14 problems

Metric cone conjecture. If the Kähler–Ricci shrinker has maximal volume growth, then $(Z_0,d^Z_0)$ is a metric cone and coincides with the metric completion of…

Polarized affine cone conjecture. The space $Z_0$, with the topology induced by $d^Z_0$, admits a polarized affine cone structure, and the map $\Phi$ realizes the polarized Fano fi…

Sun–Zhang's conjecture. Every Kähler–Ricci shrinker has a unique tangent space at infinity, and this limit is a metric cone.

Let $(X,d_X)$ be the singular Kähler–Ricci shrinker arising as the metric soliton modeled on the $\mathbb{F}$-limit described in the paper, with regular part…

Scalar-curvature conjecture. Proposition 1 should remain valid without the assumption

A gradient Ricci shrinker is a complete connected Riemannian manifold $(M,g,f)$ satisfying the gradient shrinking Ricci soliton equation. In dimension four, its scalar curvature is…

Let $\mathcal{O}_d$ denote the space of holomorphic functions on the gradient Kähler Ricci shrinker under consideration whose growth rate is at most $d$. Munteanu–Wang's conjecture…

Type-I ancient-solution conjecture. $(M^n,g(t))$ is isometric to a Ricci shrinker, up to scaling.

Let $(M^n,g,f)$ be a Ricci shrinker with weakly PIC. A Ricci shrinker with weakly PIC is expected to satisfy the following classification: Classification conjecture. $(M,g)$ is iso…

Let $n\geq 4$ be a positive integer, and let $(M,p,g,f)$ be a Ricci shrinker. Denote its scalar curvature by $R$, and write $\sup_{x\in M}R(x)$ for its supremum. Uniform scalar-cur…

Let $n\geq 4$ and $n\geq k\geq 3$ be integers. Let $(M,p,g,f)$ be a Ricci shrinker, let $\Gamma\subset O(k)$ be a finite subgroup acting freely on $S^k$, and let $g_c$ denote the c…

Let $n\geq 4$ be a positive integer. A Ricci shrinker is a tuple $(M,p,g,f)$ as in the source, and let $\Gamma\subset O(n)$ be a finite subgroup acting freely on $S^{n-1}$. Write…

Let $k$ be the extremal Kähler metric on $\mathbb{C} P^2\sharp 2\overline{\mathbb{C} P^2}$ such that $s^{-2}k$ is the Chen–LeBrun–Weber metric with Einstein constant equal to $3$,…

Let $\mathbb{C} P^2\sharp 2\overline{\mathbb{C} P^2}$ carry the Chen–LeBrun–Weber metric, viewed as a Ricci shrinker. Instability conjecture. The manifold … with the Chen–LeBrun–We…