Theory of collapsing Riemannian manifolds and geometry of Alexandrov spaces

1.研究了截面曲率和截面直径分别自下而上一致有界的坍缩四维流形，建立了三维和四维非负曲率完备开空间的几何（Yamaguchi）. 2.证明了具有小体积的下曲率界的3-流形是图流形（Yamaguchi和Shioya）. 3.我们确定了具有一致有界全绝对曲率的曲面的Gromov-Hausdorff收敛性，并详细地发展了极限pearl空间的几何学，如奇点，同伦类型，珍珠数量（Yamaguchi和Hori）. 4.我们确定了两点空间中奇点邻域的局部几何性质，曲率有界的高维奇异空间，证明它是多个Lipschitz圆盘的胶合（Yamaguchi，Nagano and Shioya）. 5.我们定义了Ricci曲率有界的奇异空间的概念，并将这种空间的能量形式引入一般度量空间。我们证明了Poincare不等式和利用它的一个紧性定理（Kaze和Shioya）。6.我们考虑了用称为网的图对空间（如黎曼流形或Alexandrov空间）的离散逼近，并证明了网的拉普拉斯算子收敛于空间的拉普拉斯算子（大津）。利用网逼近的思想，我们研究了流形上热算子的渐近性态，得到了幂零覆盖流形上热算子的一个中心极限定理。

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.