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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. 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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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. Ricci flow and diffeomorphism groups of 3-manifolds

   https://doi.org/10.1090/jams/1003  

   Bamler, Richard; Kleiner, Bruce (August 2022, Journal of the American Mathematical Society)

   We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature. 

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5. The Jacobson–Morozov Morphism for Langlands Parameters in the Relative Setting

   https://doi.org/10.1093/imrn/rnad217  

   Bertoloni_Meli, Alexander; Imai, Naoki; Youcis, Alex (September 2023, International Mathematics Research Notices)

   Abstract We construct a moduli space $$\textsf {LP}_{G}$$ of $$\operatorname {SL}_{2}$$-parameters over $${\mathbb {Q}}$$, and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism$$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$$ (where $$\textsf {WDP}_{G}$$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $$\textsf {JM}$$ is an isomorphism over a dense open of $$\textsf {WDP}_{G}$$, that it induces an isomorphism between the discrete loci $$\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$$, and that for any $${\mathbb {Q}}$$-algebra $$A$$ it induces a bijection between Frobenius semi-simple equivalence classes in $$\textsf {LP}_{G}(A)$$ and Frobenius semi-simple equivalence classes in $$\textsf {WDP}_{G}(A)$$ with constant (up to conjugacy) monodromy operator. 

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