7 problems

White's conjecture. The number of embedded minimal tori in $(S^3,g)$ is at least $9$.

Let $M$ be a compact connected spin manifold, let $D_g$ be the Dirac operator for a Riemannian metric $g$, and call $g$ $D$-minimal when the kernel of $D_g$ is no larger than force…

In the totally irrational setting, consider a minimal lamination by area-minimizing hypersurfaces with gaps, and suppose the construction produces complete, non-compact minimal hyp…

Let $M^{n+1}$ be a closed smooth manifold, let $H_n(M;\mathbb{Z})$ denote its codimension-one homology group, and let a representative of a homology class mean a smooth hypersurfac…

Let $M^{n+1}$ be a closed smooth manifold, and let $g$ be a Baire $C^\infty$-generic Riemannian metric on $M$. A min-max $g$-minimal hypersurface is a minimal hypersurface generate…

Let $M^{n+1}$ be a closed smooth manifold. A Riemannian metric $g$ is called Baire $C^\infty$-generic if it belongs to a Baire generic subset of the space of smooth Riemannian metr…

Let $M^{n+1}$ be a closed Riemannian manifold, and let $\Sigma$ be an area-minimizing hypersurface in a homology class of $M$. A perturbation of the Riemannian metric should elimin…