Research Problems in Function Theory — Problem 7.65

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.

Let $\mathcal W$ be a Greenian hyperbolic Riemann surface, with Green function $G_\omega$ at each pole $\omega$, normalized by the local singularity $-\log|z-z(\omega)|$. Fix a base point and put $\Gamma=\pi_1(\mathcal W)$. Local harmonic conjugates $G_\omega^*$ are continued along loops avoiding the pole. Let $\widetilde\Gamma$ consist of the loop classes for which the increment of this conjugate is in $2\pi\mathbb Z$ for every pole $\omega$; integrality is unchanged when a representative winds once around the pole. Write $[\Gamma,\Gamma]$ for the subgroup generated by commutators. A relatively closed subset of a surface has zero logarithmic capacity if it is locally polar: in coordinate patches its compact subsets have zero plane logarithmic capacity, equivalently they can be contained in the minus-infinity set of a nonconstant subharmonic function.

Statement recorded in the supplied source.

If $[\Gamma,\Gamma]\ne\widetilde\Gamma$, must there exist a hyperbolic Riemann surface $\mathcal V$ and a nonempty relatively closed zero-capacity subset $A\subset\mathcal V$ such that $\mathcal W$ is conformally equivalent to $\mathcal V\setminus A$? The source's special case $\widetilde\Gamma=\Gamma$ with $\mathcal V$ the disc does not replace this general structural question. [S2]

Short English statement

Must a surface with additional integral Green-conjugate periods arise by deleting a nonempty polar closed set from a hyperbolic surface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307065 (AMR-022-7065). 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 7.65, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.