[论文解读] The convex real projective orbifolds with radial or totally geodesic ends: The closedness and openness of deformations

Suhyoung Choi|arXiv (Cornell University)|Nov 4, 2010

Geometric and Algebraic Topology参考文献 45被引用 6

一句话总结

本文建立了具有径向或全测地端的凸实射影轨道丛的形变空间与基本群的 PGL(n+1,R)-表征空间之间的局部同胚关系。通过 Hessian 论证与几何收敛技术，证明了在形变下适当（分别严格）凸结构的开性与闭性，推广了 Benoist 对闭轨道丛的结果，并通过 Crampon-Marquis 与 Cooper-Long-Tillmann 的 Margulis 引理，将理论扩展至相对双曲结构。

ABSTRACT

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed $(n-1)$-dimensional orbifolds times intervals. A real projective orbifold has a {\it radial end} if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a {\it totally geodesic end} if the end can be completed to have the totally geodesic boundary. The orbifold is said to be {\it convex} if any path can be homotopied to a projective geodesic with endpoints fixed. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove the local homeomorphism between the deformation space of convex real projective structures on such an orbifold with radial or totally geodesic ends with various conditions with the $PGL(n+1, \mathbb{R})$-character space of the fundamental group with corresponding conditions. We will use a Hessian argument to show that under a small deformation, a properly (resp. strictly) convex real projective orbifold with generalized admissible ends will remain properly and properly (resp. strictly) convex with generalized admissible ends. Lastly, we will prove the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends, where we need the theory of Crampon-Marquis and Cooper, Long and Tillmann on the Margulis lemma for convex real projective manifolds. The theory here partly generalizes that of Benoist on closed real projective orbifolds.

研究动机与目标

• 建立具有径向或全测地端的凸实射影轨道丛的形变空间与基本群的 PGL(n+1,R)-表征空间之间的局部同胚关系。
• 证明具有广义可接受端的轨道丛上适当（分别严格）凸实射影结构空间在形变空间中的开性与闭性。
• 将 Benoist 对闭凸实射影轨道丛的理论推广至具有受控端结构的非紧情形。
• 利用 Hessian 论证与凸性保持性，分析小形变下 holonomy 表征的性质。

提出的方法

• 利用 Hessian 论证表明，具有广义可接受端的轨道丛的小形变可保持适当与严格凸性。
• 应用凸实射影轨道丛的几何收敛理论，依赖于万有覆叠上的 Hausdorff 与 Hilbert 度量。
• 采用对偶构造与仿射悬垂方法，分析 p-端邻域及其凸包的结构。
• 运用相对双曲性理论与 Bowditch 方法，将群作用与凸性及端行为联系起来。
• 应用 Crampon-Marquis 与 Cooper-Long-Tillmann 关于凸实射影流形的 Margulis 引理结果，以控制端部的几何结构。
• 利用表示空间的半代数性质与极限集的连续性，证明形变空间的闭性。

实验结果

研究问题

• RQ1 在何种条件下，具有径向或全测地端的轨道丛上适当凸实射影结构的小形变仍保持适当凸？
• RQ2 此类轨道丛上适当（分别严格）凸实射影结构的空间在形变空间中是否既开又闭？
• RQ3 基本群的 holonomy 表征在形变下如何变化？何种条件可确保凸性被保持？
• RQ4 具有广义可接受端的轨道丛上凸实射影结构的形变空间是否在局部同胚于基本群的表征空间？
• RQ5 p-端群及其在凸区域上的作用在轨道丛的全局凸性与拓扑结构中起何种作用？

主要发现

• 在适当条件下，具有径向或全测地端的凸实射影轨道丛的形变空间与基本群的 PGL(n+1,R)-表征空间之间存在局部同胚关系。
• 通过基于 Hessian 的分析表明，当原始结构具有广义可接受端时，小形变可保持适当与严格凸性。
• 具有广义可接受端的轨道丛上适当（分别严格）凸实射影结构的空间在形变空间中既开又闭。
• 形变空间的闭包由端群的极限集与凸包表征，维数约束确保了适当域结构。
• holonomy 作用在发展像上在形变下仍保持适当凸性，且发展映射保持为到凸域的微分同胚。
• 该理论将 Benoist 对闭凸实射影轨道丛的结果推广至具有受控端结构的非紧情形。