8 problems

Killing-form minimizer conjecture. Killing $1$-forms, equivalently Killing vector fields, are global minimizers of the quotient defining $S_1$. Consequently,

Hsiang–Lawson conjecture. The Clifford torus is the only embedded minimal torus in $S^3$.

Let $S^3$ be the three-dimensional sphere, and let $M\subset S^3$ be an embedded closed minimal surface. The complement $S^3\setminus M$ has two components, each of which determine…

Let $W_k(S^3)$ denote the successive widths in the minmax hierarchy for minimal surfaces in the round three-sphere, and let the corresponding minmax indices be $N_k$. The first thr…

Let $S^3$ be the three-dimensional sphere, and let $\xi_{2,1}\subset S^3$ denote the Lawson surface of genus $2$. An embedded minimal surface in $S^3$ is a smoothly embedded surfac…

Let $g$ be any Riemannian metric on the $3$-sphere $S^3$. Nine minimal surfaces conjecture. The metric $g$ has nine distinct minimal surfaces of genus either zero or one. This is p…

Yau's conjecture. The smallest positive eigenvalue of $-\Delta_\Sigma$ is equal to $2$.

Lawson's conjecture. In the case that $M$ is minimal in $\mathbb{S}^{3}$, $M$ divides $\Omega^{d}$ into two connected components of equal volume for the biggest such $d$, potential…