44 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…

Mondino's conjecture. $(X,\mathsf{d})$ is homeomorphic to an orbifold, possibly with boundary.

Weighted Riemannian warped-product curvature conjecture. The $N$-warped product of $B$, $F$, and $f$ satisfies $\mathsf{CD}(K(N+d-1),N+d)$ if $(B,f^{N+d}\operatorname{vol}_B)$ is a…

Let $N$ denote the synthetic dimension parameter in the dimension bound for the RCD condition discussed above. Dimension-bound conjecture. The dimension bound $N+1$ can be improved…

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…

Noncollapsed RCD warped-product conjecture. The $N$-warped product of $B$, $F$, and $f$ satisfies $\mathsf{CD}(K(N+d-1),N+d)$ if $B^\dagger$ is a noncollapsed…

Let $X$ be a non-collapsed $\operatorname{RCD}(K,N)$ space without boundary, and let … be its ramified double cover. BBP's conjecture. The space $\hat X$ is also a non-collapsed…

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 $(M^3,g)$.

An $\mathrm{RCD}(-(N-1),N)$ space $(X,d,\mathcal{H}^N)$ is considered in the setting of local bounded covering geometry, where $\mathrm{RCD}$ denotes the Riemannian curvature-dimen…

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…

Let $\Gamma_{\varepsilon}(x)$ be the fibered fundamental group, namely the image of $\pi_1(B_{\varepsilon}(x))\to\pi_1(B_1(x))$ induced by inclusion. RCD fibration conjecture. Ther…

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 $(X,\mathsf{d})$ be a noncollapsed $\operatorname{RCD}(-2,3)$ space whose tangent cones are Euclidean. Smoothability conjecture. The space $(X,\mathsf{d})$ should be bi-Hölder…

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…

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 $\partial X$.

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…

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