Bounds on the average bending of the convex hull boundary of a Kleinian group
Bounds on the average bending of the convex hull boundary of a Kleinian group
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克莱因群凸包边界的平均弯曲的界限
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发表时间：
2003
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通讯作者：
M. Bridgeman
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作者：
M. Bridgeman
本文考虑具有不可压缩凸核边界的双曲流形。我们证明了在凸核的边界上沿测地圆弧的总弯曲是由其长度的函数在上面有界的。在凸核边界的单位切丛上积分该函数，得到了凸核边界总弯曲的一个新的泛函上界。此外，我们还给出了从凸核边界到无穷远处双曲结构映射的Lipschitz常数的一个新的普适上界。这些结果改进了Bridgeman和Canary的早期界。设N=H/Γ是具有不连续域Ω(Γ)和极限集LΓ的可定向双曲流形。在本文中，我们将自己限制在Ω(Γ的所有组件都是单连接的情况下。这是一个自然的限制，并且包括拟富氏群的集合。设CH(LΓ)是Γ的凸包，βΓ是∂CH(LΓ)上的弯曲叠层。设C(N)=CH(LΓ)/Γ为凸核，βN为∂C(N)上的弯曲叠层.然后我们观察到∂C(N)是不可压缩的当且仅当Ω(Γ的所有分支都是单连通的。如果α是CH(LΓ)中的测地圆弧，则平均弯曲B(α)被定义为单位长度的弯曲，或者具体地
In this paper we consider hyperbolic manifolds with incompressible convex core boundary. We show that total bending along a geodesic arc on the boundary of the convex core is bounded above by a function of its length. Integrating this function over the unit tangent bundle of the boundary of the convex core we obtain a new universal upper bound on the total bending of the convex core boundary. Furthermore, we produce a new universal upper bound on the lipschitz constant for the map from the convex core boundary to the hyperbolic structure at infinity. These results improve on earlier bounds of Bridgeman and Canary. Let N = H/Γ be an orientable hyperbolic manifold with domain of discontinuity Ω(Γ) and limit set LΓ. In this paper we restrict ourselves to the case when all the components of Ω(Γ) are simply connected. This is a natural restriction to make and includes the set of quasi-fuchsian groups. Let CH(LΓ) be the convex hull of Γ and βΓ be the bending lamination on ∂CH(LΓ). Let C(N) = CH(LΓ)/Γ be the convex core, and βN be the bending lamination on ∂C(N). Then we observe that ∂C(N) is incompressible if and only if the components of Ω(Γ) are all simply connected. If α is a geodesic arc in CH(LΓ), the average bending B(α) is defined to be the bending per unit length, or specifically