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

Francesco Nobili, Federico Renzi, Federico Vitillaro

TL;DR

The paper develops a direct Mosco-convergence framework for Cheeger energies on sequences of pmGH-converging spaces under synthetic curvature-dimension conditions ${\sf CD}(K,N)$ and ${\sf MCP}(K,N)$. It blends a Lagrangian formulation of Sobolev/BV calculus via test plans with polygonal Wasserstein interpolations to propagate energy bounds through varying spaces, achieving semicontinuity and Mosco convergence in broad, possibly infinite-dimensional, settings. The results include sharp stability for ${\sf CD}(K,N)$ spaces and a quantified, uniform-constant stability for ${\sf MCP}(K,N)$ spaces, highlighting how a second-order curvature bound governs first-order stability under zeroth-order pmGH convergence. The methods open avenues for local curvature constraints and non-Riemannian geometries, with potential applications to stability of heat flows, variational problems, and geometric analysis on converging space sequences.

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. 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.

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

TL;DR

The paper develops a direct Mosco-convergence framework for Cheeger energies on sequences of pmGH-converging spaces under synthetic curvature-dimension conditions and . It blends a Lagrangian formulation of Sobolev/BV calculus via test plans with polygonal Wasserstein interpolations to propagate energy bounds through varying spaces, achieving semicontinuity and Mosco convergence in broad, possibly infinite-dimensional, settings. The results include sharp stability for spaces and a quantified, uniform-constant stability for spaces, highlighting how a second-order curvature bound governs first-order stability under zeroth-order pmGH convergence. The methods open avenues for local curvature constraints and non-Riemannian geometries, with potential applications to stability of heat flows, variational problems, and geometric analysis on converging space sequences.

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. 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.

Paper Structure

This paper contains 15 sections, 19 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Test plans
  4. Wasserstein space
  5. Sobolev and BV calculus
  6. pmGH-topology and convergence of functions
  7. Curvature dimension conditions
  8. An equivalent Lagrangian notion of calculus
  9. Polygonal interpolations on varying spaces
  10. The case $q<\infty$
  11. The case $q=\infty$
  12. Mosco-convergence results
  13. The infinite-dimensional setting
  14. Main results for ${\sf CD}(K,N)$ spaces
  15. Main result for ${\sf MCP}(K,N)$ spaces

Key Result

Theorem 1.1

Let $({\rm X}_n,{\sf d}_n,\mathfrak m_n,x_n)$ with $\mathfrak m_n({\rm X}_n)<\infty$ be $q$-essentially non-branching pointed metric measure spaces for all $q \in (1,\infty),n\in\mathbb{N}$. Suppose that ${\rm X}_n$ satisfies ${\sf CD}(K,N)$ for some $K\in\mathbb{R},N\in[1,\infty)$ and that ${\rm X

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: $q$-test plan
  • Proposition 2.2
  • Definition 2.3: Sobolev space
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 38 more