7 problems

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 $v\in C^4(M)$…

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

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.

Lejmi–Szekelyhidi's conjecture. Under this normalization, the $J$-equation admits a unique smooth solution $\omega$ if and only if, for every $m$-dimensional analytic subvariety…

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…