7 problems

A sphere eversion is a regular homotopy $H:\mathbb{S}^2\times[0,1]\to\mathbb{R}^3$ connecting two round spheres of opposite orientation. Kusner's 16π-conjecture. There exists a sph…

Let $\mathbb{S}^5\subset\mathbb{C}^3$ be the unit sphere, and let $\Sigma_H\subset\mathbb{S}^5$ be the hexagonal Legendrian minimal torus. For a Legendrian surface…

Let $F_t:\Sigma^n\rightarrow\mathbb R^{n+2}$ be a codimension $2$ submanifold moving by the skew-mean-curvature flow … Write $\rho=|H|^2$ for the square of its mean curvature. Will…

Compactness conjecture for minimizing sequences. There is a sequence of Möbius transformations $G_j$ of $\mathbb{R}^3$ such that the surfaces $G_j\circ\mathbf{x}_j(K)$ can be parti…

Let $\tilde\tau_{3,1}$ be Lawson's bipolar surface, and let $\varphi:\mathbb{S}^4\to\mathbb{R}^4$ be a stereographic projection. For $n\geq 4$, regard…

Let $\mathcal W(\Sigma)=\int_{\Sigma}(1+H^2)\\,d\mu$ denote the Willmore energy of a compact surface $\Sigma\subset S^3$, where $H$ is its mean curvature. Willmore's conjecture. Ev…

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…