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arXiv · 2104.03588

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

Mattia Magnabosco·Chiara Rigoni·Gerardo Sosa
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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 $d_{\mathsf{iKRW}}$, 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 $d_{\mathsf{iKRW}}$ to some metric measure space, then this limit structure is still a CD(K, N) space.

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Mattia Magnabosco, Chiara Rigoni, Gerardo Sosa. 2021-04-08. Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter. https://arxiv.org/abs/2104.03588

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