6 problems

Bonk–Heinonen–Koskela conjecture. Every Gromov hyperbolic domain in the Riemann sphere is conformally equivalent to a spherical uniform circle domain, and conversely every such con…

Let $X$ be a metric surface satisfying the transboundary Loewner property (TLP), meaning that there are lower bounds for the relevant transboundary moduli of curve families connect…

Let a domain be a connected open subset of the Riemann sphere, and let a circle domain be a domain whose boundary components are points or circles. Existing proofs of Koebe's conje…

Rigidity–removability conjecture. Rigidity of a circle domain is equivalent to $\operatorname{QCH}$-removability of its boundary.

A connected open set $\Omega$ in the Riemann sphere $\widehat{\mathbb C}$ is a circle domain if each boundary component of $\Omega$ is either a circle or a point. It is rigid if ev…

He–Schramm's removability conjecture. If every Cantor set contained in $\partial\Omega$ is conformally removable, then $\partial\Omega$ is conformally removable.