3 problems

Let $D_1,D_2\subsetneq\mathbb{C}$ be simply connected domains, let $z\in D_1\cap D_2$, and assume that $D_1\cup D_2$ is hyperbolic. The hyperbolic densities satisfy … This conjectu…

Square-lattice extremality conjecture. The square lattice should be extremal, and for the extremal function, $f(0)$ should be a center of the complement.

Let $K>1$, $n>2$, and $r\in(0,1)$. Define $\alpha=K^{1/(1-n)}$ and let $\varphi_{K,n}$ be the quasiconformal distortion function. Hyperbolic distortion conjecture. … The conjecture…