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1. Limit sets of Weil–Petersson geodesics with nonminimal ending laminations

   https://doi.org/10.1142/S1793525319500456  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (March 2020, Journal of Topology and Analysis)

   null (Ed.)

   In this paper, we construct examples of Weil–Petersson geodesics with nonminimal ending laminations which have [Formula: see text]-dimensional limit sets in the Thurston compactification of Teichmüller space. 

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2. 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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3. SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

   https://doi.org/10.1017/fms.2019.49  

   ARANA-HERRERA, FRANCISCO; ATHREYA, JAYADEV S. (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics. 

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4. Function theoretic characterizations of Weil–Petersson curves

   https://doi.org/10.4171/RMI/1398  

   Bishop, Christopher J (December 2022, Revista Matemática Iberoamericana)

   This is a companion to the paper "Weil-Petersson curves, conformal energies, beta-numbers, and minimal surfaces". That paper gives various new geometric characterizations of Weil-Petersson in the plane that can be extended to curves in all finite dimensional Euclidean spaces. This paper deals with the 2-dimensional case, giving new proofs of some known characterizations, and giving new results for the conformal weldings of Weil-Petersson curves and a geometric characterization of these curves in terms of Peter Jones's beta-numbers. 

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5. Strata separation for the Weil–Petersson completion and gradient estimates for length functions

   https://doi.org/10.1142/S1793525321500667  

   Bridgeman, Martin; Bromberg, Kenneth (January 2022, Journal of Topology and Analysis)

   World Scientific (Ed.)

   In general, it is difficult to measure distances in the Weil–Petersson metric on Teichmüller space. Here we consider the distance between strata in the Weil–Petersson completion of Teichmüller space of a surface of finite type. Wolpert showed that for strata whose closures do not intersect, there is a definite separation independent of the topology of the surface. We prove that the optimal value for this minimal separation is a constant [Formula: see text] and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for [Formula: see text] and give a lower bound on the size of the gap between [Formula: see text] and the other distances. A major component of the paper is an effective version of Wolpert’s upper bound on [Formula: see text], the inner product of the Weil–Petersson gradient of length functions. We further bound the distance to the boundary of Teichmüller space of a hyperbolic surface in terms of the length of the systole of the surface. We also obtain new lower bounds on the systole for the Weil–Petersson metric on the moduli space of a punctured torus. 

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