22 problems

Strong Ramadanov conjecture. If $\rho \in C^{n+2}(\overline{D})$, then $\partial D$ is locally spherical.

Let $M\subset \mathbb{C}^n$ be a pseudoconvex real hypersurface with $n\geq 3$, let $1\leq s\leq n-2$, let $p\in M$, and let $B$ be a smooth complex vector subbundle of…

Let $M\subset \mathbb{C}^n$ be a pseudoconvex real hypersurface with $n\geq 3$. For $1\leq s\leq n-2$ and $p\in M$, write $t^{(s)}(M,p)$, $c^{(s)}(M,p)$ and $a^{(s)}(M,p)$ for the…

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with smooth boundary of finite type, and let $f=1$ be the trivial multiplier generated by Kohn's procedures. The or…

Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex domain with smooth boundary of finite type. A $q$-multiplier is a germ of a smooth function whose product with a $(0,q)$-f…

Let $G\subset \mathbb{C}^n$ with $n\geq 2$ be a smoothly bounded pseudoconvex domain, and let $K$ be the Bergman kernel of $G$. The minimal-degree conjecture. If $K$ is algebraic,…

Let $M_t$ be the real surface under consideration, parametrized by the parameter $t$. A surface is locally polynomially convex at a point if sufficiently small compact neighbourhoo…

Let $\Omega \subset \mathbb{C}^d$ be a pseudoconvex domain of finite type and let $\xi_0 \in \partial \Omega$. A holomorphic self-map is a map $f:\Omega\to\Omega$ that is holomorph…

Extension conjecture. Then $f\in{\mathcal{S}}^1$, that is, $f$ embeds into a normalized Loewner chain. The source states that this is equivalent to the generalized Andérsen–Lempert…

Embedding conjecture.

For $n\geq 2$, let $\kappa_0$ be the largest integer such that … and define … Let $f\colon\mathbb S^n\to\mathbb S^N$ be a sufficiently smooth CR mapping. Huang–Ji–Yin Gap Conjectur…

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$, let $w\in\Omega$, let $G_{\Omega,w}$ denote the pluricomplex Green's function of $\Omega$ with pole at $w$, and let…

Let $X$ be a complex manifold endowed with a Riemannian metric, and let $f\colon X\to X$ be an automorphism acting hyperbolically on an invariant compact subset $K\subset X$. For…

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$, and let $S^0(\mathbb B_n)$ be the normalized Loewner class. For $f\in S^0(\mathbb B_n)$, $D^kf(0)$ denotes its $k$th derivative…

Let $S^0(\mathcal D_n)$ be the normalized Loewner class, and let $\varphi_{s,t}$ be an evolution family associated with an $\mathcal M(\mathcal D_n)$-Herglotz vector field. Bounded…

Let $S^0(\mathcal D_n)$ be the normalized Loewner class. Suppose that $f\in S^0(\mathcal D_n)$ is bounded by $M>0$, and let $G\in S^0(\mathcal D_n)$. Bounded-composition conjecture…

Let $S^0(\mathcal D_n)$ be the compact class of normalized mappings generated by Loewner chains, and call $f\in S^0(\mathcal D_n)$ a support point if a nonconstant real part of a c…

Let $\mathcal D_n$ denote the unit polydisc, and let $E(\mathcal D_n)$ and $F(\mathcal D_n)$ be the two classes of normalized mappings defined in the paper, with tilded versions al…

Bedford's conjecture. The basin of attraction

Let $n>2$. A quadrature domain in $\mathbb{R}^n$ is a domain satisfying a quadrature identity for the relevant class of functions, and an algebraic domain is one whose boundary is…

Let $B^n\subset\mathbb C^n$ be the unit ball, and let $a_1,\ldots,a_{n+1}\in B^n$ be vertices in general position. An $n+1$-bundle of complex lines consists of the complex lines co…

Affine-independence conjecture. Then $fcin A(cpartialcOmega)$.