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

本文作者目前正在用动力系统、李群、多复变、拟共形映射、代数几何等方法研究负曲流形和对称空间的几何和刚性的各个方面：1)将Mostow刚性推广到无限余体积的一般离散子群。作者已经取得了一些关键的结果，但仍有一些重要的公开问题需要解决，似乎必须结合遍历理论、代数几何、复几何和海森伯几何的技术才能进一步深入。2)Mostow刚性的非线性推广--非紧半单李群中格子光滑作用的刚性。特别是，作者将进一步研究光滑群作用下不变共形、射影或仿射结构存在的障碍，并将微分不变量理论与Zimmer上循环超刚性相结合。3)负曲流形周围的几何和动力学。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。有几个深入而困难的公开问题(例如，Katok熵猜想，标记长度谱问题)。动力和全球方法的潜力值得进一步探索。4)负曲线三维流形的双曲化。克莱恩集团。最近，作者通过考虑测地线流中的拟共形，证明了负曲线闭流形是双曲的，或者测地线流保持了可测的真不变分布。这似乎是朝着另一种方法迈出的不平凡的第一步，即负曲线闭三流形的双曲线化(这在瑟斯顿的程序中是一个大的公开问题)。作者还对Ahlfors面积猜想感兴趣。作者的数学研究是围绕负曲线空间的刚性和灵活性展开的。从一般的角度来看，大多数空间都是负曲线的。空间的内在性质与其测地线流的动力学密切相关。因此，为了理解这些空间(包括我们所居住的空间)--它们的大小、形状和演化，不可避免地要研究它们上各种动力系统(即群体作用)的稳定性、刚性和灵活性。

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.