Research Problems in Function Theory — Problem 7.58

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.

In $\mathbb R^2\cong\mathbb C$ let $\Delta(A,B;\mathbb R^2)$ denote all plane curves joining $A$ and $B$. Its $2$-modulus is the infimum of $\int\rho^2\,dA$ over nonnegative Borel densities with line integral at least $1$ on every rectifiable joining curve; the infimum is $+\infty$ if none exists. Let $E\subset[0,1]\subset\mathbb R$ be compact and assume positive conformal capacity in the explicit sense \[M_2(\Delta(E,\{z:|z|=2\};\mathbb R^2))>0. \] Put $F=\mathbb R\setminus E$, considered as a subset of the plane.

Statement recorded in the supplied source.

Must $M_2(\Delta(E,F;\mathbb R^2))=+\infty$? [S2]

Short English statement

Must the modulus between a compact positive-capacity subset of the unit interval and the rest of the real line be infinite?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307058 (AMR-022-7058). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.58, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.