论文标题
双曲线歧管凸子集边界的几何数据
The geometric data on the boundary of convex subsets of hyperbolic manifolds
论文作者
论文摘要
令$ n $为凸面联合式双曲线歧管$ m $具有不可压缩边界的地理凸子集。我们假设$ n $的每个边界组件要么是$ \ partial_ \ infty m $的边界组件，要么是$ m $的平滑局部凸面。我们表明，$ n $是由无穷大的边界组件上的保形结构定义的边界数据，以及归功于局部凸面表面的边界组件上的诱导度量或第三个基本形式。我们还描述了可能的边界数据。这提供了双曲线Weyl问题和Ahlfors-Bers定理的扩展。 使用此陈述对Quasifuchsian歧管，我们为凸域的类似问题获得了$ω\ subset \ hh^3 $的存在结果，这些\ subset \ hh^3 $沿quasicircle或沿quasidisk沿着quasicircle或沿quasidisk沿着无限的$ \ partial _ {\ partial _ {\ partial _ {\ partial _ {\ partial _ {\ intial _ {\ intial _ {\ intial _ {\ infty} \ hh^3 $。边界数据然后包括$ \ hh^3 $中的诱导度量或第三个基本形式，但还包括边界不同组件之间的附加“胶合”数据，以$ \ hh^3 $或$ \ partial_ \ partial_ \ infty \ infty \ hh^3 $。
Let $N$ be a geodesically convex subset in a convex co-compact hyperbolic manifold $M$ with incompressible boundary. We assume that each boundary component of $N$ is either a boundary component of $\partial_\infty M$, or a smooth, locally convex surface in $M$. We show that $N$ is uniquely determined by the boundary data defined by the conformal structure on the boundary components at infinity, and by either the induced metric or the third fundamental form on the boundary components which are locally convex surfaces. We also describe the possible boundary data. This provides an extension of both the hyperbolic Weyl problem and the Ahlfors-Bers Theorem. Using this statement for quasifuchsian manifolds, we obtain existence results for similar questions for convex domains $Ω\subset \HH^3$ which meets the boundary at infinity $\partial_{\infty}\HH^3$ either along a quasicircle or along a quasidisk. The boundary data then includes either the induced metric or the third fundamental form in $\HH^3$, but also an additional "gluing" data between different components of the boundary, either in $\HH^3$ or in $\partial_\infty\HH^3$.
