Local cut points and metric measure spaces with Ricci curvature bounded below

Abstract

A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends of such a space. We also obtain some obstruction conditions for the existence of a local cut point in a metric measure space satisfying the Bishop--Gromov inequality or the Poincaré inequality. For example, the measured Gromov--Hausdorff limits of Riemannian manifolds with a lower Ricci curvature bound satisfy these two inequalities.