Gromov-Hausdorff convergence and a theory of variational convergences

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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.