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1. 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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2. 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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3. 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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4. Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

   https://doi.org/DOI 10.1515/crelle-2020-0005  

   Bridgeman, M; Wu, Yunhui (January 2021, Journal für die reine und angewandte Mathematik)

   De Gruyter (Ed.)

   In this article we show that for every finite area hyperbolic surface X of type (g; n) and any harmonic Beltrami differential 􏰚 on X , then the magnitude of 􏰚 at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of 􏰚 over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity. 

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5. The Weil–Petersson gradient flow ofrenormalized volume and 3–dimensional convex cores

   https://doi.org/10.2140/gt.2023.27.3183  

   Bridgeman, Martin; Brock, Jeffrey; Bromberg, Kenneth (January 2023, Geometry topology)

   We use the Weil–Petersson gradient flow for renormalized volume to study the space CC(N;S,X) of convex cocompact hyperbolic structures on the relatively acylindrical 3-manifold (N;S). Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasifuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure M_geod in CC( N;S,X) with totally geodesic convex core boundary facing S. Analyzing the geometry of structures along a flow line, we show that if V_R(M) is the renormalized volume of M, then V_R(M)−V_R(M_geod) is bounded below by a linear function of the Weil Petersson distance d_WP(∂_c M,∂_cM_geod), with constants depending only on the topology of S. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3-manifolds, providing new proofs and generalizations of well-known theorems such as Storm’s result that M geod has minimal volume for N acylindrical and the second author’s result comparing convex core volume and Weil–Petersson distance for quasifuchsian manifolds. 

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