Affine Invariant Submanifolds with Completely Degenerate Kontsevich-Zorich Spectrum

TL;DR

This paper characterizes affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, showing they are arithmetic Teichmüller curves in specific genera and proving finiteness of such curves.

Contribution

It proves that completely degenerate spectra imply the submanifold is an arithmetic Teichmüller curve in genus three to five, establishing a finiteness result.

Findings

Such submanifolds are arithmetic Teichmüller curves in genus 3, 4, or 5.

There are finitely many such Teichmüller curves.

Complete degeneracy of the spectrum characterizes these special submanifolds.

Abstract

We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL$(2,\mathbb{R})$-invariant submanifold is completely degenerate, i.e. $\lambda_2 = \cdots = \lambda_g = 0$, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there are at most finitely many such Teichmueller curves.