Gromov-Hausdorff convergence and a theory of variational convergences

Gromov-Hausdorff 收敛性和变分收敛理论

基本信息

批准号：
17540058
负责人：
SHIOYA Takashi
金额：
$ 2.3万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

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.

近年来，关于度量测度空间的几何分析的研究正在兴起。首席研究员 Shioya 研究这样一个课题，他的主要兴趣是度量测度空间的曲率和收敛性，特别是 Alexandrov 空间、度量测度空间的 Ricci 曲率和度量测度空间的 Gromov-Hausdorff 收敛性。另一方面，莫斯科研究了变分收敛，这是狄利克雷能量形式收敛的泛函分析理论。我们Shioya和研究者Kuwae认为Mosco的理论与度量测度空间收敛性的研究有很深的联系，并从几何的角度扩展了该理论。我们在这个项目期间已经完成了。我们理论中的收敛概念现在被称为Mosco-Kuwae-Shioya收敛，并且被广泛应用于概率论中的有限维方法以及一些均质化问题。另一个研究是拉普拉斯比较定理和分裂定理亚历山德罗夫空间的某些条件对应于里奇曲率的下界。这件事还在继续。对于黎曼流形，下面有界的 Ricci 曲率相当于 Bishop-Gromov 不等式的无穷小版本。由于不可能在 Alexandrov 空间上定义 Ricci 曲率张量，因此我们考虑无穷小的 Bishop-Gromov 不等式而不是 Ricci 曲率界。与黎曼不同，割轨迹不一定是 Alexandrov 空间中的闭集。这甚至可能是一个密集集。因此，与黎曼流形相同的证明不起作用，我们开发了一种新的证明方法。