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

摘要岳 作者目前正致力于负曲流形和对称空间的几何和刚性的各个方面，使用动力系统，李群，多复变量，拟共形映射，代数几何的方法：1）Mostow刚性扩展到无限余体积的一般离散子群。作者已经取得了几个关键的结果，但仍然有重要的开放问题需要解决，它似乎是一个联合收割机技术从遍历理论，代数几何，复几何和海森堡几何进一步的做法。 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的共形或射影或仿射结构的障碍，并联合收割机微分不变量理论与 齐默上循环超刚性。 3)几何和动力学周围负弯曲流形。 作者利用遍历理论、整体分析等技巧，得到了一些关于负曲流形的结果。有几个深层次和困难的开放问题（例如，Katok熵猜想，标记长度谱问题）。动态和全球办法的潜力值得进一步探讨。4)负曲3-歧的双曲化。克莱因群作者最近证明了一个负弯曲的闭流形是双曲的或测地线流保持一个可测的真不变分布，通过考虑拟共形在测地线流。这似乎是第一个非平凡的一步，另一种方法的双曲化的负弯曲封闭三流形（这是一个大的开放问题，在瑟斯顿的计划）。作者还对Ahlfors面积猜想感兴趣。 作者的数学研究是围绕负曲空间的刚性和柔性展开的。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的动力学性质密切相关。因此，要了解这些空间（包括我们所居住的空间）--它们的大小、形状和演变，就不可避免地要研究它们的稳定性、刚性和柔性。 各种动力系统（即群体作用）。

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.