8 problems

Conformal-class Willmore-minimum conjecture. For all conformal classes,

Finite-genus characterization conjecture. These conditions characterize the elements of $\overline{\mathcal{M}}_{\Lambda,\eta,\mathscr{W}}$ for all $\mathscr{W}>0$.

Conformal-class minimum conjecture. The lower-bound function describes the minimum of the restriction of the Willmore functional to conformal immersions of two-dimensional tori…

Let ${\cal W}$ be the Willmore functional on tori, and consider deformations generated by the first modified Novikov–Veselov (mNV) equation that preserve tori. Stationarity conject…

Let $\Sigma$ range over tori in $\mathbf{R}^3$ within a fixed conformal class, let ${\cal W}(\Sigma)$ be their Willmore functional, and let the spectral genus be the genus of the n…

Taimanov's sphere inequality conjecture. The inequality ${\cal W}\geq 4\pi N^2$ holds for all spheres, not only for spheres with a one-dimensional potential. The text presents this…

Let a torus evolve under the modified Novikov–Veselov (mNV) flow, and regard two surfaces differing by translations as equivalent. A torus is nonstationary if its mNV evolution is…

Let $n\geq 2$ be an integer, let $E\subset \mathbb{R}^n$ have boundary $\Sigma:=\partial E$ of class $C^2$, and let $\Omega\subset \mathbb{R}^n$ be open. For $\lambda>0$, define ……