Progress toward classifying Teichmuller disks with completely degenerate Kontsevich-Zorich spectrum

We present results toward resolving a question posed by Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich. They asked for a classification of all SL₂ ($R$)-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum. Let $Dg$(1) be the subset of the moduli space of Abelian differentials $Mg$ whose elements have period matrix derivative of rank one. There is an SL₂($R$)-invariant ergodic probability measure ν with completely degenerate Kontsevich-Zorich spectrum, i.e. λ₁ = 1 > λ ₂ = ··· = λ_g = 0, if and only if ν has support contained in $Dg$(1). We approach this problem by studying Teichmüller disks contained in $Dg$(1). We show that if (X, ω) generates a Teichmüller disk in $Dg$(1), then (X, ω) is completely periodic. Furthermore, we show that there are no Teichmüller disks in $Dg$(1), for g = 2, and the known example of a Teichmüller disk in $D3$(1) is the only one. Finally, we present an idea that might be able to fully resolve the problem.