Mathematical Sciences: Linear and Nonlinear Rigidity of Discrete Subgroups Lie Groups and Manifolds of Negative Curvature

本文主要研究负弯曲流形和对称空间的几何和刚性问题，主要包括：1)将Mostow刚性推广到无限协体积的一般离散子群。作者已经取得了一些重要的成果，但仍有一些重要的开放性问题有待解决，似乎必须结合遍历理论、代数几何、复几何和海森堡几何的技术来进一步研究。2) Mostow刚性的非线性扩展——非紧半单李群中格的光滑作用的刚性。特别地，作者将进一步研究在光滑群作用下存在不变共形、射影或仿射结构的障碍，并将微分不变量理论与Zimmer的循环超刚性结合起来。3)负弯曲流形周围的几何和动力学。作者运用遍历理论、全局分析等方法，得到了一些关于负弯曲流形的结果。有几个深刻而困难的开放问题（例如，Katok熵猜想，标记长度谱问题）。动态和全球方法的潜力值得进一步探索。负弯曲3-maifold的夸张化。克莱因理论的组。作者最近通过考虑测地线流中的拟共形性，证明了负弯曲闭流形要么是双曲的，要么是测地线流保持了可测的固有不变分布。这似乎是对负弯曲闭三流形（Thurston的方案中一个大的开放问题）的夸张化的另一种方法迈出的重要的第一步。作者对阿尔福斯面积猜想也很感兴趣。作者的数学研究主要围绕负弯曲空间的刚性和柔性展开。从一般的角度来看，大多数空间都是负弯曲的。空间的内在性质与其测地线流动的动力学密切相关。因此，为了理解这些空间（包括我们生活的空间）——它们的大小、形状和演变，就不可避免地要研究各种动力系统的稳定性、刚性和灵活性。对他们采取集体行动。

Abstract Yue The author is currently working on various aspects of the geometry and rigidity of negatively curved manifolds and symmetric spaces using methods from dynamical systems, Lie groups, several complex variable, quasiconformal mapping, algebraic geometry: 1) Extensions of Mostow rigidity to general discrete subgroups of infinite covolume. The author has achieved several key results but still there are important open problems to be solved and it seems that one has to combine techniques from ergodic theory, algebraic geometry, complex geometry and Heisenberg geometry for further approach. 2) Nonlinear extensions of Mostow rigidity--the rigidity of smooth actions of lattices in noncompact semisimple Lie groups. In particular, the author will further study the obstructions to the existence of an invariant conformal or projective or affine structure under a smooth group action and to combine the theory of differential invariants with Zimmer's cocycle superrigidity. 3)Geometry and Dynamics around negatively curved manifolds. The author has obtained a number of results concerning negatively curved manifolds by using techniques from ergodic theory, global analysis. There are several deep and difficult open problems(for example, the Katok entropy conjecture, the marked length spectrum problem). The potential of the dynamical and global approach deserves to be further explored. 4)Hyperbolization of negatively curved 3-maifolds. Kleinian groups. The author proved recently that a negatively curved closed manifold is either hyperbolic or the geodesic flow preserves a measurable proper invariant distribution by considering quasiconformality in the geodesic flow. This seems to be a first nontrivial step towards another approach to the hyperbolization of negatively curved closed three-manifolds(which is a big open problem in Thurston's program). The author is also interested in the Ahlfors area conjecture. The author's mathematical research are centered around the rigidity and flexibilit y of negatively curved spaces. From a generic point of view, most spaces are negatively curved. The intrinsic property of spaces are closely related to the dynamics of its geodesic flow. Therefore to understand these spaces(including the one we are living in)--their size, shape, and evolution, it is inevitable to study the stability, the rigidity and flexibility of various dynamical systems(i.e. group actions) on them.