14 problems

Let $\xi_{g,1}\subset S^3$ be the Lawson minimal surface of genus $g$ indexed by $(g,1)$, and consider its stereographic projection to $\mathbb{R}^3$. Kusner's conjecture. For surf…

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…

Conjecture 5.4. The Willmore-type bound should force finite-time round-point convergence under mean curvature flow and spherical topology. The source presents this as a stronger ve…

Sharp-constant conjecture. The constant $\pi/2$ can be replaced with $\pi$:

Consider the diffuse Willmore functional and its Gamma limit for phase-field approximations. In the sharp-interface limit, the relevant one-dimensional configurations are represent…

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

Let $\Sigma$ be a surface in $\mathbb R^4$ with Willmore energy $W(\Sigma)$. Kusner's conjecture. If … then $\Sigma$ must be a sphere. This remains open in the context of the paper…

Odd-dimensional failure conjecture. The naive conjecture that such powers yield integrands for conformally invariant energy functionals is false when $d$ is odd, because of the lea…

Let the physical spacetime have dimension $d$, and let $S^m\times S^n$ be a product surface with $m+n+2=d$. For surfaces $\Sigma$ of the same topology as $S^m\times S^n$, let…

Kusner–Pinkall conjecture. The area $A(\Sigma_{g,L})$ is monotonically increasing as a function of $g$, and

Kusner's conjecture. Stereographic projections of Lawson's $g$-holed tori in $\mathbb{S}^3$ should be global minimizers of $\mathcal{W}$ among all genus-$g$ surfaces.