11 problems

Let $\phi$ be a smooth, strictly convex norm on $\mathbf{R}^{n+1}$, with Wulff shape $\mathcal{W}^\phi=\{x\in\mathbf{R}^{n+1}:\phi^*(x)=1\}$. Let $X_0:M\to\mathbf{R}^{n+1}$ be a cl…

Let $M_t\subset\mathbb R^{n+1}$ be a smooth, closed, uniformly convex hypersurface evolving by the curvature flow whose speed is the $\alpha$-power of the Gauss–Kronecker curvature…

Let $M_0$ be a starshaped and $k$-convex initial hypersurface in the sphere, and consider the locally constrained curvature flows … and … Here $u$, $\phi'$, $\nu$, and $H_j$ denote…

Let $m\geq0$ and $N_0\geq2$, where $N_0$ is the rotation number of the initial immersed curve. Consider the area-preserving curvature flow of order $m$. Finite-time blowup conjectu…

Let $m\geq1$. Consider the area-preserving curvature flow of order $m$ for embedded initial curves with length $L_0$ and enclosed area $1$. Immortal-solution conjecture. For every…

Let $\Omega$ be a domain with star-shaped hypersurface boundary, and let $\widetilde W_j(\Omega)$ denote the modified quermassintegrals and $h_{\alpha,j}$ the functions appearing i…

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…

Translating surface-patch convergence conjecture. A general class of initial data will converge to this translating grim reaper shape for large times.

Grim reaper convergence conjecture. The profile of the curve approaches this grim reaper solution in the limit.

Let $c0_1$ and $c0_2$ be the flow parameters appearing in the curvature flow model. A conjecture on singular profiles. For some values of $c0_1$ and $c0_2$, embeddedness is not pre…

A vanishing function is a function whose constant terms $C_v(a,b)$ in maximum-principle-function condition (IV) vanish for all $0<a,b$. A maximum-principle function (MPF) is a func…