12 problems

Let $\mathbb{B}^2$ be the unit ball in $\mathbb{C}^2$. Let $X$ be a relatively closed complex manifold of dimension one in $\mathbb{B}^2$, smooth up to the boundary and transversal…

Polynomial-convexity and analytic-disc conjecture. The surface $M_t$ is locally polynomially convex if $t>1$. The local polynomial hull of $M_t$ contains an analytic disc for $t<1$…

Polynomial convexity conjecture. The domain ${\mathcal C}$ is polynomially convex.

Second conjecture. The disc $K$ is polynomially convex if

First conjecture. Any totally real disc lying in a nonsingular Levi-flat hypersurface in $\mathbb{C}^2$ is polynomially convex.

Let $\Omega\subset\mathbb{C}^2$ be a domain and let $z\in\partial\Omega$. The point $z$ is a local MDM point for $\Omega$ if there exists a ball $B$ centered at $z$ such that, for…

Analytic-disc conjecture. The complementary set $\widehat{X}\setminus X$ must contain an analytic disc. This conjecture was disproved by Gabriel Stolzenberg in 1963, who constructe…

Let $\gamma$ be a polynomially convex simple closed curve in $\mathbb{C}^N$. A simple closed curve is locally polynomially convex if each of its points has a neighborhood whose int…

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 $E$ be a compact set in $\mathbb C^n$, and let $\widehat E$ denote its polynomially convex hull. Analytic-disc conjecture. If $\widehat E$ is strictly larger than $E$, then…

Let $\Omega$ be a strictly pseudoconvex domain, and consider the equivalence of the following four conditions for the graph $\mathcal G_h(b\Omega)$ and functions $h_1,\ldots,h_N$:…

Let $W$ be a Stein domain of complex dimension $n\geq 3$ that is polynomially convex when embedded in $\mathbb{C}^n$. Polynomial-convexity flexibility conjecture. Polynomial convex…