Boundaries of Groups and Kleinian Groups — Problem 36

Definitions and mathematical statement

Let $B^n=\{x\in\mathbb R^n:|x|<1\}$, with $n\geq2$, and let $f:B^n\to B^n$ be a quasiconformal homeomorphism. One analytic definition is that $f\in W^{1,n}_{\mathrm{loc}}$ and there is a finite constant $K_f$ such that \[ \lVert Df(x)\rVert^n\leq K_f\,|\det Df(x)| \quad\text{for almost every }x\in B^n. \] Here the norm is the operator norm, and the Sobolev condition means that the coordinate functions and their first weak derivatives are locally $n$-integrable.

Can $f$ be approximated, uniformly on compact subsets of $B^n$, by smooth diffeomorphisms $f_j:B^n\to B^n$ which are globally quasiconformal? Globally means each $f_j$ has one finite distortion bound valid throughout the ball, rather than bounds which may diverge near its boundary. The stronger question asks whether one can choose a single finite $K$, independent of $j$, such that \[ \lVert Df_j(x)\rVert^n\leq K\,|\det Df_j(x)| \quad\text{for every }j\text{ and almost every }x. \] The bound may depend on $f$ and dimension; it is not stipulated to equal the optimal distortion of $f$. No equality of boundary values is imposed. Smoothing that changes the image domain, loses injectivity or yields only local distortion bounds does not satisfy the requested approximation.

Short English statement

Can quasiconformal maps of a ball be approximated by globally quasiconformal diffeomorphisms, with uniformly bounded distortion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200036 (AMR-061-0036), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 36, PDF page 12. Original source indexed by AMR-061-0036. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.