Research Problems in Function Theory — Problem 7.11

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\geq2$, a quasiconformal map $f:\mathbb R^n\to\mathbb R^n$ is a homeomorphism in $W^{1,n}_{\mathrm{loc}}$ with \[\|Df(x)\|^n\leq K|\det Df(x)|\quad\text{for almost every }x\] for some finite $K$; $\|Df\|$ is the Euclidean operator norm. The Sobolev condition means that every coordinate and its first weak derivatives are locally integrable to the nth power. Identify $\mathbb R^n$ with the hyperplane $\{(x,0):x\in\mathbb R^n\}$ in $\mathbb R^{n+1}$.

Statement recorded in the supplied source.

Does every quasiconformal $f:\mathbb R^n\to\mathbb R^n$ extend to a quasiconformal homeomorphism $F:\mathbb R^{n+1}\to\mathbb R^{n+1}$ satisfying \[F(x,0)=(f(x),0)\qquad(x\in\mathbb R^n)?\] The extension may have a different finite dilatation. The source's update states that Tukia and Väisälä established the assertion in all dimensions under consideration. [S2]

Short English statement

Extend any global quasiconformal map to one additional Euclidean dimension while preserving its restriction to the original hyperplane.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307011 (AMR-022-7011). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 7.11, complete problem and update. https://arxiv.org/abs/1809.07200.