68 problems

Let $(M,g,f)$ be a compact gradient Ricci soliton with positive curvature operator. A gradient Ricci soliton is a Riemannian manifold whose Ricci tensor satisfies an equation of th…

Let $(M^n,g,f)$, with $n\geq 4$, be a complete $n$-dimensional gradient shrinking Ricci soliton. A gradient shrinking Ricci soliton is rigid if it is isometric to a quotient of an…

A four-dimensional κ-noncollapsed steady gradient Ricci soliton is a complete steady gradient Ricci soliton satisfying the stated curvature conditions: it has nonnegative curvature…

Cao's conjecture. A Ricci shrinker has constant scalar curvature if and only if it is isometric to a finite quotient of

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…

Bryant soliton conjecture. The four-dimensional steady Ricci soliton in that theorem is actually a Bryant soliton.

Hamilton's Type-III asymptotic soliton conjecture. Type-III solutions asymptotically approach soliton metrics that are locally homogeneous.

Generalized Alekseevskii conjecture. Every expanding homogeneous Ricci soliton is isometric to a simply-connected solvmanifold.

Let $M^n$ be a compact $n$-dimensional Kähler manifold with positive first Chern class $C_1(M)$, and consider the normalized Ricci flow … A gradient Kähler–Ricci soliton is a solut…

Noncompact Type I limit conjecture. Any noncompact Type I limit is also a gradient shrinking soliton.

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…

Pinching-off decay conjecture. Either the Koiso metric or the Page metric decays to $\mathbb{CP}^2$ via a $\mathbb{CP}^1$ pinching off.

Let $(M^4,g(t))$ be a simply-connected $4$-dimensional $\kappa$-solution of Ricci flow. A shrinking gradient soliton is a round sphere $S^4$ or one of the cylinders…

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 $(M^n,g,f)$ be a non-collapsed steady gradient Ricci soliton with non-negative curvature operator. Suppose that its tangent flow at infinity is cylindrical, of the form … Here,…

Let $(M^n,g,f)$ be an $n$-dimensional steady gradient Ricci soliton with positive curvature operator. Here, $O(2)$-symmetry means that the soliton admits an isometric $O(2)$ action…

The Ricci flow under consideration is $U(2)$-invariant on $\mathcal{O}(-1)$, with the zero section as its bolt, and develops a finite-time singularity when the bolt shrinks to zero…

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…

For integers $m,k\geq 1$, let $\mathcal{A}_{m,k}$ be the class of solutions $(f,a,b):[0,\infty)\to\mathbb{R}^3$ defining Appleton's steady gradient Ricci solitons on the total spac…

Asymptotic dichotomy conjecture. Any noncompact four-dimensional shrinker must fit one of these two behaviors, at least asymptotically.

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…

Let $G_{\mathrm{FS}}$ be the Fubini–Study Einstein metric, and consider unstable perturbations that are conformal but not Kähler. Let $\mathcal L^2_{-1}$ be the Kähler blowdown sol…