36 problems

Let $(M^n,g)$ be a steady gradient Ricci soliton with positive Ricci curvature. Suppose there are a ball $B_R(0)\subset\mathbb R^n$ and a diffeomorphism from…

Hamilton–Tian conjecture. For any global solution $\omega(t)$ as above, any sequence $(X,\omega(t))$ along the Kähler–Ricci flow contains a subsequence converging to a $\mathbb{Q}$…

Let $(\mathcal{M}^{n},g(t))$ be a solution of Ricci flow such that $g(0)$ is sufficiently close, in a suitable norm, to a (locally) homogeneous metric on a quotient of a nilpotent…

Let $M$ be a compact Riemannian manifold with a gradient shrinking Ricci soliton metric $g$, so that for some function $f$ and $\tau>0$, … The metric has positive curvature operato…

Let $(M^4,g,f)$ be a Kähler Ricci shrinking soliton orbifold singular at $p_o$, with singularity group in $U(2)$. Desingularization conjecture. There exists a smooth ancient Kähler…

Let $\mathcal{S}$ be one of the spherical space forms appearing in Table 1, and consider asymptotically $\mathcal{S}$-cylindrical steady gradient Ricci solitons on smooth Riemannia…

Classification conjecture. If a compact simply-connected $4$-dimensional Ricci soliton admits an invariant cohomogeneity one action, then it is isometric to a rescaling of one of t…

Non-solvsoliton conjecture. If the one-loop deformation has $c>0$ and dimension $4n>4$, then every hypersurface orbit of its cohomogeneity-one action, with the induced metric, is n…

Ricci-soliton obstruction conjecture. Such $(M_o,g_o)$ cannot be the limit of smooth Ricci solitons in the pointed Gromov--Hausdorff sense.

Bryant-quotient conjecture. There exist immortal Ricci flows approaching any quotient of Bryant's steady soliton by a subgroup of $\operatorname{SU}(2)$, bubbling off any hyperkähl…

A steady gradient Kähler–Ricci soliton is a complete Kähler manifold with a metric satisfying the steady gradient Ricci soliton equation. Cao's conjecture. Every complete steady gr…

Let $\mathfrak{l}$ be an $n$-dimensional Lie algebra, and let $L$ be the connected and simply connected Lie group with Lie algebra $\mathfrak{l}$. The group…

Let $\mathcal{X}$ be a metric-flow limit of Type I blow-ups of a Kähler-Ricci flow, with time slices $(\mathcal{X}_t,d_t)$ as in Theorem 2main1. A quasi-projective normal variety i…

Fang–Man–Zhang conjecture. Any shrinking Ricci soliton has finite topological type.

Let $(M,g,f)$ be a gradient shrinking Ricci soliton with constant scalar curvature. In the terminology recalled in the source, such a soliton is rigid when it is a flat bundle…

Let $\operatorname{Bl}_{p}(\mathbb{C} \times \mathbb{P}^{1})$ be the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at a fixed point $p$ of the standard torus action, and let…

Let $(M,g,f)$ be a complete non Ricci flat steady gradient Ricci soliton with $|\operatorname{Rm}|\to 0$ as $x\to\infty$, where $r$ denotes the distance function on $M$. Munteanu–S…

Uniqueness conjecture. Any such solution is the round sphere, up to diffeomorphism.

A gradient shrinking soliton is a Riemannian manifold $(M,g)$ with a potential function $f$ satisfying … A soliton may be degenerate, and its singular set may have codimension at l…

A gradient shrinking soliton is a Riemannian manifold $(M,g)$ equipped with a function $f\in C^\infty(M)$ satisfying … Such a soliton generates a self-similar Ricci flow and is inv…

Dimension-reduction conjecture. If $(M,g,f)$ is a $4$-dimensional steady gradient Ricci soliton singularity model, then it dimension reduces to $3$-manifolds.

Perelman's conjecture. Then $(M,g(t))$ is the Bryant soliton.

Let the non-collapsed steady solitons be the complete steady solitons described in the paper's main theorem, and let a singularity model mean a blow-up limit arising from Ricci flo…

Let $L_k$ be a line bundle over $(\hat{M},\omega)$, with $k>p(\hat{M},\omega)$, and consider the class of metrics specified in the paper, normalized by $b(0)=1$. A steady soliton i…

Let $\bar{E}>0$, and let $(M,g,f)$ be a four-dimensional shrinking Ricci soliton whose Euler characteristic satisfies $|\chi(M)|\leq \bar{E}$. Write $\mathcal{R}_g$ for the scalar…