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1. TEICHMÜLLER CURVES IN HYPERELLIPTIC COMPONENTS OF MEROMORPHIC STRATA

   https://doi.org/10.1017/S1474748025000088  

   Möller, Martin; Mullane, Scott; Bakker, Benjamin; Mullane, Scott (July 2025, Journal of the Institute of Mathematics of Jussieu)

   Abstract We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two. 

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2. Limit Sets of Weil–Petersson Geodesics

   https://doi.org/10.1093/imrn/rny002  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (February 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we prove that the limit set of any Weil–Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil–Petersson geodesics with minimal non-uniquely ergodic ending laminations and limit set a circle in the Thurston compactification. 

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3. Carathéodory's metrics on Teichmüller spaces and L-shaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497–535.

   Makovic, Vladimir (July 2018, Duke Math. J)

   One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each Teichmüller space of a closed surface of genus g≥2. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential. 

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4. Stable Strata of Geodesics in Outer Space

   https://doi.org/10.1093/imrn/rnx269  

   Algom-Kfir, Yael; Kapovich, Ilya; Pfaff, Catherine (January 2018, International Mathematics Research Notices)

   Abstract In this article, we propose an Outer space analog for the principal stratum of the unit tangent bundle to the Teichmüller space $${\mathcal{T}}(S)$$ of a closed hyperbolic surface $$S$$. More specifically, we focus on properties of the geodesics in Teichmüller space determined by the principal stratum. We show that the analogous Outer space “principal” periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmüller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmüller space context. 

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5. Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

   https://doi.org/10.1515/crelle-2017-0024  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (January 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension {3g-3} , for example. We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex. 

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