Research Problems in Function Theory — Problem 2.55

Definitions and mathematical statement

Let $f_1,f_2,f_3:\mathbb C\to\mathbb C$ be nonconstant entire functions, and define the complex analytic set \[ V=\{(z_1,z_2,z_3)\in\mathbb C^3: f_1(z_1)+f_2(z_2)+f_3(z_3)=0\}. \] Let $F:\mathbb C^3\to\mathbb C$ be entire, meaning holomorphic in all three complex variables, and assume that its restriction to $V$ is bounded: \[ \sup_{z\in V}|F(z)|<\infty. \] Must there be one constant $c\in\mathbb C$ such that $F(z)=c$ for every $z\in V$? The conclusion concerns only the restriction of $F$ to the analytic set: $F$ need not be constant throughout $\mathbb C^3$. Neither smoothness nor irreducibility of $V$ is assumed. Consequently an argument valid on one smooth connected piece must still justify a single constant on the complete set. The three separated-variable summands are arbitrary nonconstant entire functions; replacing them all by polynomials changes the problem to a special case. This is a Liouville-type question for a family of possibly nonalgebraic surfaces. Ordinary Liouville's theorem in the ambient space cannot be applied directly because no bound is assumed away from $V$.

Short English statement

Is every ambient entire function bounded on a separated-variable surface necessarily constant when restricted to that whole surface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302055 (AMR-022-2055), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.55. Original source indexed by AMR-022-2055. https://arxiv.org/abs/1809.07200.