Examples of Ricci limit spaces with infinite holes

Let n 3, R, and (X, h) be an n-dimensional smooth complete Riemannian manifold with Ricₕ >. In this paper, we construct, for each given >0, a sequence of (n+2) -dimensional manifolds (M₈, gᵢ) GH (X_, d_) with Ric₆㶁 >, such that d₆₇ (X, X_), and X_ is homeomorphic to the space obtained by removing an infinite number of balls from X. Hence X_ has dense boundary with an infinite number of connected components. Moreover, X_ has no open subset which is topologically a manifold. This generalizes Hupp-Naber-Wang's result (arxiv: 2308. 03909) from 4-dimensional case to the general case of dimension n 3. Notably, the restriction n 3 is essential, as Ricci limit spaces in 1 and 2 dimensions always have a dense open subset which is topologically a manifold. Our construction differs from that of Hupp-Naber-Wang. In their approach, Hupp-Naber-Wang considered doing an infinite number of blow-ups on the local complex surface structure of X, thus relying on the 4-dimensional condition. However, our method involves removing an infinite number of balls from X, allowing us to construct in the general case of dimensions greater than or equal to 3. As a corollary, we provide a solution to an open problem posed by Naber in the 3-dimensional case. Note that the 3-dimensional case is the only remaining unresolved case in Naber's problem. Consequently, Naber's question is resolved in all dimensions.