论文标题
Weil-Petersson完成的地层分离和长度功能的梯度估计值
Strata Separation for the Weil-Petersson Completion and Gradient Estimates for Length Functions
论文作者
论文摘要
通常，很难在Teichmüller空间上测量Weil-Petersson指标中的距离。在这里，我们考虑了有限型表面的Teichmüller空间的Weil-Petersson完成中的地层之间的距离。沃尔珀特（Wolpert）表明，对于闭合没有相交的地层，存在明确的分离，而与表面拓扑独立。我们证明，这种最小分离的最佳值是常数$δ_{1,1} $，并证明它是由阶层相交一次的层次实现的。我们还给出了$δ_{1,1} $的几乎清晰的估计，并在$δ_{1,1} $和其他距离之间的间隙大小和其他距离之间给出了一个下限。本文的主要组成部分是Wolpert在$ \ langle \ nabla \ell_α，\ nabla \ell_β\ rangle $的有效版本，这是长度函数的Weil-Petersson梯度的内部产物。我们进一步将双曲线表面的Teichmüller空间的边界绑定到表面的收缩期。我们还在刺穿的圆环的模量空间上的Weil-Petersson度量的收缩期上获得了新的下限。
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 $ \langle \nabla \ell_α,\nabla \ell_β\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.
