27 problems

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space, and let $({\mathcal R}_{N})_{\varepsilon}$ denote its $\varepsilon$-regular set. Then the interior…

Let $X$ be an $n$-dimensional projective variety with log terminal singularities, and let $\theta_X$ be a smooth Kähler metric on $X$. Let $(X,\omega)\in\mathcal{RK}(n)$, so that…

RCD versus smooth curvature conjecture. Such an $n$ and $M^n$ exist.

RCD no-$\mathbb{RP}^2$ cone conjecture. Then $Y\approx S^2$ or $Y\approx \overline{D}^2$. If the $X_i$ have empty boundaries, only the first possibility can occur.

RCD stability conjecture. There exists $i_0\in\mathbb{N}$ such that $X_i$ is homeomorphic to $X$ for every $i\ge i_0$.

Brue–Pigati–Semola conjecture. $(X,\mathsf{d})$ is locally biHölder homeomorphic to a smooth, complete Riemannian manifold $(M^3,g)$.

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 $X$ be an $n$-dimensional normal compact Kähler space, and let $\omega_{KE}$ be a singular Kähler–Einstein metric on $X$ satisfying … on the regular set $X^{reg}$, with induced…

For $K\rrbracketin\mathbb{R}$ and $N\in\mathbb{N}$, let an $\operatorname{RCD}(K,N+\varepsilon)$ space mean a metric-measure space satisfying the RCD condition with parameters…

Let $(X_i,d_i,\mathfrak{m}_i)$ be a sequence of $RCD^{\ast}(K,N;D)$ spaces, and let $(\widetilde{X}_i,\widetilde{d}_i,\widetilde{\mathfrak{m}}_i)$ be their universal covers. Assume…

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(K,N;D)$ space. If its fundamental group $\pi_1(X)$ is finite, then Fukaya–Yamaguchi conjecture. It contains an abelian subgroup $A$ of in…

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(0,N)$ space, where $RCD^{\ast}(0,N)$ denotes the relevant synthetic non-negative Ricci-curvature condition. Milnor's conjecture. The fund…

Quantified asymptotic Euclideanity conjecture. There exists $\delta(N,\epsilon)>0$ such that, for every $\delta\in(0,\delta(N,\epsilon)]$, there exists $R_0\geq 1$ for which…

RCD gluing conjecture. The glued metric measure space

Let $(X,\mathsf{d},\mathfrak{m})$ be the RCD space under discussion, let $k=\dim(X,\mathsf{d},\mathfrak{m})$, and let $\mathcal{R}_k^*$ be the full-measure regular set. Let $\phi$…

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 $(X,\mathsf{d},x,\mathfrak{m})$ 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 $K\in\mathbb{R}$, $N\in\mathbb{N}$, and $v>0$. Let $\{(X_i,\mathsf d_i,\mathcal{H}^{N}_{\mathsf d_i})\}_{i\in\mathbb{N}}$ be a sequence of non-collapsed $\mathsf{RCD}(K,N)$ spa…

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space. Let $\boldsymbol\partial^{*}X$ denote its reduced boundary. Boundary rectifiability conjecture. Th…

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space. Denote by ${\mathcal R}_{N}$ its $N$-regular set. Codimension-three conjecture. There exists an op…

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