24 problems

Let $\Omega\subset\mathbb C^n$ be a smoothly bounded pseudoconvex domain with a non-compact automorphism group, and let $p\in\partial\Omega$ be a boundary orbit accumulation point,…

Let $D\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. The Bergman metric of $D$ is the canonical Kähler metric associated with its Bergman kernel. A domain is biholomorphic…

Yau's conjecture. If the Bergman metric of $D$ is Kähler–Einstein, then $D$ is homogeneous.

Mergelyan property characterization. The domain $D$ has the Mergelyan property if and only if there are pseudoconvex domains $D_j$ with $\bar D\subset D_j$ such that

Let $D$ be a worm domain whose $\overline\partial$-Neumann problem is not globally hypoelliptic. Let $A$ be its boundary annulus, let $V\subset\mathbb C^2$ be an arbitrarily small…

Let $\Omega$ be a smoothly bounded pseudoconvex domain of finite D'Angelo type in $\mathbb C^n$, with $n\geq 3$. The Levi non-flatness conjecture. The Levi determinant, regarded as…

Let $\Omega$ be a smoothly bounded pseudoconvex domain of finite D'Angelo type in $\mathbb C^n$, with $n\geq 3$. The weak-locus measure conjecture. The collection of all weakly pse…

Cheng–Yau conjecture. The domain $\Omega$ is Bergman–Einstein if and only if it is biholomorphic to the unit ball of the same dimension.

Yau's conjecture. The Bergman metric of $\Omega$ is complete and Einstein if and only if $\Omega$ is biholomorphic to a bounded homogeneous domain.

Let $\Omega$ be a bounded domain that is not a product domain. A domain is homogeneous if its automorphism group acts transitively, meaning that for any $x,y e Omega$ there is a…

Let $D$ and $G$ be nice domains, and let $F:D\to G$ be a holomorphic mapping with the real slice rigidity property. Assume additionally that $D$ is simply connected. Simple-connect…

Let $D$ and $G$ be nice domains, for example bounded pseudoconvex domains, and let $F:D\to G$ be a holomorphic mapping with the real slice rigidity property. Real slice rigidity co…

Let $\Omega$ be a bounded pseudoconvex domain, and suppose its Bergman metric is Kähler–Einstein. A domain is homogeneous if its automorphism group acts transitively on it. Yau's h…

Let $\Omega$ be the bounded strongly pseudoconvex domain with smooth boundary under consideration, let $P(z,\zeta)$ denote its Poisson kernel, and let $VMO(\partial\Omega)$ be the…

Let $\Omega \subset \operatorname{\mathbb{C}}^d$ be a bounded pseudoconvex domain with $\operatorname{\mathcal{C}}^{1,1}$ boundary, and let $\Gamma \leq \operatorname{Aut}(\Omega)$…

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with $\mathcal C^2$ smooth boundary. The $p$-Bergman kernel of $\Omega$ is called exhaustive when it has the exhaus…

Let $\widetilde{\Omega} \subset \mathbb{C}^n \times \Delta$ be a pseudoconvex domain. For each $t \in \Delta$, define the fiber … Fix $w\in\mathbb{C}^n$, and assume that every…

Let $\Omega$ be a bounded pseudoconvex domain with $C^\infty$ boundary in $\mathbb C^d$, where $d\geq 1$. A complete Kähler metric with pinched negative holomorphic bisectional cur…

Infinite-time existence conjecture. Under these assumptions, the Kähler–Ricci flow has a solution for infinite time. This conjecture extends the cited existence result under the ad…

Exhaustion conjecture. The domain $D$ can be exhausted by domains biholomorphic to convex domains.

Let $D\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with $C^\infty$ boundary. A domain $D$ is $c$-complete when its Carathéodory pseudodistance $c_D$ is a complete metric.…

Let $\Omega$ be a bounded pseudoconvex Lipschitz domain in $\mathbb{C}^n$, let $\delta(z)$ denote the distance from $z\in\Omega$ to the boundary of $\Omega$, and let $S(z,z)$ be th…

Let $D$ be a smooth bounded pseudoconvex domain, let $d_D(z)$ denote the distance from $z$ to the boundary of $D$, and let $\b7$ denote the Kobayashi-Royden metric. Fu's conjecture…

Let $\mathcal O_n$ be the ring of germs of holomorphic functions at the origin in $\mathbb C^n$, and let $I$ be the ideal generated by $f_1,\dots,f_l\in\mathcal O_n$, where…