(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.