Research Problems in Function Theory — Problem 7.67

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 domain is a connected open subset. Let $\Omega\subset\mathbb C^2$ be strictly pseudoconvex: near each boundary point it has a $C^2$ defining function $\rho$ with nonzero gradient and positive Levi form $\sum_{j,k}\rho_{z_j\bar z_k}v_j\bar v_k$ on every nonzero complex tangent vector $v$. Let $f\in\mathcal O(\Omega)$ and set $V=\{z\in\Omega:f(z)=0\}$. Finite area means $\mathcal H^2(V)<\infty$ inside $\Omega$, using ordinary real two-dimensional Hausdorff measure. Equality of zero sets below means equality of sets, not an imposed equality of vanishing multiplicities.

Statement recorded in the supplied source.

Does finite area of $V$ imply that there exists a bounded holomorphic $g$ on $\Omega$ with $\{g=0\}=V$? The domain is arbitrary within the strictly pseudoconvex class in $\mathbb C^2$; no ball or additional cohomology hypothesis is imposed. The source records an affirmative answer for the ball as a special case. [S2]

Short English statement

Does every finite-area holomorphic zero set in a strictly pseudoconvex domain in two complex dimensions have a bounded holomorphic defining function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307067 (AMR-022-7067). 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 7.67, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.