6 problems

Let $f$ be holomorphic on a neighborhood of the closure of the unit disk $\mathbb{D}$. An injective holomorphic function is a holomorphic map that is one-to-one. Conformal polynomi…

Let $M>0$ be a positive real number. Consider exterior quadrilaterals with prescribed angles and vertices $A_1=1$, $A_2=0$, $A_3$, and $A_4$, and let their conformal modulus be $M$…

Suppose $F:\mathbb D\to\Omega$ is a conformal map of the unit disk onto a Jordan–Lipschitz domain $\Omega$, and let $L(\Omega)$ denote the perimeter of $\Omega$. For $\alpha>0$, th…

Let $t_1,t_2$ satisfy … For a real parameter $\lambda$, let $f$ be the solution of … normalized by $J_f(0)=(0,1,0)$, and call $f$ a gear mapping when its image is a gear-shaped dom…

Let $\Omega$ and $\Omega^*$ be bounded doubly connected domains in $\mathbb{C}$, and suppose that there is an energy-minimal diffeomorphism … Write $\operatorname{Mod}\Omega$ and…

Let $\Omega^*$ be a bounded doubly connected domain in $\mathbb{C}$, and let $\mathsf{E}(\tau,\Omega^*)$ denote the infimum of the Dirichlet energy over deformations from a doubly…