Splitting theorem and localization for RCD(0,∞) and negative-dimension cases

Establish an isometric splitting theorem and a localization (needle decomposition) for RCD(0,∞) spaces, and for RCD(0,N) spaces with N in (−∞,−1), to enable extensions of the generalized Grünbaum inequality beyond the smooth weighted Riemannian setting.

Background

The main results rely crucially on Gigli’s splitting theorem and Cavalletti–Mondino’s localization, both currently available in RCD(0,N) with N in (1,∞). To treat N=∞ and N<−1 in the synthetic framework, analogous splitting and localization results are needed.

In the smooth setting (weighted Riemannian manifolds), splitting and localization are available for N=∞ and N<−1, but for RCD spaces these results are not yet established, limiting generalization of the paper’s inequalities.