42 problems

The curvature-dimension condition $\operatorname{CD}(K,N)$ concerns metric measure spaces with a lower Ricci curvature bound $K\in\mathbb{R}$ and an upper dimension bound…

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

Measure contraction conjecture. Carnot–Carathéodory spaces do not possess any type of measure contraction property.

Shioya–Takatsu's conjecture. Any weak limit of compact homogeneous Riemannian manifolds is not mm-isomorphic, that is, not isomorphic as a metric measure space, to the abstract Wie…

Let $N\in(1,\infty)$ and let $X$ be a proper, complete, geodesic metric space with a Radon measure $\mathfrak{m}$. Consider the Minkowski $N$-cone $[0,\infty)\times_r^N X$. Minkows…

Generalized-cone characterization conjecture. The space $Y$ satisfies $\textnormal{\textsf{TCD}}(-\kappa N,N+1)$ if and only if $f”-\kappa f\leq0$ in the distributional sense in…

Let $(X,\omega_X)$ be an $n$-dimensional compact normal Kähler space with smooth Kähler metric $\omega_X$, and let $\omega=\omega_X+\sqrt{-1}\partial\bar\partial u$ be a singular K…

A sub-Finsler Carnot group is a stratified, nilpotent Lie group equipped with a left-invariant distance. Its homogeneous dimension is denoted by $\mathcal{Q}$, and…

Let $X$ be an $RCD(K,N)$ space, and let $\Omega$ be a geodesically convex open subset of $X$ with $\operatorname{m}(\partial\Omega)=0$. Let $\hat{\operatorname{m}}$ be the measure…

Let $X$ be a noncollapsed $RCD(K,n)$ space with nonempty boundary $\partial X$. Doubling conjecture. The doubled space $\operatorname{Dbl}(X)$ satisfies the condition $RCD(K,n)$. T…

Let $X_0$ and $X_1$ be noncollapsed $RCD(K,n)$ spaces with nonempty boundaries $\partial X_0$ and $\partial X_1$, where the boundaries are understood in the sense of Gigli and Pasq…

Let $({\rm X},{\sf d},\mathcal{H}^N)$ be a non-collapsed $\mathrm{RCD}(K,N)$ space, and let $x\in\partial {\rm X}$. Here $\partial {\rm X}$ denotes the boundary and $B_r(x)$ the me…

Let $X$ be a non-collapsed RCD space, and let $W^{1,2}_C$ and $H^{1,2}_C$ denote the two Sobolev spaces of vector fields considered in the paper. Boundary characterization conjectu…

RCD characterization conjecture. The metric-measure space $(X,d,\mu)$ is $RCD(K,\infty)$ if and only if the five gradients inequality holds.

Boundary-equality conjecture. The two boundary notions agree:

De Philippis–Gigli's conjecture. The interior $X\setminus\partial X$ is strongly convex: every geodesic joining two points of $X\setminus\partial X$ avoids $\partial X$.

Let $u\colon\mathbb{R}^n\to\mathbb{R}^m$ be a $1$-Lipschitz map with respect to the Euclidean norms, and let $(\mathbb{R}^n,\lVert\cdot\rVert,\mu)$ be a weighted Riemannian manifol…

Let $(X,\mathsf{d},o)$ be the pointed Gromov–Hausdorff limit of a sequence of closed Riemannian manifolds $\{(M_\alpha^n,\mathsf{d}_{g_\alpha},o_\alpha)\}_\alpha$ satisfying the un…

Weak convergence conjecture. Under this assumption, there exists a unique $L^{\infty}$-tensor $T$ such that, as $t\to0^+$,

RCD gluing conjecture. The glued metric measure space

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space. For $f\in D(\Delta)$, let $\mathrm{Hess}_f$ be its Hessian, viewed as a $(0,2)$-type $L^2$ tensor, and let…

RCD normalization conjecture. Then $(X,\mathsf{d},\mathcal{H}^k)$ is a non-collapsed $\operatorname{RCD}(K,k)$ space. This is the RCD formulation of the paper's first conjecture an…

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space, and let $\dim(X,\mathsf{d},\mathfrak{m})$ be its essential dimension. De Philippis--Gigli conjecture. If ……

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…