Calculus of Variations and Geometric Measure Theory

E. Caputo

Geometric characterizations of ${\sf PI}$ spaces: an overview of some modern techniques

created by caputo on 03 Feb 2025

[BibTeX]

Submitted Paper

Inserted: 3 feb 2025
Last Updated: 3 feb 2025

Year: 2025

ArXiv: 2501.19132 PDF

Abstract:

We survey recent results on the study of metric measure spaces satisfying a Poincar\'e inequality. We overview recent characterizations in terms of objects of dimension 1, such as pencil of curves, modulus estimates and obstacle-avoidance principles. Then, we turn our attention to characterizations in terms of objects of codimension 1, such as relative isoperimetric inequalities and separating sets, the last one obtained in collaboration with N. Cavallucci in arXiv:2401.02762. We propose a strategy to provide examples using our characterization in the toy-model of the Euclidean case. We also discuss a more geometric relation between separating sets and obstacle-avoidance principles, obtained in IMRN, Vol. 2025, Issue 1, Jan. 2025, rnae276. Finally, we recall some open questions in the field.