Convergence of energy functionals and stability of lower bounds of Ricci curvature via metric measure foliation

TL;DR

This paper explores how metric measure foliations influence the convergence and stability of Ricci curvature bounds in metric measure spaces, extending understanding of geometric analysis in unbounded dimension contexts.

Contribution

It introduces a convergence theory for sequences of metric measure spaces with unbounded dimensions using metric measure foliations, linking curvature conditions and quotient space properties.

Findings

Curvature-dimension conditions are preserved under metric measure foliations.

Established convergence results for sequences of spaces with unbounded dimensions.

Demonstrated stability of lower Ricci curvature bounds via foliation techniques.

Abstract

The notion of the metric measure foliation is introduced by Galaz-Garc\'ia, Kell, Mondino, and Sosa. They studied the relation between a metric measure space with a metric measure foliation and its quotient space. They showed that the curvature-dimension condition and the Cheeger energy functional preserve from a such space to its quotient space. Via the metric measure foliation, we investigate the convergence theory for a sequence of metric measure spaces whose dimensions are unbounded.