Several Complex Variables Problem List — Problem 38

Definitions and mathematical statement

For interpreting the terminology, let $n\geq 1$ and let $\Omega\subset{\mathbb{C}}^n$ be a bounded domain, meaning a bounded, connected open set. The source does not specify the complex dimension. Call $\Omega$ pseudoconvex if it admits a continuous plurisubharmonic exhaustion function $\rho\colon\Omega\to{\mathbb{R}}$, so that $\{z\in\Omega:\rho(z)<c\}$ is relatively compact in $\Omega$ for every $c\in{\mathbb{R}}$.

The symbol $B$ is not defined in the supplied record. A possible interpretation is the Bergman metric. Under this interpretation, define \[ A^2(\Omega)=\left\{f\colon\Omega\to{\mathbb{C}}:\ f\text{ is holomorphic and }\int_\Omega |f|^2\,{\,\mathrm{d}} V<\infty\right\}, \qquad K_\Omega(z,z)=\sup_{\substack{f\in A^2(\Omega)\\ \|f\|_{L^2(\Omega)}\leq 1}}|f(z)|^2. \] Its metric tensor is \[ g_{j\bar k}(z)=\frac{\partial^2\log K_\Omega(z,z)}{\partial z_j\,\partial\bar z_k}, \qquad B_\Omega(z;v)^2=\sum_{j,k=1}^n g_{j\bar k}(z)v_j\overline{v_k}. \] Let $d_{B,\Omega}$ be the distance obtained by taking the infimum of the $B_\Omega$-lengths of piecewise continuously differentiable curves joining two points of $\Omega$. Completeness means that every $d_{B,\Omega}$-Cauchy sequence converges in this distance to a point of $\Omega$.

Source question, with unresolved qualifications. Is $B$ complete on bounded pseudoconvex domains with boundary of regularity "less than $C^1$"? If $B$ means the Bergman metric and $\mathcal D$ denotes the intended class of such domains, the proposed assertion is \[ \text{for every }\Omega\in\mathcal D,\quad (\Omega,d_{B,\Omega})\text{ is complete}. \] The supplied record neither identifies $B$ nor defines $\mathcal D$: it gives no precise boundary regularity threshold and does not specify whether all boundary regularity assumptions are to be dropped. Thus this conditional formulation requires clarification before it becomes a definite mathematical assertion.

Short English statement

The question asks whether bounded pseudoconvex domains with boundary regularity below $C^1$ have a complete metric denoted by $B$. If $B$ is the Bergman metric, completeness means that every Cauchy sequence for its induced distance converges inside the domain. The metric notation, intended boundary class, and dimension require clarification.

Sources

• Local UnsolvedMath registry: permanent ID $800241$, exact source code catalogue entry, title Several Complex Variables Problem List --- Problem 38. The supplied registry labels the record open; this label is not an independently verified mathematical status.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 38. Supplied source URL: https://arxiv.org/abs/math/9509202. The registry attributes the collection to these authors; the question itself is signed J. McNeal, with supplied contact mcneal@math.princeton.edu.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance statements, not a claim of independent source access or verification for this draft.