Mosco-convergence of Cheeger energies on varying spaces satisfying curvature dimension conditions

TL;DR

This paper investigates the Mosco-convergence of Cheeger energies on spaces converging in the Gromov-Hausdorff sense, with applications to eigenvalue continuity, using a Lagrangian approach that unifies stability and nonsmooth calculus.

Contribution

It introduces a novel Lagrangian method to analyze Cheeger energy convergence on varying metric measure spaces with curvature conditions, including infinite-dimensional cases.

Findings

Proves Mosco-convergence under curvature dimension conditions.

Establishes continuity of Neumann eigenvalues in converging spaces.

Extends analysis to functions of bounded variation.

Abstract

We study the Mosco-convergence of Cheeger energies on Gromov-Hausdorff converging spaces satisfying different types of curvature dimension conditions. The case of functions of bounded variation is also considered. Applications to the continuity of Neumann eigenvalues are obtained. Our method, covering possibly infinite dimensional settings, is based on a Lagrangian approach and combines the stability properties of Wasserstein geodesics with the characterization of the nonsmooth calculus in duality with test plans.