51 problems

Let $d\zeta^{2}$ be an everywhere finite quadratic differential on a surface $\frak{F}$, different from $0$. At each point of $\frak{F}$, consider the direction in which…

Let $\Omega\subset\mathbb R^2$ be a domain and let $\varphi$ be a $K$-quasiconformal mapping, with $K\geq 1$. Let $\beta_0$ denote the exponent appearing in the generalized integra…

Let $d\in(0,2]$, let $E\subset\mathbb C$ be compact, and let $h$ be a principal $K$-quasiconformal mapping conformal on $\mathbb C\setminus E$. Here $\mathcal M^d$ denotes $d$-dime…

Let $E\subset \mathbb C$ be compact, let $0<t\leq 2$, and let $h$ be normalized and conformal in $\mathbb C\setminus E$, admitting a $K$-quasiconformal extension to $\mathbb C$. He…

Let $F$ be a reflection of the sphere $\widehat{\pmb{\mathbb R}}^3$, with fixed-point set $\bf T$ a topological sphere and complementary domains $\bf D$ and $D'$. A reflection is t…

Koskela–Wildrick's conjecture. The space $\widetilde{X}$ might support some weaker Poincaré inequality, but still better than a $2$-Poincaré inequality.

Conjectured order formula.

Let $f=h+\overline{g}\in\mathcal{S}^0_H(K)$, with $h$ and $g$ having the series representations specified in the source. Let $A(n,k)$ and $B(n,k)$ be the quantities defined in the…

Covering-radius conjecture. Every function $f\in\mathcal{S}^0_H(K)$ contains the disk

Let $T=T(\alpha,d)$ be an isosceles trapezoid of unit height with acute angle $\pi\alpha$ and base lengths $2c$ and $2d$, where $c\leq d$, $0<\alpha\leq 1/2$, and…

Covering-radius conjecture. The radius satisfies

Let $\mathcal{S}_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Harmonic K-quasiconformal Hardy-order conjecture. On…

Let $\mathcal{S}^0_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Second-coefficient conjecture. One has … The bound…

For $K\geq1$, let $f=h+\overline{g}\in\mathcal{S}^0_\mathcal{H}(K)$, where … Set $k=(K-1)/(K+1)$. HQC Bieberbach conjecture. For every $n\geq2$, … where $A(n,k)$ and $B(n,k)$ are t…

Let $\mathcal{S}^0_\mathcal{H}(K)$ be the class of harmonic $K$-quasiconformal mappings in the normalized univalent harmonic class, and let the harmonic $K$-quasiconformal Koebe fu…

Borderline integrability conjecture. If $p=n-1$, then $\varphi$ is continuous and either open or constant.

Let $0\le \theta<1$ and let $E\subset\mathbb{R}^n$. Write $\dim_A^\theta(E)$ for the Assouad spectrum with parameter $\theta$, and let the global quasiconformal Assouad spectrum di…

For a domain in $\mathbb{R}^n$, let $p_O^{{\scriptstyle{RH}}}(n,K)$ and $p_O^{{\scriptstyle{Sob}}}(n,K)$ denote the reverse Hölder and Sobolev higher integrability exponents associ…

Let $G$ be a Carnot group other than $\mathbb{R}^n$ or the Heisenberg group $\mathbb{H}_n$ for any $n$, and let $U,U'$ be open subsets of $G$. Let $f:U\to U'$ be a quasiconformal h…

Let $G$supset U$and$U' ormal?" Wait. Cannot introduce invalid notation. Let $G$ be a rigid Carnot group, and let $U,U' e othing$ be open subsets of $G$. A quasiconformal homeomor…

Uniqueness conjecture. If $\hat{\Phi}$ is holomorphic, then $\hat{h}=\operatorname{identity}$.

Let $f$ be an extremal mapping for the relevant distortion functional, and suppose that its distortion satisfies ${\Bbb K}(z,f)\in L^p({\Bbb D})$. Let $\mu$ denote the Beltrami coe…

A Carnot group $G$ is rigid in the sense of Ottazzi–Warhurst if, for every connected open subset $U\subset G$, the family of smooth contact embeddings $U\rightarrow G$ is finite di…

Let $\mathfrak L$ be a stratified Lie group equipped with its canonical Carnot–Carathéodory distance function, and let $\Upomega$ be a domain. Assume that $\mathfrak L$ is neither…

Let $G$ be a Carnot group, and let $U,U'\subset G$ be open sets. Xie's conjecture. If $G$ is neither $\mathbb{R}^n$ nor a Heisenberg group $\mathbb{H}_n$, then every quasiconformal…