11 problems

Boundary maximum principle conjecture. A similar boundary maximum principle should hold for varifolds with boundary in general codimension.

Let a sequence of sets in $\mathbb{R}^{n}$ consist of smoothly embedded connected submanifolds without boundary and have uniformly bounded curvature everywhere. Consider their lift…

Unique continuation conjecture. If $V=V'$ in $M\setminus {\rm An}$ and $V'$ is a stable minimal surface in ${\rm An}$, then $V=V'$.

Let $M$ be the closed manifold and $F$ the anisotropic integrand considered above, and let $(u_\varepsilon)$ be any sequence of critical points of $E_\varepsilon$ with bounded ener…

Dimension-bound conjecture. The singular set of a stationary integral varifold has dimension at most $m-1$.

Let $\Omega_b\subset\mathbb{B}^3$ be a free boundary minimal surface of genus $0$ with $b$ boundary components. Convergence conjecture. There is a sequence…

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…

Let $V$ be a varifold satisfying the assumptions of the paper except for assumption 2, which excludes classical singularities. Write $\operatorname{sing} V$ for its singular set. O…

Let $F$ be a Lagrangian, let $H\subseteq\mathbb R^n$ be a closed set, and let $\mathcal P(H)$ be a good class. Consider the Plateau problem of minimizing the anisotropic energy…

Let $X$ be a compact Riemannian manifold of dimension $n$ isometrically embedded in $\mathbb{R}^N$, and let $1\leq k\leq n$. For $\varepsilon>0$, let $[\Pi]$ be a homotopy class of…

Let $V$ be a stable codimension $1$ integral varifold on a smooth Riemannian manifold, with support $\operatorname{spt}(V)$. An embedded $C^{1,\alpha}$ hypersurface-with-boundary i…