22 problems

Perelman's ancient oval conjecture. Every compact ancient $\kappa$-solution must either be a quotient of a family of shrinking round spheres or be isometric to a so-called ancient…

Let an ancient oval be an ancient compact noncollapsed mean curvature flow that is not self-similar, and let $1\leq k\leq n$. An ancient oval is…

A complete ancient solution to the Ricci flow on a surface is a solution defined on a time interval $(-\infty,\omega)$, with bounded curvature and satisfying the Type II condition…

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 $(M^4,g(t))$ 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 $(M,g_t)_{t\in(-\infty,0]}$ be a noncollapsed ancient Ricci flow with bounded curvature within each compact time interval. Suppose … on $M\times(-\infty,0]$, and suppose every…

Let $M_t\subset\mathbb{R}^{n+1}$ be an ancient oval whose tangent flow at $-\infty$ is $\mathbb{R}^{k}\times S^{n-k}$. A $k$-oval is an ancient oval with this tangent flow whose hi…

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

Let $G$ be the curvature function defining the flow, and consider convex ancient $G$-flows that are noncollapsing, uniformly two-convex, and compact. Uniqueness conjecture. Up to p…

Standard-flow simplex conjecture. The invariant set

Simplex conjecture. The space of all convex ancient solutions with a point symmetry in ${\mathbb R}^{n+1}$ should be homeomorphic to an $(n-1)$-dimensional simplex.

Let $n\leq 6$ and let $\Sigma\subset\mathbb{R}^{n+1}$ be a smoothly embedded self-shrinker whose ends satisfy the nice ends hypothesis: each end is either smoothly asymptotic with…

Let $(M,g)$ be a complete, $n$-dimensional Riemannian manifold with bounded geometry. An ancient, positive solution is a positive solution $u$ of … that is defined for all past tim…

Let an ancient oval be an ancient compact noncollapsed mean curvature flow that is not self-similar, and let $1\leq k\leq n$. An ancient oval is…

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

Let $(M^n,g(t))_{t \in (-\infty,0]}$ be a complete ancient solution to the Ricci flow with weakly PIC, meaning the stated curvature condition from the source. Weakly PIC2 conjectur…

Let $g(t)$ be a compact ancient $\kappa$-solution on $S^3$ to the Ricci flow in dimension $3$. Compact ancient-solution conjecture. Then $g(t)$ is either a family of contracting sp…

Existence conjecture. There exists an $n$-dimensional, connected, full ancient solution to the mean curvature flow in $\mathbb{R}^N$ with finite entropy.

A mild bounded ancient solution of the Navier–Stokes equations on $\mathbb{R}^3$ is a solution defined for all backward times, bounded and smooth, satisfying the integral formulati…

Let $Q_-^+=\mathbb R^3_+\times(-\infty,0)$, where $\mathbb R^3_+=\{x=(x_1,x_2,x_3)\in\mathbb R^3:x_3>0\}$. A mild bounded ancient solution is a bounded ancient solution of the Navi…

Let $u$ be a mild bounded ancient solution of the Navier–Stokes equations in the half space $\mathbb R^3_+$, meaning a bounded ancient solution satisfying the pressure decompositio…

Conjecture (L). The velocity field of any bounded mild ancient solution of the Navier–Stokes equations is constant.