Research Problems in Function Theory — Problem 6.61

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.

A Jordan domain is the interior of a simple closed curve. A rectifiable curve has finite arclength. Let $D_1,D_2$ be Jordan domains whose boundary curves $C_1,C_2$ have the same finite length. An isometric sewing identifies their boundary points by an orientation-reversing arclength-preserving homeomorphism. It is conformally admissible everywhere if the sewn topological sphere has a Riemann-surface structure compatible with the original complex structures on both interiors. Uniformize this surface conformally by the Riemann sphere; denote the image of the seam by $C$. Rectifiability of $C$ is measured in the round spherical metric (equivalently in a smooth planar coordinate avoiding $C$).

Statement recorded in the supplied source.

For every everywhere conformally admissible isometric sewing as above, must the seam $C$ be rectifiable? [S2]

Short English statement

Does sewing two equal-length rectifiable Jordan boundaries conformally always give a rectifiable seam on the uniformized sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306061 (AMR-022-6061). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.61 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.