15 problems

The $16\pi$ conjecture. The number $\beta_0$ is attained by a path $\omega\in C^0([0,1],\operatorname{Imm}(\mathbb{S}^2,\mathbb{R}^3))$ such that both restrictions…

Let $g>1$ be an integer, and consider smooth surfaces of $\mathbb{R}^3$ of genus $g$ and their Willmore energy. Willmore conjecture in codimension 1. The only minimizers are the co…

Montiel–Urbano conjecture. The Clifford torus achieves the minimum of the functional $\mathcal{W}^-$, and hence of $\mathcal{W}$, either among all tori in $\mathbb{C}P^2$ or among…

Grand uniqueness conjecture. The genus-$g=0$ or $1$ Canham problem with any isoperimetric ratio constraint $v_0\in(0,1)$ has a unique solution up to homothety. Moreover: (i) for…

Let ${\vec{\Phi}}_1,\ldots,{\vec{\Phi}}_p,\vec{\Psi}_1,\ldots,\vec{\Psi}_q$ be the Willmore spheres given by the decomposition in equation (12), where the preceding discussion conc…

Let $\beta_0$ be the min-max width associated with a nontrivial loop in the space of immersions of the round sphere $S^2$ into $\mathbb{R}^3$, equivalently the infimum of the maxim…

Let $X:\Sigma\setminus\{p_1,\ldots,p_m\}\to\mathbb{R}^3$ be a complete minimal surface with $m$ embedded planar ends $\{p_1,\ldots,p_m\}$. Let $n_X(p_i)$ denote the normal at the e…

Let $g$ be a genus and let a surface of genus $g$ have a conformal type called rectangular in the sense used for the quotient by the cyclic $(g+1)$-fold symmetry of Lawson's minima…

A Willmore torus is stable when its Willmore second variation is nonnegative, and it is isothermic when it admits the corresponding isothermic structure in conformal surface geomet…

A Willmore torus is called stable when the second variation of the Willmore functional is nonnegative, equivalently when the Jacobi operator satisfies … The Clifford torus is the s…

A constrained Willmore torus is a torus that is critical for the Willmore functional under conformal-class-preserving variations, and it is embedded when its immersion is embedded.…

Let $\mathcal W$ denote the Willmore functional, and fix a conformal class of tori. The $(1,2)$-equivariant constrained Willmore tori are those constrained Willmore tori of orbit t…

For $b\geq 1$, consider tori in $\mathbb{R}^3$ with rectangular conformal type … The relevant candidates are the homogeneous tori $S^1\times S^1(b)$ in the $3$-sphere and the $2$-l…

Let $\vec{\Phi}:T^2\to\mathbb{R}^m$ be an immersion, and let $\vec{H}_{\vec{\Phi}}$ and $d\operatorname{vol}_{\vec{\Phi}^*g_{\mathbb{R}^m}}$ denote its mean curvature vector and in…

Complex-holomorphic-data conjecture. Then $f$ is given by complex holomorphic data.