72 problems

Let $(X^n,\omega_0)$ be a compact Kähler manifold with semiample canonical bundle $K_X$. Let $\omega(t)$ solve the normalized Kähler–Ricci flow … Let $f:X\to Y$ be the canonical-mo…

Let $X$ be a smooth Kähler manifold with nef canonical bundle, and let $g_0$ be any initial Kähler metric on $X$. Let $g(t)$ be the solution of the unnormalized Kähler–Ricci flow.…

Let $X^n$ be a compact complex manifold with $K_X$ nef and big, and let $tilde\omega(s)$ be the solution of the Chern-Ricci flow … starting at an arbitrary Hermitian metric…

Let $\lambda_j$ map the isometry class of a closed Riemannian manifold to the $j$-th eigenvalue of its Laplacian. The measured Gromov–Hausdorff topology is the convergence topology…

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…

Tangent-cone product conjecture. These spaces converge in pointed Gromov–Hausdorff topology to a product of copies of $\mathbb{R}$ and $\mathbb{C}$, compact Calabi–Yau varieties, a…

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle, let $\pi:X\rightarrow X_{can}$ be the associated holomorphic Calabi–Yau fibration, and let $S$…

La Nave–Tian's conjecture. As $t\to\infty$,

Let $(M_i,p_i)$ be $n$-dimensional complete pointed Riemannian manifolds with $\operatorname{Ric}_{M_i}\ge -(n-1)$, and suppose that a subsequence converges in the pointed Gromov--…

Let $(M_i,p_i)$ be $n$-dimensional complete pointed Riemannian manifolds with a uniform lower Ricci-curvature bound, and suppose that a subsequence converges in the pointed Gromov-…

Let $\mathcal{M}$ be the family of source spaces and $\mathcal{Y}$ the family of pointed target spaces. Suppose that a net $M_i\in\mathcal{M}$ converges in the measured Gromov–Haus…

Stretching cusps conjecture. The sequence $\{(M,g_i)\}$ Gromov–Hausdorff converges, in the appropriate sense, to a disjoint union of two isometric copies of $N$.

Let $(X_j,d_{X_j})$ and $(Y_j,d_{Y_j})$ be sequences of compact metric spaces such that … Suppose that $F_j:X_j\to Y_j$ are uniformly bounded $K$-Lipschitz functions for some $K>0$…

Let $X$ be a nonsingular algebraic variety with semi-ample canonical line bundle $K_X$, canonical model $X_{can}$ of dimension $\kappa$ satisfying $0<\kappa<\dim X=n$, and let…

Codimension 4 conjecture. The pointed Gromov–Hausdorff limit spaces of non-collapsed Riemannian manifolds with bounded Ricci curvature are smooth away from a closed set of real Hau…

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

Let $X$ be a Calabi–Yau $3$-fold with a K3-fibration $\pi:X\rightarrow Y$, let $\omega_X$ and $\omega_Y$ be Kähler forms on $X$ and $Y$, respectively, and let…

Let $K\in\mathbb{R}$, $n\in\mathbb{N}$, and $v,D>0$. For every $\epsilon>0$, consider Riemannian $n$-tori $M$ satisfying $\operatorname{ric}_M\geq K$,…

Let $M$ be a closed Riemannian manifold as in part (2) of the characterization theorem: it is $\delta$-Gromov-Hausdorff close to a flat $n$-torus $T$ with…

Metric-completion conjecture. $(\hat X_T,\hat d_T)$ and $(X_T,d_T)$ are isomorphic, and they are both homeomorphic to the original variety $Y$.

Let $\mathcal M(n,r,V)$ be the class of $n$-dimensional complete Riemannian manifolds with sectional curvature at most $1$, injectivity radius at least $r>0$, and volume at most…

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 the sequence of warped product spacetimes in Example … with the taxi space portion removed. The conjecture concerns the discrepancy between Gromov–Hausdorff and Sormani-Wenger…

Consider a sequence of pointed semi-flat $ALH_d^*$-gravitational instantons satisfying the assumption of Theorem, with parameters $c_i$ and $\tau_i$. Collapsing limits conjecture.…

Let $M_i$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass tending to zero. Huisken–Ilmanen's conjecture. There is a set $Zi e Mi$…