Koebe's circle-domain conjecture (Kreisnormierungsproblem)

Claims that every domain in $\hat{\mathbb C}$ is conformally equivalent to a circle domain, with no countability or geometric condition on the complementary components. The method marks countably many components, collapses unmarked ones to points with small energy barriers, and shows marked ones become exactly round disks, using transboundary extremal length and a modulus-probability duality. Combined with Ntalampekos-Rajala it also gives a convergent exhaustion by finitely connected domains. It does NOT give uniqueness of the circle-domain model; the rigidity question is the companion's subject and is answered there only under conformal removability of the boundary.

Claims that every domain in $\hat{\mathbb C}$ is conformally equivalent to a circle domain, with no countability or geometric condition on the complementary components. The method marks countably many components, collapses unmarked ones to points with small energy barriers, and shows marked ones become exactly round disks, using transboundary extremal length and a modulus-probability duality. Combined with Ntalampekos-Rajala it also gives a convergent exhaustion by finitely connected domains. It does NOT give uniqueness of the circle-domain model; the rigidity question is the companion's subject and is answered there only under conformal removability of the boundary.