45 problems

Boundary curvature-integral conjecture. One has

Alexandrov empty-boundary conjecture. Every Alexandrov space with empty topological boundary has vanishing metric measure boundary.

Let $\Omega$ be a bounded open domain in an $n$-dimensional Alexandrov space with curvature bounded from below by $k$, and let $N$ be a compact smooth Riemannian manifold. Suppose…

Naber's measure-convergence conjecture. As measures,

Let … be a non-collapsed Gromov–Hausdorff-convergent sequence of compact Alexandrov spaces of dimension $n$ with curvature bounded below by $-1$. Perelman's bi-Lipschitz stability…

Let $Y^3_\infty$ be a $3$-dimensional compact, simply connected, non-negatively curved Alexandrov space without boundary, and suppose that $Y^3_\infty$ is a topological manifold. S…

Synthetic curvature-measure conjecture. There exists a locally finite curvature measure

Converse curvature conjecture. The distributional sectional curvature of $(M,g)$ is bounded from below (respectively, above) by $k$.

Finiteness conjecture. The class of all such submetries contains only finitely many foliated homeomorphism types.

Let $X$ be a closed, simply-connected Alexandrov $3$-space with positive curvature. Write $\mathrm{Susp}(P^2)$ for the suspension of the projective plane $P^2$. Mitsuishi–Yamaguchi…

Let $X_i\in\text{Alex}^n(-1)$ be a sequence of Alexandrov spaces converging to $X$ in the Gromov–Hausdorff sense without collapsing. Let $\mu_{X_i}$ and $\mu_X$ denote their synthe…

Let $X\in\text{Alex}^n(-1)$, and let … be the synthetic curvature measure, where ${\operatorname R}$, $\mathcal{K}$, and $\theta$ are the corresponding density functions. Local fin…

Let $X\in\text{Alex}^n(-1)$. For $0\le k\le n-1$, define the $k$-singularity function by setting $\theta_k(x)=2\pi$ for $x\in\mathcal{S}^{k-1}(X)$ and…

Let $\text{Alex}^n(-1)$ denote the $n$-dimensional Alexandrov spaces with curvature at least $-1$. For $0\le k\le n-1$, let $\mathcal{S}^k_\epsilon(X)$ be the set of points whose t…

Alesker's conjecture. Under these hypotheses, the asserted existence and, in the collapsing case, the stated stratified expression for the limit of each intrinsic volume $V_i(M_j)$…

Harmonic maps are considered from metric spaces with sectional curvature bounded from below to metric spaces with non-positive curvature. Local Lipschitz regularity conjecture. Suc…

Torus collapse conjecture. $X$ is homeomorphic to an $ell$-dimensional torus.

Let $Y$ be an $n$-dimensional length metric space. For a subset $\mathcal{U}\subseteq Y$, set $\mathcal{A}=Y\setminus\mathcal{U}$, and write…

Let $X\in\operatorname{Alex}_{\amalg}^n(\kappa)$ satisfy the paper's condition, let $Y$ be a connected length metric space, and let $f\colon X\to Y$ be a 1-Lipschitz onto map. Glui…

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, let $(X,p)\in\operatorname{Alex}^n(-1)$, and set … Here $\mathcal{S}^k_{\epsilon_i}$ denotes the $\epsilon_i$-singular strat…

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, and let $(X,p)\in\operatorname{Alex}^n(-1)$, where $\mathcal{S}^k_\epsilon$ denotes the $\epsilon$-singular stratum. Hausdor…

An Alexandrov $3$-space is a three-dimensional Alexandrov space; it is aspherical when all its higher homotopy groups vanish, equivalently when its universal cover is contractible.…

Let $M$ be an $n$-dimensional compact Alexandrov space with nonnegative curvature, and let $\tau_n$ denote the number of extremal subsets of $M$. Extremal-subset counting conjectur…

Let $T^1$ act isometrically and effectively on $X^4$, where $X^4$ is a $4$-dimensional, closed, positively curved, orientable Alexandrov space. Classification conjecture. Up to equ…

Let $X$ be a positively curved Riemannian orbifold of dimension $4$ with an isometric circle action. If the action has only isolated fixed points, then its underlying topological s…