46 problems

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 $X^n$ 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 $(X^n,\omega_0)$ 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 $(X_j,d_{X_j})$ 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 $K>0$…

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 … 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…

Song–Tian's Gromov–Hausdorff conjecture.

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 $\mathcal M(n,r,V)$ 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 $(M_i,g_i)$ be a sequence of $n$-dimensional manifolds with lower Ricci or sectional curvature bounds, and suppose that $M_i$ Gromov–Hausdorff converge without collapsing to…

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 $Zi e Mi$…

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 $M^m$ be a compact Kähler manifold with semiample canonical bundle and Kodaira dimension $n=\kappa(M)$ satisfying $0<n<m$. Let $f:M\to N$ be its Iitaka fibration onto a normal…

Let $M^m$ be a projective Calabi–Yau manifold with $K_M\cong\mathcal{O}_M$, equipped with a fiber space structure $f:M\to N$, where $N^n$ is a normal projective variety and $0<n<m$…

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle $K_X$, and let $g(t)$ be the long-time solution of the normalized Kähler–Ricci flow. For a seque…

Essential-skeleton metric limit conjecture. As $t\to0$, $(X_t,\widetilde{\omega}_t)$ converges in the Gromov–Hausdorff topology to a compact metric space $(Z,d)$ homeomorphic to th…

Metric SYZ conjecture. As $t\to0$, $(X_t,\widetilde{\omega}_t)$ converges in the Gromov–Hausdorff topology to a compact metric space $(Z,d)$ with an open dense subset $Z_0$ such th…

Let $X$ be a $K3$ surface and let $[\alpha]\in\partial\mathcal{C}_X$ be as in Setup I.B.2. Point-limit conjecture. The Ricci-flat manifolds $(X,\omega_t)$ converge to a point in th…

Compactification conjecture. The following hold:

In Setup I.B.1, let $f:X\to Y$ be the fiber space and let $S$ be the singular-fiber locus. Smooth convergence conjecture. The Ricci-flat Kähler metrics satisfy … in…

In Setup I.A, let $(X,\omega_t)$ be the corresponding family of Ricci-flat manifolds and let $Z$ denote its Gromov–Hausdorff limit. Gromov–Hausdorff limit conjecture. The limit $Z$…