Convergence Theory for Alexandrov Spaces

在曲率下有界的Alexandrov空间的奇点不大的情况下，在空间收敛下。我们能够构造空间之间的Lipschitz同胚。特别地，亚历山德罗夫空间的体积的连续性从这个结果得出。此外，我们证明了亚历山德罗夫空间的奇异集的Hausdorff测度为零，并且可以在正则集上定义一个自然的黎曼结构。我们还证明了曲率有界的Alexandrov空间的等距群是李群，这在黎曼几何中有一些应用。另一方面，我们将Gromov不变量的概念推广到Alexandrov空间，并阐明了曲率、体积与Gromov不变量之间的关系。首先，利用Alexander-Spanier上同调理论证明了X的基本类[X]的存在性，并定义了X的Gromov不变量。其次，证明了基本类[X]的质量与X的体积一致。在这方面的证明中，我们利用几何测度理论将质量拓扑中的链表示[X]近似为具有良好性质的Lipschitz链，并发展了一种可以被认为是斯托克斯定理的替代的抵消技术。证明了负曲率Alexandrov空间的Gromov不变量可以在曲率上界和体积下估计。在Alexandrov曲面的情况下，我们得到了一个尖锐的估计的Gromov不变量的类型的奇点。对于曲率下有界的Alexandrov空间，我们给出了Gromov不变量的体积和曲率下界的估计。因此，它证明了这样一个空间的奇点的出现并不影响格罗莫夫不变量这么多。这显示了两种情况之间的巨大差异，上面弯曲的空间和下面弯曲的空间。

In the case where the singularities of Alexandrov spaces with curvature bounded below are not so big, under convergence of spaces. we were able to construct Lipschitz homeomorphisms between spaces. In particalar, the continuity of volumes of Alexandrov spaces follows from this result. Moreover, we proved that the Hausdorff measure of the singular set of an Alexandrov space is zero, and that one can define a natural Riemannian structure on the regular set. We also proved that the isometry group of an Alexandrov space with curvature bounded below is a lie group, which has some applications to Riemannian geometry. On the other hand, we extended the notion of the Gromov invariant to Alexandrov spaces, and clarified the relation between the curvature, volume and the Gromov invariant. First, making use of the Alexander-Spanier cohomology theory, we proved the existence of the fundamental class [X] of X, and defined the Gromov invariant of X.Next, we proved that the mass of the fundamental class [X] coincides with the volume of X.In the proof of this face, we used geometric measure theory to approximate a chain representing [X] in the mass topology by a Lipschitz chain with nice properties, and developed a cancellation technique which might be considered as a replacement of Stokes' theorem. And we proved that the Gromov invariant of a negatively curved Alexandrov space can be estimated below interms of the upper bound of curvature and the volume. In the case of Alexandrov surfaces, we obtained a sharp estimate for the Gromov invariant with the type of singularities. For Alexandrov spaces with curvature bounded below, we bave an estimate for the Gromov invariant from above in terms of the volume and the lower bound of curvature. Thus it turned out that the appearanceo of singularities of such a space does not affect the Gromov invariant so much. This shows the big difference between the two cases, spaces curved above and spaces curved below.