Curvature and rank of Teichmüller space
Curvature and rank of Teichmüller space
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DOI：
10.1353/ajm.2006.0003
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发表时间：
2001-09
影响因子：
1.7
通讯作者：
J. B. Brock;B. Farb
中科院分区：
文献类型：
--
作者：
J. B. Brock;B. Farb
设S为亏格为g，边界分支为n的曲面，d（S）= 3g - 3 + n表示S的任意Pants分解中的曲线个数.利用Pants分解图P（S）的度量性质，证明了Teichmiiller空间Teich（S）上的Weil-Petersson度量是Gromov-双曲的当且仅当d（S）3，Weil-Petersson度量在Gromov意义下有较高的秩（它允许Rk的拟等距嵌入，k > 2）;当d（S）> 2时，模空间M（S）上不存在具有pinched负曲率的完备黎曼度量. 1.导论. Teichmiiller空间Teich（5）上的Weil-Petersson度量有许多奇怪的性质。它是一个具有负截面曲率的黎曼度量，但它的曲率不是远离零或负无穷大的。它是测地线凸的，但它不是完全的。在本文中，我们表明，尽管表现出负曲率行为，Weil-Petersson度量不是粗负弯曲，除了拓扑简单的曲面S。我们的主定理回答了鲍迪奇的一个问题（Be，问题11.4）。
Let 5 be a surface with genus g and n boundary components, and let d(S) = 3g - 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions P(S) to prove that the Weil-Petersson metric on Teichmiiller space Teich (S) is Gromov-hyperbolic if and only if d(S) 3, the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of Rk,k > 2); when d(S) 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) > 2. 1. Introduction. The Weil-Petersson metric on Teichmiiller space Teich (5) has many curious properties. It is a Riemannian metric with negative sectional curvature, but its curvatures are not bounded away from zero or negative infinity. It is geodesically convex, but it is not complete. In this paper we show that in spite of exhibiting negative curvature behavior, the Weil-Petersson metric is not coarsely negatively curved except for topologically simple surfaces S. Our main theorem answers a question of Bowditch (Be, Question 11.4).