Research Problems in Function Theory — Problem 5.70

Definitions and mathematical statement

Let $\mathbb D=\{z:|z|<1\}$ and let $f$ be a nonconstant bounded holomorphic function on $\mathbb D$, so $\sup_{\mathbb D}|f|<\infty$. For a number $c>0$ define its modulus level set \[ L_c(f)=\{z\in\mathbb D:|f(z)|=c\}. \] A level-set component means a complete connected component of $L_c(f)$, not a selected subarc of it. Call a component unbranched if it has no branching points as a real analytic level curve. For $c>0$, this is equivalent to $f'(z)\ne0$ at every point of that component; critical points of a nonzero modulus level create intersecting local level arcs. Does there exist a bounded holomorphic $f$, a level $c>0$, and an unbranched component $C$ of $L_c(f)$ with \[ \mathcal H^1(C)=\infty? \] Here $\mathcal H^1$ is Euclidean one-dimensional Hausdorff measure, agreeing with arclength on smooth arcs. The component is locally smooth but may approach the boundary of the disc in a complicated way, allowing infinite total length in a bounded region. The source records constructions of infinite-length components with extensive branching; the requested example removes branching entirely.

Short English statement

Can a bounded analytic function have a complete smooth, unbranched modulus-level component of infinite Euclidean length?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305070 (AMR-022-5070), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.70. Original source indexed by AMR-022-5070. https://arxiv.org/abs/1809.07200.