In the appropriate Fukaya-category setting for a compact Calabi–Yau manifold X, if a Lagrangian submanifold L⊂ X defines an object E_L that is stable with respect to the relevant Bridgeland stability condition, then there should exist a special Lagrangian submanifold L'⊂ X whose Fukaya-category object is isomorphic to E_L, i.e. E_L'cong E_L. The conjectures also predict that mean-curvature flow starting from an unstable Lagrangian evolves through controlled singularities and converges, after the corresponding surgeries, to a configuration of special Lagrangians.

Thomas–Yau–Joyce conjectures

In the appropriate Fukaya-category setting for a compact Calabi–Yau manifold $X$ , if a Lagrangian submanifold $L\subset X$ defines an object $E_L$ that is stable with respect to the relevant Bridgeland stability condition, then there should exist a special Lagrangian submanifold $L'\subset X$ whose Fukaya-category object is isomorphic to $E_L$ , i.e. $E_{L'}\cong E_L$ . The conjectures also predict that mean-curvature flow starting from an unstable Lagrangian evolves through controlled singularities and converges, after the corresponding surgeries, to a configuration of special Lagrangians.

References

Primary source

arXiv

Additional references

A Thomas-Yau-Joyce result for Lagrangian spheres in K3 surfaces — arXiv — Jacopo Stoppa

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A new preprint reports conditional progress toward the conjectures in K3 surfaces, but the general problem remains open.

The Thomas–Yau–Joyce conjectures seek stability criteria for when a Lagrangian has a special-Lagrangian representative and predict its behavior under mean-curvature flow. The general conjectures are not solved.

Known results

Stoppa (2025) proved a restricted equivalence for Lagrangian sections in certain Calabi–Yau fibrations mirror to toric weak Fano manifolds, including selected weak del Pezzo cases.
In specified hyperkähler $4$ -manifolds, mean-curvature flow was shown to undergo finitely many neck-pinches and converge to chains of special-Lagrangian spheres; the full Thomas conjecture remains open.

September 2026 K3 development

Jacopo Stoppa’s preprint claims a stability-to-special-Lagrangian implication for a class of K3 objects, with unconditional results for selected quartic and sextic surfaces. It presents a controlled compact-manifold realization of the conjectural picture, but its principal theorem assumes strong expected properties of homological mirror symmetry; the claim is unverified.

Current status (as of October 2026): restricted and conditional results are available, while the general Thomas–Yau–Joyce conjectures remain open and the K3 preprint is an unverified progress claim.

Sources

No solutions have been posted yet.