Strata Separation for the Weil-Petersson Completion and Gradient Estimates for Length Functions

Abstract

In general, it is difficult to measure distances in the Weil-Petersson metric on Teichm\"uller space. Here we consider the distance between strata in the Weil-Petersson completion of Teichm\"uller 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 $\delta_{1,1}$ 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 Teichm\"uller 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.