Gromov-Hausdorff convergence and a theory of variational convergences

近年来，关于度量空间的几何分析的研究方兴未艾。首席研究员Shioya主要研究度量度量空间的曲率和收敛，特别是Alexandrov空间、度量度量空间的Ricci曲率和度量度量空间的Gromov-Hausdorff收敛。另一方面，莫斯科研究了变分收敛，这是一种关于狄里克莱特能量形式收敛的泛函分析理论。我们Shioya和研究者科威特认为Mosco的理论与度量度量空间的收敛研究有很深的联系，并从几何的角度对该理论进行了推广。我们已经在这个项目期间完成了它。我们的理论中的收敛概念现在被称为Mosco-科威特-Shioya收敛，并被广泛地应用于概率论中的有限维方法以及一些齐次化问题。另一种研究是关于Alexandrov空间上的拉普拉斯比较定理和分裂定理，其条件对应于Ricci曲率的下界。这件事仍在继续。对于黎曼流形，下面有界的Ricci曲率等价于Bishop-Gromov不等式的无穷小版本。由于在Alexandrov空间上不可能定义Ricci曲率张量，所以我们考虑无穷小的Bishop-Gromov不等式而不是Ricci曲率的界。与黎曼空间不同，在Alexandrov空间中，割轨迹不一定是闭集。这甚至可能是一个密集的集合。由于这个原因，与黎曼流形相同的证明不起作用，我们发展了一种新的证明方法。

In these days, the study of geometric analysis on metric measure spaces is going around. The head investigator, Shioya, Studies such a subject and his main interest is curvature of metric measure spaces and convergence, especially Alexandrov spaces, Ricci curvature of metric measure spaces, and Gromov-Hausdorff convergence of metric measure spaces. On he other hand, Mosco studied variational convergences, which is a functional analytic theory of convergence of Dirichlet energy forms. We, Shioya and the investigator, Kuwae, thought that Mosco's theory is deeply related with the study of convergence of metric measure spaces, and have extended the theory in the geometric viewpoint. We have completed it in the period of this project. The concept of convergence in our theory is nowadays called the Mosco-Kuwae-Shioya convergence and is being widely applied to the finite dimensional method in probability theory and also to some homogenization problems.Another study is on a Laplacian comparison theorem and a splitting theorem on Alexandrov spaces with some condition corresponding to a lower bound of Ricci curvature. This is still on going. For Riemannian manifods, the Ricci curvature being bounded below is equivalent to an infinitesimal version of the Bishop-Gromov inequality. Since it is impossible to define the Ricci curvature tensor on Alexandrov spaces, we consider the infinitesimal Bishop-Gromov inequality instead of the Ricci curvature bound. Different from Riemannian, the cut-locus is not necessarily a closed set in an Alexandrov space. That may even be a dense set. By this reason, the same proof as for Riemannian manifolds does not work and we develop a new method of proof.