83 problems

Let $n\ge 4$. An ancient Ricci flow is a Ricci flow defined for all sufficiently negative times. Consider the solution on $S^n$ constructed in the paper's main existence theorem, a…

Let $(M^n,g)$ be a Riemannian manifold with $n\geq 2$, and write $g=(g_{ij})$ in local coordinates $(x^1,\ldots,x^n)$. Let $t$ denote time, and let $\xi^1,\ldots,\xi^n$ be arbitrar…

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…

Let $(M,g(t))$ be a solution to the Ricci flow on a compact $3$-manifold with arbitrary initial metric. Hamilton's Harnack-estimate conjecture. A Harnack-type estimate holds for th…

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 an irreducible compact Kähler manifold with positive orthogonal bisectional curvature. Projective-space conjecture. Is $M$ necessarily biholomorphic to…

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…

A complete noncompact ancient solution to the Ricci flow is a Ricci flow $(\mathcal{M}^{n},g(t))$ defined for $-\infty<t<\omega$ on a complete noncompact manifold. It is Type-I lik…

Let $(M,g(t))_{t\in[0,T)}$ be a solution to Ricci flow on a closed smooth $n$-dimensional Riemannian manifold, with $T$ its maximal time. Hamilton's scalar-curvature blow-up conjec…

Let $(M,g_t)_{t\in(-\infty,0]}$ be a noncollapsed ancient Ricci flow with bounded curvature within each compact time interval. Suppose … on $M\times(-\infty,0]$, and suppose every…

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 $(Z,d_Z,\mathfrak{t})$ be a noncollapsed Ricci flow limit space arising as the pointed Gromov--Hausdorff limit of a sequence in $\mathcal{M}(n,Y,T)$. Let…

Let $M$ be a compact manifold and $g(t)$, $t\in [0,T)$ be a smooth Ricci flow on $M$, which may become singular at $t=T$. A Ricci flow is Type I if, for some constant $C_I<+\infty$…

Let $C_s,v_0>0$ and $n\geq 3$. Let $(M^n,g_0)$ be a complete non-compact Riemannian manifold. Assume that … … … and $\mathcal{R}(g_0)\geq 0$. Positive scalar-curvature rigidity con…

Let $k\geq3$. Consider Ricci flow on $S^1\times S^3/\mathbb{Z}_k$, and call a singularity a quotient-neck singularity when its tangent flow is $\mathbb{R}\times S^3/\mathbb{Z}_k$.…

Consider an ancient asymptotically cylindrical 4-dimensional Ricci flow, meaning a Ricci flow existing for all sufficiently negative times and asymptotic at negative infinity to a…

Let $T=(V,E)$ be a tree with an initial metric $w\in\mathbb{R}_{>0}^{E}$, and let the metric $w(t)$ evolve under the normalized Ricci flow. The flow converges to a metric with cons…

Dynamical Alekseevskii conjecture. If a homogeneous space has an immortal Ricci flow, then its universal cover is diffeomorphic to $\mathbb{R}^n$.

Equivalence conjecture for infinite ideal polyhedra. The following statements are equivalent, with the Euclidean alternatives applying to IIP and the hyperbolic alternatives applyi…

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…

Böhm–Wilking's conjecture. For every $a\in[0,n/4]$, the curvature cone $\Omega(a)$ is invariant under the Hamilton ODE and preserved by Ricci flow.

Stable ALE desingularization conjecture. There exists an ancient, respectively immortal, renormalized Ricci flow on $M_o\#_{\mathbb{S}^3/\Gamma}N$ with limit $(M_o,g_o)$ at…

Higher-dimensional Calabi-bubble conjecture. In any dimension $2n\geq 6$, there exist:

Uniqueness conjecture. Up to time reparametrization and diffeomorphism, the ancient solutions to Ricci flow from Theorem $…

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…