(The Cone Space $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$) The main result of this paper, Theorem 1.1, states that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$, with $\beta<2\pi$, is not the noncollapsed Gromov-Hausdorff limit of a sequence of manifolds with bounded Ricci curvature. However, it is clear that this space is the Gromov-Hausdorff limit of a a sequence of noncollapsed manifolds with a uniform lower Ricci curvature bound. Indeed, by rounding off $C(S^{1}_{\beta})$ we see that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ can appear as a noncollapsed limit of manifolds with nonnegative sectional curvature.

In this example, let us just consider the two dimensional cone $C(S^{1}_{\beta})$ with $\beta<2\pi$. Regard $S^{1}_{\beta}$ as $0\leq\theta\leq 2\pi$, with the end points identified. Then the Laplacian on $S^{1}_{\beta}$ is $(\frac{2\pi}{\beta})^{2}\cdot\frac{\partial^{2}}{\partial\theta^{2}}$. The eigenfunctions are of the form $e^{ik\theta}$, where $k$ is an integer. Written in polar coordinates, a basis for the bounded harmonic functions on $C(S^{1}_{\beta})$ is $\{r^{\frac{2\pi}{\beta}|k|}\cdot e^{ik\theta}\}$. In particular, we see from this that if $\beta<2\pi$ then $|\nabla(r^{\frac{2\pi}{\beta}|k|}\cdot e^{ik\theta})|\to 0$ as $r\to 0$. As a consequence, every bounded harmonic function has vanishing gradient at the vertex, which is a set of positive $(n-2)$-dimensional Hausdorff measure. By considering examples with more vertices, we can construct limit spaces where bounded harmonic functions $h$ must have vanishing gradient on bounded subsets sets of arbitrarily large, or even infinite, $(n-2)$-dimensional Hausdorff measure. This set can even be taken to be dense.

(The Eguchi-Hanson manifold) The Eguchi-Hanson metric $g$ is a complete Ricci flat metric on the cotangent bundle of $S^{2}$, which at infinity, becomes rapidly asymptotic to the metric cone on $\mathds{R}\mathds{P}(3)$ or equivalently to $\mathds{R}^{4}/\mathds{Z}_{2}$, where $\mathds{Z}_{2}$ acts on $\mathds{R}^{4}$ by $x\to-x$. When the metric $g$ is scaled down by $g\to r^{2}g$, with $r\to 0$, one obtains a family of Ricci flat manifolds whose Gromov-Hausdorff limit is $C(\mathds{R}\mathds{P}(3))=\mathds{R}^{4}/\mathds{Z}_{2}$. This is the simplest example which shows that even under the assumption of Ricci flatness and noncollapsing, Gromov-Hausdorff limit spaces can contain codimension 4 singularities.

(Infinitely many topological types in dimension 4) Let $T^{3}$ denote a flat $3$-torus. According to Anderson [A93], there is a collapsing sequence of manifolds $(M^{4}_{j},d_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}T^{3}$ satisfying

where $b_{2}(M^{4}_{j})$ denotes the second Betti number of $M^{4}_{j}$. In particular, Theorem 1.4 , the finiteness theorem in dimension $4$, does not extend to the case in which the lower volume bound is dropped.