17 problems

Finite local fundamental group conjecture. There exist constants $t=t(n,v)>0$ and $C=C(n,v)>0$ such that, for every $(Z,d,p)\in\mathcal{P}(n,v)$,

Finite local fundamental group conjecture. There exist constants $t=t(n,v)>0$ and $C=C(n,v)>0$ such that, for every $(X,d,p)\in\mathcal{M}(n,v)$,

Let $(M^n,g,p)$ be a pointed Riemannian manifold with $\operatorname{Ric}_M\geq-\lambda$ and $\operatorname{Vol}(B_1(p))\geq v$. Naber's conjecture. There is a constant…

Let … be a non-collapsed sequence of closed Riemannian manifolds with $\mathrm{Ric}\ge -(n-1)$, converging in the Gromov–Hausdorff sense to a compact Ricci limit space $X^n$. Let…

Let $(X^4,\mathsf{d})$ be a noncollapsed Ricci limit space, and consider the set of points whose tangent-cone cross-sections are not homeomorphic to $S^3$. Discrete singularity con…

Let $(X^n,\mathsf{d})$ be a noncollapsed Ricci limit space. For $k$-symmetry, require a tangent cone to split off an isometric factor $\mathbb{R}^k$ (as in the source). Naber's con…

Let $(M_i,p_i)$ be a sequence of simply connected complete $n$-manifolds with Ricci curvature satisfying $\mathrm{Ric} \geq -(n-1)$, and let $G_i$ be discrete groups acting isometr…

Let $(X,\mathsf{d})$ be the metric limit under consideration, let $k=\dim_{\mathcal{H}}(X,\mathsf{d})$, and let $\mathcal{H}^k$ denote its $k$-dimensional Hausdorff measure. For…

Let $(X,\mathsf{d},x,\mathfrak{m})$ be a collapsed Ricci limit space of Hausdorff dimension $k$, with its limit measure satisfying … for some $b,k\in(0,\infty)$. Collapsed Ricci li…

Let $X$ be a noncollapsed Ricci limit space of dimension $n$, with singular set ${{\cal S}}$. For each $k\in\{0,\ldots,n-1\}$, let ${{\cal S}}_k$ be the subset of ${{\cal S}}$ cons…

Let $(M_i,g_i,y_i)$ be a sequence of pointed Riemannian manifolds whose pointed Gromov–Hausdorff limit is a metric space $(X,d,y_0)$. A subset of $X$ has codimension 4 if its compl…

Let $\overline{M(n,K,d)}$ denote the closure of the moduli space of compact Riemannian manifolds with dimension $n$, Ricci curvature bounded below by $K$, and diameter bounded abov…

Homeomorphism-type conjecture.

Let $M_\infty$ be a limit space of Riemannian manifolds with a uniform lower Ricci curvature bound. Lie-group conjecture. The isometry group of any limit space $M_\infty$ is a Lie…

Let $M_\infty$ be a limit space of Riemannian manifolds with a uniform lower Ricci curvature bound, and interpret its dimension as the dimension of its regular set or tangent cones…

The cut-off covering spectrum conjecture. Theorem 7, namely the corresponding cut-off covering spectrum conclusion established earlier in the paper, holds for such limit spaces $X$…

The flat normal bundle conjecture. If $X$ is a limit space as above, then either $X$ has the loops to infinity property or it has a split double cover and it is the flat normal bun…