26 problems

Grayson's conjecture. There exists an example of curve shortening flow whose limit geodesics are not unique.

Grid-peeling convergence conjecture. As the grid is refined, the $m$-th convex layer of a convex curve converges to the ACSF after time $T$ when

Sharp constant conjecture. The sharp value of the universal constant $C_1$ in the Colding–Minicozzi codimension bound is

Let $\Omega\subset\mathbb R^2$ be a bounded, strictly convex domain, and let $\mathbb T_t$ be the flow by curvature of an embedded triod in $\Omega$. Assume that the lengths of the…

Let $\mathbb T^\infty_{\mathfrak t}$ be an eternal evolution by curvature of an unbounded embedded triod, curve, or curve with a single endpoint. Translation conjecture. The triods…

Let $\mathbb T^\infty_{\mathfrak t}$ be an eternal evolution by curvature of an unbounded embedded triod, curve, or curve with a single endpoint, and let its curvature be defined a…

Let $\mathbb T_t$ be a smooth evolution of an embedded triod with fixed endpoints $P^1,P^2,P^3$ in a domain $\Omega\subset\mathbb R^2$. Denote by $L^i$ the lengths of its three cur…

For a closed curve evolving by the curve shortening flow, a homothetic curve is a solution that changes only by scaling and reparametrization. Abresch–Langer's saddle-point conject…

Complete classification conjecture. Up to similarity, the area-preserving curve-shortening-flow shrinkers are precisely $\mathbb{S}^1$ and, for each coprime $m,n\in\mathbb{N}$ sati…

J.-E. Chang's conjectures. If $\lambda\geq 0$, then $E\mapsto\Theta_\lambda(E)$ is strictly monotonically decreasing. If $\lambda<0$, there is a value $V_\lambda^\ast>V_\lambda^0$…

Let GAPF denote Gage's area-preserving flow for plane curves. A smooth, embedded initial curve is star-shaped if it is star-shaped with respect to the relevant center. Global exist…

Consider a symmetric figure-eight curve evolving by the curve shortening flow, and normalize its curvature as the curve shrinks to a point. A doubly covered segment is a segment tr…

Let $X_0$ be a smooth initial datum with added “hairs”, meaning added degenerate loops. Under the curve shortening flow, the curve may contain doubly covered segments or degenerate…

A curve $X_0$ is a curve of finite total curvature, and its evolution under the curve shortening flow may have zero extinction time $t_e(X_0)=0$. A doubly or multiply covered segme…

Uniqueness conjecture. CSF is unique on $(\mathbb{R}^2,g)$ if and only if $g$ does not allow blooming at infinity.

Uniqueness conjecture. CSF is unique on the flat plane.

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow, and let $\kappa$ denote its signed curvature. Write $n_\kappa(t)$ for the number of critical points…

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow. Denote by $n_U(t)$ the number of critical points of its radial function relative to the fixed point…

Homotopic curve shortening (HCS) is a discrete curve-shortening process defined for curves in the plane with obstacles, in which each iteration replaces suitable subpaths by shorte…

Let $\delta$ be an initial curve contained in a convex region $R$ of area $A$. For $t>0$, write $\delta'=\delta(t)$ for its affine curve-shortening flow whenever it is defined. For…

Let $\gamma_0:[0,1]\to\mathbb{R}^2$ be the particular initial curve constructed in Example Two, with $\sigma_0$ the connection of $\gamma_0(1)$ to $\gamma_0(0)$ along $\Sigma$, and…

No higher-multiplicity static-line tangent-flow conjecture. At a singular time of the planar network flow, no tangent flow that is a static line of higher multiplicity can develop.

Let $\mathcal D$ be a simply connected, sufficiently smooth domain in $\mathbb R^2$. Start the zero-temperature heat-bath dynamics for the nearest-neighbor Ising model on…

Let $\mathcal D\subset[-1,1]^2$ be a compact, simply connected set with smooth closed boundary. For $L\in\mathbb N$, let $\mathcal A_L(t)$ be the union of unit squares centered at…

The curve-shortening flow is the evolution of a plane curve by its curvature; in Cartesian graph coordinates $y(x,t)$ it is governed by … The paper studies Lie symmetries and separ…