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Strata separation for the Weil-Petersson completion and gradient estimates for length functions. (English) Zbl 07870899

Summary: 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. S. A. Wolpert [Surv. Differ. Geom. 8, 357–393 (2003; Zbl 1049.32020)] 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 \(\delta_{1,1}\) and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for \(\delta_{1,1}\) and give a lower bound on the size of the gap between \(\delta_{1,1}\) and the other distances. A major component of the paper is an effective version of Wolpert’s upper bound on \(\langle\nabla \ell_\alpha,\nabla \ell_\beta\rangle \), 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.

MSC:

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 1049.32020

References:

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