17 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…

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 $(M,g_t)_{t\in(-\infty,0]}$ 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

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…

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…

Perelman's compact ancient solution conjecture. Then $g(t)$ is either a family of shrinking spheres or Perelman's ancient solution.

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

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

Let $(S^3,g(t))$ be a compact, ancient $\kappa$-noncollapsed solution to the Ricci flow on $S^3$, where $\kappa$-noncollapsed means that there is some $\kappa>0$ such that every me…

Let $(S^3,g(t))$ be a compact ancient $\kappa$-noncollapsed solution to the three-dimensional Ricci flow … existing for $t\in(-\infty,T)$ and shrinking to a round point at $T$. Per…