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May 5, 2026Open Access

Rigidity and Stability of RCD (K, ) Path Spaces under Singular Kato–Voigt Perturbations

Key Points

This research aims to analyze the rigidity and stability of RCD(K,∞) path spaces under singular Kato–Voigt perturbations.
Utilized projective limits to construct RCD(K,∞) path spaces.
Applied L^2-normed modules and Gigli's differential calculus for analysis.
Demonstrated stability via Mosco convergence and characterized curvature shifts using the EVI on Wasserstein space.
Established closure of the Γ2-operator on ℳ2,2, indicating robustness of these spaces.
Identified the intrinsic Hessian as a Riesz representative, enhancing the understanding of curvature.
Proved stability under singular Kato–Voigt perturbations, linking geometric properties with spectral changes.

Abstract

We study the rigidity and stability properties of RCD (K, ) path spaces constructed over metric measure spaces via projective limits. The analysis is performed within the framework of L²-normed modules and Gigli’s differential calculus. We establish the closure of the ₂-operator on D^2, 2, identify the intrinsic Hessian as a Riesz representative, and prove stability under singular Kato–Voigt perturbations via Mosco convergence. The main rigidity result characterizes curvature shifts through the Evolution Variational Inequality (EVI) on Wasserstein space P₂ (), linking geometric curvature bounds with spectral perturbations of the associated Dirichlet form.

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