Kirby Problem 4.105

Definitions and mathematical statement

A positive cooriented contact three-manifold $(Y,\xi)$ is planar if it has a supporting open book with genus-zero pages. An open book decomposes the complement of a binding link into interiors of compact surface pages; support means that a defining contact form is positive on the binding and its derivative is positive on each page.

A Stein filling is a compact Stein domain $W$ of complex dimension two whose induced boundary contact structure is contactomorphic to $(Y,\xi)$. Concretely, $W$ is a regular sublevel of a strictly plurisubharmonic exhausting function on a Stein surface; complex tangencies on its strictly pseudoconvex boundary give the contact planes.

Can one planar contact manifold admit infinitely many distinct Stein fillings? The source intentionally allows several meanings of distinct, which must be separated. The strongest existence version asks for infinitely many pairwise nondiffeomorphic underlying compact four-manifolds. A finer comparison asks for infinitely many inequivalent Stein or symplectic deformation classes, even if the underlying fillings are diffeomorphic.

For deformation equivalence, identify the smooth fillings by a diffeomorphism and require a continuous smooth family of the corresponding Stein or filling symplectic structures with contactomorphic boundary throughout. Boundary markings can either be fixed or forgotten, but the same convention must be used for every comparison.

The contact structure $\xi$, not just the three-manifold $Y$, remains fixed across the family. Arbitrarily many presentations of one filling or infinitely many choices of a plurisubharmonic function producing equivalent fillings do not supply the required distinct examples.

Short English statement

Can a fixed planar contact three-manifold have infinitely many Stein fillings, by diffeomorphism type or by deformation class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2981 (KP-4.105), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.105. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.