6 problems

Fleming's Bernstein conjecture. The Bernstein result should be valid in any dimension: every such graph must be an affine hyperplane. Fleming established this property for $N=2$; t…

Let $M$ be a minimal hypersurface in $\mathbb{R}^{n+1}$ that is $\frac{n-2}{n}$-stable, meaning that … for every $\varphi\in C_0^\infty(M)$. Flat-or-catenoid conjecture. For…

Let $u$ be a solution of an anisotropic minimal hypersurface equation, with $B_r$ denoting the ball of radius $r$ and $abla u$ its gradient. Mooney–Yang's conjecture. If … for som…

Trudinger–Wang's conjecture. If $n\leq 9$, then $u$ is an elliptic quadratic polynomial. If $n\geq 10$, then there exists a smooth, locally uniformly convex solution of the equatio…

Li–Jia's conjecture. Then $f$ must be a quadratic polynomial. This is a Bernstein property for the Abreu equation on an entire strictly convex potential; the source attributes the…

Let $\Sigma^3\subset\mathbb{R}^4$ be a complete immersed $2$-sided stable minimal hypersurface. Schoen's Bernstein conjecture. $\Sigma$ is flat. The conjecture extends the Bernstei…