Calculus of Variations and Geometric Measure Theory

E. Pasqualetto - T. Schultz

Ultralimits of pointed metric measure spaces

created by pasqualetto on 24 Feb 2021



Inserted: 24 feb 2021
Last Updated: 24 feb 2021

Pages: 65
Year: 2021

ArXiv: 2102.11365 PDF


The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured Gromov-Hausdorff convergence. We also introduce a weaker variant of pointed measured Gromov-Hausdorff convergence, for which we can prove a version of Gromov's compactness theorem by using the ultralimit machinery. This compactness result shows that, a posteriori, our newly introduced notion of convergence is equivalent to the pointed measured Gromov one. Another byproduct of our ultralimit construction is the identification of direct and inverse limits in the category of pointed metric measure spaces.

Tags: GeoMeG