Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter

2021/04/08 by Mattia Magnabosco, Magnabosco, Mattia, Chiara Rigoni +3
Mathematics · #Geometric Analysis and Curvature Flows #Fixed Point Theorems Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2104.03588

Abstract

We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we introduce an appropriate setting to introduce the CD(K, N)-condition for N < 0, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we introduce the distance d_iKRW, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K, N)-condition with N < 0 is converging with respect to the distance d_iKRW to some metric measure space, then this limit structure is still a CD(K, N) space.

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