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We bound the derivative of complex length of a geodesic under variation of the projective structure on a closed surface in terms of the norm of the Schwarzian in a neighborhood of the geodesic. One application is to cone-manifold deformations of acylindrical hyperbolic 3-manifolds. 

We consider the Weil–Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the unique structure $$M_{\rm geod}$$ with minimum convex core volume. 

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. 

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. 

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.