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

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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.