Concordance Classes of Finite-Energy Foliations

Definitions and mathematical statement

A cooriented contact structure on a three-manifold $M$ is a plane field $\xi=\ker\alpha$ for a one-form with $\alpha\wedge{\,\mathrm{d}}\alpha\ne0$. It is tight if it has no overtwisted disk, and overtwisted otherwise. Finite-energy holomorphic foliations in this setting are built from punctured pseudoholomorphic curves in the symplectization ${\mathbb{R}}\times M$, with contact-adapted almost complex structure and finite symplectization energy, using the admissibility conditions of the report.

Let $\mathscr F(M,\xi)$ denote the source's class of such foliations and $\sim_{\mathrm{conc}}$ its concordance relation. The two requested assertions are \[ \#\left(\mathscr F(S^3,\xi_{\mathrm{tight}})/\!\sim_{\mathrm{conc}}\right)=1, \] and \[ \exists\text{ overtwisted }\xi\text{ on }S^1\times S^2: \#\left(\mathscr F(S^1\times S^2,\xi)/\!\sim_{\mathrm{conc}}\right)\ge2. \] The first concerns the tight contact three-sphere. The second is existential: the report says "a certain" overtwisted structure, without specifying an invariant or canonical distinguished structure. It does not assert multiplicity for every overtwisted contact structure on $S^1\times S^2$.

The excerpt does not define concordance, the precise foliation admissibility and energy conditions, or which contact forms and almost complex structures are fixed or allowed to vary. These choices determine the quotient above. Concordance cannot be replaced without verification by isotopy of plane fields, a path of contact forms, or an arbitrary smooth foliation equivalence. The displayed quotient records the exact target while retaining its source-specific relation.

The literature assessment checked on 21 August 2026 says both uniqueness and multiplicity were labelled conjectural and that no later primary resolution of this exact classification was found. That dated search outcome does not establish current openness or identify the missing overtwisted structure. All claims remain source-qualified.

Short English statement

Is there exactly one concordance class of finite-energy holomorphic foliations on the tight three-sphere, while at least one overtwisted structure on $S^1\times S^2$ has more than one? The source's precise concordance and admissibility conventions are essential.

Sources

• Dynamical Systems (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/31, pp. 1743--1798, https://doi.org/10.4171/owr/2005/31; proposed year supplied as 2005.
• Literature assessment checked 21 August 2026: original report, https://doi.org/10.4171/owr/2005/31; "Wendl holomorphic foliations notes," https://www.mathematik.hu-berlin.de/~wendl/pub/Koeln_20051010.pdf.
• Local UnsolvedMath registry, permanent ID $30000272$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Supplied verification explicitly preserves only "a certain" overtwisted structure and removes invented notation for a distinguished one.