Theory of collapsing Riemannian manifolds and geometry of Alexandrov spaces

1.我们完成了截面曲率和直径分别从下到上均匀有界的坍缩4流形的研究，建立了三维和四维非负曲率完全开空间的几何（Yamaguchi）.2。我们证明了具有较小体积的下曲率界的3-流形是图流形（Yamaguchi and Shioya）。我们确定了具有一致有界的绝对曲率曲面的Gromov-Hausdorff收敛性，并详细地发展了极限珍珠空间的几何，如奇点、同伦类型、珍珠数（Yamaguchi和Hori）。在曲率有界的二维奇异空间中，我们确定了一个奇点的邻域的局部几何性质，证明了它是几个Lipschitz盘（Yamaguchi， Nagano和Shioya）的胶合。我们定义了里奇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。我们用它证明了庞加莱不等式和一个紧性定理（Kuwae和Shioya）。我们考虑了空间如黎曼流形或亚历山德罗夫空间的离散逼近，并证明了网络的拉普拉斯算子与空间的拉普拉斯算子的收敛性（Otsu）。利用上述净逼近的思想，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.