On $\mathsf{CD}$ spaces with nonnegative curvature outside a compact set

Abstract

In this paper we adapt work of Z.-D. Liu to prove a ball covering property for non-branching CD spaces with nonnegative curvature outside a compact set. As a consequence we obtain uniform bounds on the number of ends of such spaces.Theorem 1.1. Let M n be a complete Riemannian manifold with nonnegative Ricci curvature outside a compact set B. Assume that Ric M ≥ (n − 1)H and that B ⊂ B D0 (p 0 ) for some p 0 ∈ M and D 0 > 0. Then for any µ > 0 there exists C = C(n, HD 2 0 , µ) > 0 such that for any r > 0, the following property is satisfied:We state and prove this result in the more general context of non-branching metric measure spaces satisfying the curvature-dimension condition introduced by Lott-Sturm-Villani [11,13,14] (see the section on preliminaries below for the definitions). This class of spaces contains the class of RCD spaces, as it was recently shown that these are non-branching (see [5, Theorem 1.3]), so, a fortiori, it also includes Alexandrov spaces [12,16] and weighted Riemannian manifolds. More precisely, we prove the following theorem.Theorem A. Let (X, d, m) be a non-branching metric measure space satisfying the CD(K, N ) condition for some N > 1 and K ∈ R. Assume that B is a compact subset of X with B ⊂ B D0 (p 0 ) for some p 0 ∈ M and D 0 > 0 and such that the CD loc (0, N ) condition is satisfied on X \ B (see Definition 2.5). Then for any µ > 0, there exists C = C(N, KD 2 0 , µ) > 0 such that for any r > 0, the following property is satisfied: If S ⊂ B r (p 0 ), there exist p 1 , . . . , p k ∈ S with k ≤ C and such thatThe proof follows the arguments of [9,10] almost verbatim, albeit with some needed adaptations to account for the more general hypotheses. The main tools we need are a version of the local-toglobal theorem for the CD condition (see Lemma 2.2) and a Bishop-Gromov inequality for certain