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.

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H odd (4) the only GL + (2, R) orbit closures are closed orbits, the Prym locusQ(3, −1 3 ), and H odd (4).Together with work of Matheus-Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in H odd (4) outside of the Prym locus.

We reduce a question of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich, which asks for a classification of all SL2(R)-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum, to a conjecture of Möller's. 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 > λ2 = · · · = λg = 0, if and only if ν has support contained in Dg(1). We approach this problem by studying Teichmüller discs contained in Dg(1). We show that if (X, ω) generates a Teichmüller disc in Dg(1), then (X, ω) is completely periodic. Furthermore, we show that there are no Teichmüller discs in Dg(1), for g = 2, and the two known examples of Teichmüller discs in Dg(1), for g = 3, 4, are the only two such discs in those genera. Finally, we prove that if there are no genus five Veech surfaces generating Teichmüller discs in D5 (1), then there are no Teichmüller discs in Dg(1), for g = 5, 6. matrix, cf. Section 3.2, and a technical lemma concerning the limit of a surface with cylinders that do not fill the surface under the Teichmüller geodesic flow, cf. Lemma 4.3. These results quickly yield some applications, cf. Proposition 6.4.Next we show that the closure of every Teichmüller disc in D g (1) must contain a (possibly degenerate) surface that is a Veech surface, cf. Theorem 7.4. This leads to an analysis of punctures on a Veech surface with the goal of excluding more and more configurations of the punctures until the remainder of the results follow. Theorem 1.1 summarizes Proposition 6.4, Theorem 8.10, Theorem 9.10, and Proposition 8.16.Acknowledgments: This work was done in partial fulfillment of the requirements for the Ph.D. at the University of Maryland -College Park. The author would like to express his sincere gratitude to his advisor Giovanni Forni, for his patience and the generosity of his time throughout the entirety of the research process. The author would also like to thank Scott Wolpert for his insight into the geometry of the moduli space. The author is grateful to Martin Möller for expressing interest in his work at an early stage. Finally, the author would like to thank Matt Bainbridge and Kasra Rafi for their helpful discussions. Preliminaries The Moduli Space of Riemann SurfacesLet X be a Riemann surface of genus g with n punctures (i.e. marked points). Let R(X) denote the Teichmüller space of X or simply R g,n when X is understood. The surface X admits a pants decomposition, X = P 1 ∪ · · · ∪ P 3g−3+n , into 3g − 3 + n pairs of pants, where each pair of pants is homeomorphic to the sphere with a total of three punctures and disjoint boundary curves. The Fenchel-Nielsen coordinates for Teichmüller space describe surfaces in terms of the lengths and twists of curves in a pants decomposition of X. A point in Teichmüller space is given by ( 1 , . . . , 3g−3+n , θ 1 , . . . , θ 3g−3+n ) ∈ R 3g−3+n + ...

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