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

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

2006-10-05

arxiv.org/abs/math/0610170v1

Mathematics

Differential Geometry

26 pages, 6 figures

Scientific paper

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 Poincar\'{e} inequality. For example, the measured Gromov--Hausdorff limits of Riemannian manifolds with a lower Ricci curvature bound satisfy these two inequalities.

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