Seminar

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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 δ_(1,1) and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for δ_(1,1) and give a lower bound on the size of the gap between δ_(1,1) and the other distances. A major component of the paper is an effective version of Wolpert's upper bound on 〈∇ℓα,∇ℓβ〉, 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. This is joint work with Ken Bromberg.

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Strata Separation For The Weil-Petersson Completion And Gradient Estimates For Length Functions