19 problems

A domain in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is a connected open set. A circle domain is a domain whose complementary components are points or close…

Circle patterns are configurations of circles with prescribed combinatorial structures. A conformal mapping is a structure-preserving map between planar domains that preserves angl…

Let $\mathrm B_{\mathcal K}(t)$ denote the Kraetzer version of the integral means spectrum, and let $t$ be a real parameter. Kraetzer's conjecture. … The conjecture encompasses the…

Let $n$ be a positive integer, and let $\Omega$ be an $n$-connected domain in the plane such that no boundary component is a point. A domain has algebraic kernel functions when its…

Logarithmic action conjecture. The formula above remains true for general smooth subordination evolution. This is posed as an open question concerning the extension of the variatio…

Let $\mathbb{U}=\{z\in\mathbb{C}:|z|<1\}$ be the unit disk, let $0\leq a<1$, and define the Möbius transformation … For $z_1,z_2\in\mathbb{U}$, write $w_1=f(z_1)$ and $w_2=f(z_2)$.…

Let $\varphi:\bD\to\bC$ be analytic and injective, where $\bD$ is the unit disk and $\bT$ its boundary circle. Anderson's conjecture. There exists $\zeta\in\bT$ such that … The con…

Let $\mathbb{D}$ be the unit disk and let $\partial \mathbb{D}^{+}=\{z\in\partial\mathbb{D}:\operatorname{Im}z>0\}$ be its upper semi-circle. Suppose $f$ is a conformal mapping of…

Conjectured modulus values.

Let $T_a:\mathbb{B}^2\to\mathbb{B}^2$ be the Möbius transformation … For $x,y,a\in\mathbb{B}^2$ and $\alpha>0$, the generalized point pair function conjecture. … Computer tests sug…

Let a Jordan domain be approximated by inversive distance circle packings, and let the associated discrete conformal maps be the maps induced by those packings. Bowers–Stephenson's…

Discontinuity conjecture. In general, neither $\log f'$ nor $\log(h^{-1})'$ depends continuously on $\Gamma$ (or on $b$).

Functional derivative conjecture. The fact that, for Nehari functionals with poles only at the origin, the functional derivative at an extremal is the pull-back of the original qua…

Let $f\colon\mathbb{D}\to\Omega$ be a bounded conformal map from the unit disk, and define the universal integral means spectrum by … where $\beta_f(t)$ is the integral means spect…

Let $Omega subseteq \mathbb{C}$ be a simply connected domain and let $varphi:Omegato Delta$ be a conformal mapping from $Omega$ to the unit disk. For a normalized point…

The conjectured value of $L(a)$. For $n=2$, numerical experiments suggest that

Let $\Omega\subset\mathbb C$ be a simply connected plane domain with nonempty boundary, let $\mathbb D$ be the unit disc, and let $\varphi:\mathbb D\to\Omega$ be a conformal homeom…

Geometric Brennan's conjecture. The $p$-distortion is integrable in degree $\alpha$ for all $\frac{4}{3(p-2)}<\alpha<\frac{4}{p-2}$ if $2<p<+\infty$, and for all…

Let $\Omega\subset \mathbb R^n$ be a bounded open connected domain with $n\geq 3$. Let $u\in W^{1,n}(\Omega)$ and $v\in W^{1,1}(\Omega)$ satisfy … almost everywhere, and suppose th…