5 problems

Let $f_0:\mathbb{S}^2\to\mathbb{R}^3$ be a smooth immersion with $W(f_0)\leq 12\pi$. An avoidable-singularity conjecture asserts that there exists such an $f_0$ for which the Willm…

Consider the evolution of an axisymmetric toroidal surface, with the singularity occurring when the toroidal surface closes up at the origin. Let $t$ denote time along the evolutio…

Consider the numerical scheme $\left(\mathcal{P}_{\mathcal{S}}^m\right)$ for generalized Willmore flow, and suppose that the mesh contains long elements at the boundary, specifical…

Let $\mathcal A_\varepsilon$ be the Cahn--Hilliard diffuse-interface energy and let its squared $L^2(\Omega)$-gradient be integrated against the diffuse area measure. De Giorgi's c…

In the numerical experiment, let the initial curve be the union of a circle of radius $0.5$ and two quarter circles of radius $2$, and let $(\mathcal{W}_m)^h$ denote the finite ele…