Research Problems in Function Theory — Problem 6.114

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Write $\mathcal H^1$ for one-dimensional Hausdorff measure (length). A plane curve $\Gamma$ is regular in the stated sense if there is $C<\infty$ such that \[\mathcal H^1(\Gamma\cap B(w,r))\leq Cr\qquad(w\in\mathbb C,\ r>0),\] where $B(w,r)$ is an open disk. Let $f:\mathbb D\to f(\mathbb D)$ be a conformal map, that is, a holomorphic injective map; $f^{-1}(\Gamma)=\{z\in\mathbb D:f(z)\in\Gamma\}$.

Statement recorded in the supplied source.

Must $\mathcal H^1(f^{-1}(\Gamma))<\infty$ hold for every regular curve $\Gamma$ and every such conformal map $f$? [S2]

Short English statement

Does a conformal inverse image of every curve with uniformly linear local length have finite length inside the disk?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306114 (AMR-022-6114). 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 6.114, complete problem and update. https://arxiv.org/abs/1809.07200.