Convergence of group actions in metric measure geometry

2021/04/01 by Hiroki Nakajima, Nakajima, Hiroki, Takashi Shioya +1
Computer Science · Mathematics · #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities #Topological and Geometric Data Analysis #math.DG #math.MG #msc:53C23

paper · pdf · doi:10.48550/arxiv.2104.00187

18 pages

openalex publication_date 2021/04/01 · arxiv created 2021/04/20 · arxiv updated 2021/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize the box and observable distances to those between metric measure spaces with group actions, and prove some fundamental properties. As an application, we obtain an example of a sequence of lens spaces with unbounded dimension converging to the cone of the infinite-dimensional complex projective space. Our idea is to use the theory of mass-transport.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

Related

• Topological aspects of the space of metric measure spaces
• Convergence of metric transformed spaces
• Box distance and observable distance via optimal transport
• Metric measure limits of spheres and complex projective spaces
• Group actions on metric spaces: fixed points and free subgroups