12 problems

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…

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 $V_\lambda^\ast>V_\lambda^0$…

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.

A balanced figure-eight is a figure-eight curve whose lobes bound regions of equal area, and the curve-shortening flow is the evolution of a plane curve by curvature. Grayson's con…

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…

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 $\delta$ be an initial curve, and let $\gamma_0$ be the shortest curve homotopic to it among curves avoiding the obstacle set $P_n=(\mathbb Z/n)^2$. For $t>0$, write…