Several Complex Variables Problem List — Problem 79

Definitions and mathematical statement

Let $M$ be a Kähler manifold. Write $\mathcal O(M)$ for its holomorphic complex-valued functions, and define \[ H^\infty(M)=\left\{f\in\mathcal O(M):\sup_{p\in M}|f(p)|<\infty\right\}. \] The source assumes that "all curvatures are between $-a^2$ and $-b^2$" and asks for the construction of nontrivial bounded holomorphic functions on $M$.

If "nontrivial" means nonconstant, the requested conclusion is \[ \text{construct }f\in H^\infty(M)\quad\text{such that}\quad \forall c\in{\mathbb{C}},\ f\not\equiv c. \] This interpretation of "nontrivial" requires confirmation. The curvature hypothesis also remains unresolved: the source does not identify the curvature notion or specify the domains, signs, or ordering of $a,b$. For example, if real sectional curvature and the displayed endpoint order are intended, the hypothesis would read \[ -a^2\le K_g(p,\sigma)\le-b^2 \quad\text{for every }p\in M\text{ and every real two-dimensional subspace }\sigma\subset T_pM, \] where $g$ is the Kähler metric and $K_g$ its sectional curvature. This is only a conditional interpretation, not an additional source hypothesis. Completeness, simple connectivity, connectedness, and complex dimension are not specified. No such conditions are added here; these omissions require review before the intended universal assertion can be made precise.

Short English statement

The source asks for nontrivial bounded holomorphic functions on a Kähler manifold with curvature between $-a^2$ and $-b^2$. Reading "nontrivial" as nonconstant gives the displayed target, but the curvature notion, parameter restrictions, and intended geometric hypotheses remain unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800279$, exact source problem code catalogue entry; title: Several Complex Variables Problem List --- Problem 79. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 79. Supplied source URL: https://arxiv.org/abs/math/9509202. The registry lists Buzzard, Fornaess, and Krantz in its proposer field and gives the proposed year as 1995; the extracted problem itself is signed N. Sibony, with contact sibony@anh.matups.fr.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; the source was not independently visited or verified for this draft. The registry labels the problem open; that label is supplied curation, not an independently verified assessment of its mathematical status.