Kirby Problem 4.123

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 $M$ be an oriented smooth four-manifold and $\mathcal F$ a nonsingular smooth foliation by two-dimensional leaves. Coorientability is an orientation of the rank-two normal bundle $TM/T\mathcal F$; together with the orientation of $M$ this orients its leaves. The source calls $\mathcal F$ taut if there is a smooth two-form $\omega$ whose restriction to each leaf is a positive area form and \[ d\omega(v_1,v_2,z)=0 \quad(v_1,v_2\in T\mathcal F,\ z\in TM). \] This does not require $\omega$ to be a symplectic form. A closed leaf $L$ is a compact connected oriented surface without boundary. Its class $[L]$ lies in integral singular homology $H_2(M;\mathbb Z)$.

Statement recorded in the supplied source.

For every such $M,\mathcal F$ and every closed leaf $L$, is $L$ smoothly genus-minimizing in its homology class? Explicitly, for every closed connected oriented smoothly embedded surface $S\subset M$ with $[S]=[L]$, must $\operatorname{genus}(S)\geq\operatorname{genus}(L)$? [S2]

Short English statement

Must every compact leaf of a coorientable smooth taut two-dimensional foliation in an oriented four-manifold have the least genus among smooth surfaces representing its homology class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2999 (KP-4.123). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 4.123, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.