13 problems

Let a boundary curve lie in the boundary of a mean-convex domain, and let the Douglas solution denote the solution of the Plateau problem spanning that curve. Douglas–Morrey embedd…

Continuous-family conjecture. A minimal surface with flat points is still contained in a continuous family of twice differentiable constant-ratio-of-principal-curvatures surfaces i…

Non-collapsing conjecture. The non-existence of singular solutions to the homotopic Plateau problem should be sufficient to exclude collapsing; equivalently, it should guarantee no…

White's conjecture. The union of the boundary and interior singular points of $\Sigma$ is discrete. In particular, the overall singular set is finite, $\Sigma$ has finite genus $g$…

Meeks's Convex Curves conjecture. The surface $\Sigma$ is a topological annulus.

Let $E$ be a sliding almost minimal set near a curve $\Gamma$, and suppose that a blow-up limit at a point of $E\cap\Gamma$ is a cone $Y\in\mathbb{Y}(\Gamma)$, meaning that $Y$ con…

Large-aspect-ratio convergence conjecture. As $h\to+\infty$, the rescaled minimizer $\Sigma_{\min}$ converges to $\Sigma$.

Nonwetting conjecture. The surface $M_{h,s}$ satisfies condition (NW).

Invisible-wire nonwetting conjecture. Numerical evidence suggests that the absolute minimizer has the required topology and does not wet the invisible wires.

Let $\Gamma$ be a Jordan curve. Rellich's conjecture. For all sufficiently small $H>0$, the curve $\Gamma$ bounds at least two solutions of the $H$-Plateau problem. The paper prese…

Let $\Gamma$ be a simple closed curve in $\mathbf{R}^3$. It is extreme if it lies in the boundary of its convex hull. Embeddedness conjecture. If $\Gamma$ is extreme, then the area…

Let $\Gamma$ be a simple closed curve in a Riemannian manifold, and call it extreme if it lies in the boundary of its convex hull. Consider an area-minimizing disk spanning…

Let $K>0$, let $k\in]0,K[$, and let $\Gamma\subseteq\mathbb{R}^{n+1}$ be a smooth, compact, codimension $2$ curve bounding a compact, locally strictly convex immersed hypersurface…