18 problems

A closed set $E\subset \mathbb{R}^n$ is removable under bounded $K$-quasiregular mappings if every bounded $K$-quasiregular mapping defined on an open set minus $E$ extends to a…

Let $U \subset \mathbb R^n$ be open with $n \ge 3$, let $f:U\to f(U)\subset\mathbb R^n$ be a non-constant quasiregular mapping, let $K_I(f)$ denote its inner dilatation, and let…

Let $n\geq 2$, let $\Omega\subset\mathbb R^n$ be a domain, and let $F\colon\Omega\to\mathbb R^n$ be an orientation-preserving mapping in…

Let $cuOmega\textbackslashsubset\textbackslashmathbb R^n$ be a domain, let $K\textbackslashge 1$, and let…

Let $cuOmega\textbackslashsubset\textbackslashmathbb R^n$ be a domain, let $K\textbackslashge 1$, and let $p\textbackslashge 1$. A mapping…

Iwaniec–Martin removability conjecture. Sets of Hausdorff $d$-measure zero, where

Let $n\geq 3$ and $K>1$. A $K$-quasiregular mapping has higher integrability exponent $p(n,K)$, meaning that its derivative belongs locally to $W^{1,pn}$ for every exponent below t…

Let $f\colon \mathbb{R}^n\to\mathbb{R}^n$ be a solution of the higher-dimensional inequality … where $\Sigma$ is locally integrable. The inequality is a higher-dimensional analogue…

Let $n\geq 2$, and let $f:\mathbb{R}^n\to\mathbb{R}^n$ be an entire quasiregular Yosida mapping, meaning that the family of translates … is normal. Order conjecture. The order of g…

Let $n$ be the dimension and $K$ the distortion of a quasiregular mapping. Its sharp higher-integrability exponent is denoted by $p(n,K)$, meaning the largest exponent for which th…

Inner-dilatation conjecture. Any quasiregular mapping satisfying $K_I(f)<2$ is a local homeomorphism.

Martio–Rickman–Väisälä conjecture. For $n\geq 3$ and $\mathsf{K}\geq 1$, there exists a constant $c=c(n,\mathsf{K})\geq 1$ such that, for every $\mathsf{K}$-quasiregular mapping…

Let $K$ satisfy $1<K<\infty$, and let $f$ be a weakly $K$-quasi-regular mapping in $W^{1,q}_{\operatorname{loc}}$, where … A mapping is quasi-regular when it belongs to…

Let $f:B^n\to\mathbb{R}^n$ be a proper quasiregular mapping, with $n\geq 3$, and let $B^n(r)\subset B^n$ denote the ball of radius $r$. For $y\in f(B^n)$ and $r\in(1/2,1)$, conside…

Sharpness conjecture. There exists a sequence of positive numbers $\epsilon_k\rightarrow 0$ and a sequence of pairs of $Q$-quasiregular maps $u_k:B_1\rightarrow \mathbb{R}^2$,…

Analytic-properties conjecture. Analogous properties hold for mappings between equiregular sub-Riemannian manifolds.

Eremenko's conjecture. There exists a version of Bloch's theorem for $K$-quasiregular mappings on the unit ball, as well as for quasimeromorphic mappings.

Nonvanishing imaginary-part conjecture. Either $\operatorname{Im}(f_z)$ is constant, or