Theory of collapsing Riemannian manifolds and geometry of Alexandrov spaces

1.完成了截面曲率和直径分别自下有界和上有界的4-流形的折叠研究，建立了3维和4维完备的非负曲率开空间(Yamaguchi)的几何；2.证明了具有小曲率下界的3-流形是图流形(Yamaguchi和Shioya)；3.确定了全曲率一致有界曲面的Gromov-Hausdorff收敛，并发展了极限珍珠空间的几何，如奇点、同伦型、4.我们确定了曲率在上面有界的二维奇异空间中奇点邻域的局部几何性质，证明了它是若干Lipschitz圆盘(Yamaguchi，Nagano和Shioya)的粘合.5.我们定义了Ricci曲率在下面有界的奇异空间的概念，并将这种空间的能量形式引入到一般的度量空间.我们用它证明了Poincare不等式和一个紧性定理(科威特和Shioya)。6.我们考虑了黎曼流形或Alexandrov空间的离散逼近，证明了网的拉普拉斯收敛到空间的拉普拉斯收敛(Otsu)。利用上述网逼近的思想，我们研究了流形上热算子的渐近行为，得到了幂零覆盖流形上热算子的中心极限定理。

1.We have completed the study of collapsing 4-manifolds whose sectional curvature and diameter are uniformly bounded from below and above respectively, and established the geometry of 3-dimensional and 4-dimensional complete open spaces of nonnegative curvature (Yamaguchi).2.We have proved that a 3-manifold with a lower curvature bound having a small volume is a graph manifold (Yamaguchi and Shioya).3.We have determined the Gromov-Hausdorff convergence of surfaces with uniformly bounded total absolute curvature, and developed geometry of limit pearl spaces in detail such as singularities, homotopy types, number of pearls (Yamaguchi and Hori).4.We have determined local geometric properties of a neighborhood of a singular point in an two-dimensional singular spaces with curvature bounded above proving that it is a gluing of several Lipschitz disks (Yamaguchi, Nagano and Shioya).5.We have defined the notion of singular spaces with Ricci curvature bounded below, and introduced energy forms from such spaces to general metric spaces. We have proved the Poincare inequality and a compactness theorem using it (Kuwae and Shioya).6.We have considered discrete approximations of spaces like Riemannian manifolds or Alexandrov spaces by graphs called nets, and proved that the convergence of Laplacians of nets to that of the space (Otsu). Using the idea of net-approximation above, we have studied the asymptotic behavior of heat operators on manifolds and obtained a central limit theorem for heat operator on nilpotent covering manifolds.