18 problems

Carnot tangent characterization. The following are equivalent:

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…

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

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…

De Philippis–Gigli conjecture. If $(X,\mathsf{d},\mathfrak{m})$ is a weakly non-collapsed $\operatorname{RCD}(K,N)$ space, then

Uniform volume-doubling conjecture. There is a constant $D(K)$ such that

The conjecture on uniform continuity and weakly Euclidean points. Under these assumptions,

Cheeger's conjecture.