[论文解读] THE CONVEX REAL PROJECTIVE MANIFOLDS AND ORBIFOLDS WITH RADIAL ENDS I: THE OPENNESS OF DEFORMATIONS

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

Geometric and Algebraic Topology参考文献 44被引用 9

一句话总结

本文通过分析其基本群和端点结构，建立了具有径向端点的凸实射影轨道丛的形变开放性，证明在特定条件下，小形变可保持严格凸性。该研究通过证明此类轨道丛当且仅当其端点处严格凸时为相对双曲，推广了Benoist在闭轨道丛上的结果。

ABSTRACT

A real projective orbifold is an n-dimensional orbifold modeled on RP n with groups PGL(n + 1;R). We concentrate on orbifolds with a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, dieomorphic to (n 1)-dimensional orbifolds times intervals. A real projective orbifold has radial end if each of its end is foliated by projective geodesics concurrent to each other. It is said to be convex if any path can be homotoped to a projective geodesic with endpoints xed. A projective structure sometimes admit deformation to inequivalent parameters of real projective structures. We will prove that local homemorphism between the deformation space of projective structures on such an orbifold with radial ends with various conditions with the representation space of the fundamental group with corresponding conditions. We will use a Hessian argument to show that a small deformation of a real projective orbifold with ends and without essential annuli will remain properly and strictly convex in a generalized sense if so is the beginning real projective orbifold provided that the ends behave in a convex manner. Here, we have to restrict each end to have a fundamental group isomorphic to a nite extension of a product of hyperbolic groups and abelian groups. The understanding of the ends is not accomplished in this paper as this forms an another subject. We will prove the closedness of the convex real projective structures on orbifolds with irreducibilty condition. One theorem of note is that a convex irreducible real projective orbifold with radial ends with some condition is relatively hyperbolic if and only if it is strictly convex with respect to ends, generalizing the result of Benoist for closed orbifolds.

研究动机与目标

• 建立具有径向端点的轨道丛上实射影结构形变空间的开放性。
• 根据端点行为和基本群结构，刻画凸性与严格凸性。
• 将Benoist关于闭轨道丛的结果推广至具有径向端点的轨道丛情形。
• 研究相对双曲性与端点处严格凸性之间的关系。
• 对端点基本群施加限制，使其为双曲群与交换群乘积的有限扩张，以实现结构控制。

提出的方法

• 使用Hessian方法分析小形变下凸性条件的二阶变分。
• 将轨道丛建模为紧致核心，其端点邻域微分同胚于(n−1)维轨道丛与区间的乘积。
• 要求每个端点被一族在一点相交的射影测地线所叶状覆盖，从而定义径向端点。
• 对每个端点的基本群施加条件，限制其为双曲群与交换群乘积的有限扩张。
• 在实射影结构的形变空间与基本群的表示空间之间建立局部同胚。
• 利用不可约性条件，确保轨道丛结构在拓扑与几何上的控制。

实验结果

研究问题

• RQ1 在何种条件下，具有径向端点的凸实射影轨道丛的小形变仍保持严格凸性？
• RQ2 端点的基本群如何约束实射影结构的形变空间？
• RQ3 在这些轨道丛中，相对双曲性与端点处的严格凸性之间存在何种关系？
• RQ4 Hessian方法如何确保形变空间的开放性？
• RQ5 端点的结构——特别是其基本群——如何影响轨道丛的整体凸性？

主要发现

• 在给定条件下，具有径向端点的轨道丛上实射影结构的形变空间是开放的。
• 若原结构为严格凸且端点行为凸，则小形变可保持严格凸性。
• 一个不可约的凸实射影轨道丛若具有径向端点，则当且仅当其在端点处严格凸时，为相对双曲。
• 为使结论成立，每个端点的基本群必须同构于双曲群与交换群乘积的有限扩张。
• 在实射影结构的形变空间与基本群的表示空间之间建立了局部同胚。
• Hessian方法提供了一个关键的分析工具，用以验证凸性在小扰动下得以保持。