Affine Manifolds and Zero Lyapunov Exponents in Genus 3

Abstract

In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curve in genus three has two zero Lyapunov exponents in the Kontsevich-Zorich cocycle, then it lies in the principal stratum and has at most quadratic trace field. Moreover, there can be at most finitely many such Teichmüller curves.

40 pages, To appear in GAFA

References in corpus (3)

• Cylinder deformations in orbit closures of translation surfaces
• The field of definition of affine invariant submanifolds of the moduli space of abelian differentials
• Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum