Study of Conformal Geometry and Group C^*-bundle from the Viewpoint of Global Analysis

(1)Yamabe不变量的研究我们利用逆平均曲率流和格林函数族估计并确定了一些正的三维流形的Yamabe不变量。特别地，我们对所有Yamabe不变量大于Rp^3的3-流形进行了完全分类。我们还研究了黎曼积的正Yamabe常数以及它们在一个因子放大下的行为。我们正在研究无限覆盖的正Yamabe常数的Aubin型引理，得到了一些新的结果。(2)共形/仿射几何和群C^*-代数的研究我们给出了共形几何和射影几何的新发展。特别地，我们得到了由爱因斯坦度规诱导的仿射联络的一个有趣的变分刻画。研究了扭曲K-理论和群C^*-代数，证明了Gromov-Lawson定理在分片空间中的推广。(3)几何中非线性分析的研究我们研究了完备流形上的特征值问题、离散群上的特征值问题和平均曲率上的验证问题。得到了非正曲率完备流形上不存在特征值的结果，以及Hadamard空间上离散群作用的不动点定理。

We have studied the following :(1)Study of Yamabe InvariantsWe estimated and determined the Yamabe invariant of some positive 3-manifolds, by using the inverse mean curvature flow and families of Green's functions. In especial, we classified completely all 3-manifolds with Yamabe invariant greater than that of RP^3. We also studied the positive Yamabe constants of Riemannian products and the behavior of them under magnifying one factor. We are now studying Aubin's type lemma for the positive Yamabe constants of infinite coverings, with some new results.(2)Study of Conformal/Affine Geometry and group C^*-algebraWe gave new developments on conformal and projective geometry. In particular, we obtained an interesting varitional characterization of affine connections induced from Einstein metrics. We studied on twisted K-theory and groupoid C^*-algebras, and then proved a generalization of Gromov-Lawson theorem for foliated spaces.(3)Study on Non-linear Analysis in GeometryWe studied on eigenvalue problem on complete manifolds, discrete groups and valiational problems on mean curvature. We then obtained results on non-existence of eigenvalues on complete manifolds of non-positive curvature, and a fixed-point theorem for discrete-group actions on Hadamard spaces.