Rank two affine submanifolds in $\mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$

TL;DR

This paper classifies all rank two affine submanifolds in specific genus three strata of translation surfaces, confirming a conjecture by Mirzakhani and providing technical results that may aid in a full classification.

Contribution

It provides a complete classification of rank two affine submanifolds in certain genus three strata, confirming Mirzakhani's conjecture for these cases.

Findings

Confirmed Mirzakhani's conjecture for these strata.

Classified all rank two affine manifolds in the specified strata.

Proved technical results applicable to genus three strata.

Abstract

We classify all rank two affine manifolds in strata in genus three with two zeros. This confirms a conjecture of Maryam Mirzakhani in these cases. Several technical results are proven for all strata in genus three, with the hope that they may shed light on a complete classification of rank two manifolds in genus three.