33 problems

Boundary curvature-integral conjecture. One has

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

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 $X_i\in\text{Alex}^n(-1)$ 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 $(M_i,g_i)$ be a sequence of $n$-dimensional manifolds with lower Ricci or sectional curvature bounds, and suppose that $M_i$ Gromov–Hausdorff converge without collapsing to…

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…

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…

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…

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 $T^1$ 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…

Comparison obtuse constant conjecture. The conclusions of the relevant compact results would remain valid with $\widetilde{\rm ob}(M)$ in place of ${\rm ob}(M)$ when $M$ is compact…

Double smoothness conjecture. If $M$ is compact and ${\rm ob}(M)=\pi/2$, then $D(M)$ has no singular points. Similarly, if $M$ is noncompact and ${\rm ob}_{\infty}(M)=\pi/2$, then…

Fix a finite field $\mathbb{F}$ and let $\mathcal{L}_{\mathbb{F}}(n,D,\kappa)$ consist of pairs $(X,[\mathcal{F}])$, where $X$ is an Alexandrov space of Hausdorff dimension at most…

Fix a finite field $\mathbb{F}$. Let $X$ be a compact Alexandrov space, and let $D^b_c(\operatorname{Sh}_{\mathbb{F}}(X))$ denote the bounded derived category of constructible shea…

Let $X$ be a compact weakly smoothable Alexandrov space, meaning that it is a Gromov–Hausdorff limit of closed smooth connected Riemannian manifolds with uniformly bounded above di…

Let $\{M_l^n\}_{l=1}^{\infty}$ be smooth $n$-dimensional closed connected Riemannian manifolds with a uniform lower bound on sectional curvature, converging in the Gromov–Hausdorff…

Positive synthetic Ricci curvature conjecture. Under these assumptions,

Let $(X,\mu)$ be a weighted compact finite-dimensional Alexandrov space of $\operatorname{CD}(0,\infty)$, and let $k$ be a natural number. Eigenvalue ratio conjecture. There exists…

Let $D>0$. Let $A$ be a bounded subset with $\operatorname{diam}(A)\leq D$ of an $n$-dimensional complete Alexandrov space $X^n$ with curvature $K\geq -1$. Alexandrov-space embeddi…

Funano and Shioya's conjecture. For every natural number $k$, there exists a positive constant $C_k$ depending only on $k$ such that