14 problems

Rigidity conjecture. Any such solution $u$ must be a quadratic polynomial.

Székelyhidi's conjecture. Such a function $u$ exists if and only if, for every $p$-dimensional irreducible analytic subvariety $V\subset M$ with $n-l\leq p\leq n-1$,

Let $\Omega\subset\mathbb{R}^n$ be a bounded uniformly convex domain. Consider the Neumann problem for the $k$-Hessian equation, with $2\leq k\leq n-1$, obtained from … with prescr…

Rotational symmetry conjecture. Every such solution $u$ on $\mathbb{R}^n$ is rotationally symmetric.

Murakami's conjecture. The class $[\chi]$ contains a smooth $k$-admissible representative if and only if, for every $t\geq0$ and every $p$-dimensional irreducible analytic subvarie…

Ivochkina–Trudinger–Wang conjecture. The condition $f^{\frac{1}{k-1}} \in C^{1,1}(\overline{\Omega})$ should be sufficient for obtaining an existence theorem for the degenerate $k$…

Let $1\leq l<k$ and consider the general Hessian quotient equation … Let $u$ be a solution on a closed Riemannian manifold $(M,g)$ endowed with a symmetric $(2,0)$ tensor $\chi$. C…

Let $M$ be a connected, closed Riemannian manifold with metric $g$, and let $\chi$ be a symmetric $(2,0)$ tensor on $M$. Suppose that there is an admissible function $v\in C^4(M)$…

Let $u$ be a $W^{2,p}$ solution of the $k$-Hessian equation with smooth right-hand side, where $p>\frac{k(k-1)}{2}$. Urbas's conjecture. Such a solution is smooth. This conjecture…

Singular J-equation conjecture. If $(\alpha,\beta)$ is $J$-positive, then this equation admits a unique solution $\omega$ as a Kähler current with bounded local potentials.

The concavity conjecture. For every $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n$, the quadratic form

The quadratic-form conjecture. For every $n4$-dimensional vector $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n4$, the quadratic form

Let $u\in C^4(\mathbb{R}^n)$ be an entire 2-convex function, meaning that its Hessian eigenvalues lie in the closure of the $2$-Hessian ellipticity cone, and suppose that … through…

Let $(X,\alpha)$ be a compact Kähler manifold, and consider the positivity conditions denoted by $(\mathrm{SW1})$ in the source, together with their generalizations. Lejmi–Székelyh…