface. Nous utilisons cette description des différentielles dans le sous-espace de Forni afin de les évaluer sur les courbes d’homologie absolue et appliquons le “jump problem”, dû à Hu et au troisième auteur, aux différentielles près du bord de l’adhérence de l’orbite. Ceci implique un critère géométrique simple qui exclut l’existence d’un sous-espace de Forni.

Classification : 32G20, 37D40, 14H15, 14G35, 37D25
Keywords: Teichmüller geodesic flow, Kontsevich-Zorich cocycle, Abelian differentials, translation surfaces, Lyapunov exponents, period matrices, variational formulas

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