The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
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DOI：
10.1090/s0894-0347-03-00424-7
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发表时间：
2001-09
影响因子：
3.9
通讯作者：
Jeffrey F. Brock
中科院分区：
文献类型：
--
作者：
Jeffrey F. Brock
利用Hatcher和瑟斯顿提出的裤子分解图，我们给出了两个有限区域双曲Riemann曲面X和Y之间的Weil-Petersson距离Dwp(X，Y)的粗略解释。裤子图的组合学揭示了黎曼曲面与瑟斯顿猜想的双曲三维流形之间的联系：以X和Y为共形边界的拟富氏流形Q(X，Y)的凸核的体积与Weil-Petersson距离Dwp(X，Y)相当。在应用中，我们将Weil-Petersson距离与Q(X，Y)的极限集的Hausdorff维数和拉普拉斯算子的最低本征值联系起来，给出了几何极限的一个新的有限判据。芝加哥大学数学系，邮编：5734.伊利诺伊州芝加哥S大学大道，邮编：60637电子邮件地址：broc@math.uchicago.edu
We present a coarse interpretation of the Weil-Petersson distance dWP(X,Y ) between two finite area hyperbolic Riemann surfaces X and Y using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y ) with X and Y in its conformal boundary is comparable to the Weil-Petersson distance dWP(X,Y ). In applications, we relate the Weil-Petersson distance to the Hausdorff dimen- sion of the limit set and the lowest eigenvalue of the Laplacian for Q(X,Y ), and give a new finiteness criterion for geometric limits. Mathematics Department, University of Chicago, 5734 S. University Ave., Chicago, IL 60637 E-mail address: brock@math.uchicago.edu