12 problems

Let $C^{m,n}_s$ denote a fractional Lawson cone, and let $C^{n,n}(1)$ denote the symmetric cone. A fractional Lawson cone is free boundary in a ball if it is a free boundary $s$-mi…

Let $s\in(0,1)$, and consider stable $s$-minimal hypercones in $\mathbb{R}^n\setminus\{0\}$. Flatness conjecture. For every integer $n$ with $3\leqslant n\leqslant 7$, there exists…

For $s\in(0,1)$, let $n_s^*$ be the minimum dimension $n\geq 3$ for which there exists a smooth, stable $s$-minimal cone in $\mathbb{R}^n\setminus\{0\}$ that is not a hyperplane. C…

Let the minimizers of the variational problem in the paper be periodic nonlocal-CMC surfaces, with the volume serving as the constraint. Straight cylinders are the configurations d…

Let $\Omega = (-1,1)^2 \setminus (0,1)\times (-1,0)$ be the L-shaped domain considered in the numerical examples, and let $\partial\Omega$ denote its boundary. The examples concern…

Smoothness-threshold conjecture. Nonlocal minimal surfaces should be smooth up to dimension $6$, while completely new phenomena may arise in dimension $7$.

For $s_0\in(0,1)$ and $s\in[s_0,1)$, let $C(n,s_0)>0$ be a constant for which the quantitative fractional isoperimetric inequality … holds, where $A(E)$ is the Fraenkel asymmetry o…

Let $E$subset$\mathbb R^2$ be a planar nonlocal minimal graph as in the surrounding regularity results, and let the Hölder exponent be the one obtained in Theorem. Optimality conje…

Let $E$ be an $s$-minimal surface in ambient dimension $n$. Singularity conjecture. When $s$ is sufficiently small, $s$-minimal surfaces may develop singularities in dimension $7$…

Bernstein-type conjecture. The only stable embedded minimal (hyper)surfaces in $cmathbb{R}^n$ are hyperplanes as long as the dimension of the ambient space is less than or equal to…

Stability-threshold conjecture. There is a number $s_0$ such that $\Sigma$ is stable for

Let the cone be the nonlocal Lawson cone considered in the paper, parametrized by $s\in[0,1)$. Stability threshold conjecture. There is a value $s_0$ such that the cone is stable f…