Relaxing compactness to completeness

Determine whether the flatness rigidity theorem for compact Kähler manifolds carrying regular Riemannian foliations with Lagrangian minimal leaves remains valid when compactness is replaced by completeness.

Background

The paper proves that a compact Kähler manifold carrying a regular Riemannian foliation whose leaves are Lagrangian and minimal must be flat. The proof relies essentially on compactness: compactness yields the relevant global integral identities, including the horizontal analogue of Ros’s formula, and allows nonnegative terms in the resulting integral formula to be shown to vanish.

The author explicitly leaves unresolved whether the compactness hypothesis can be weakened to completeness. The text notes that a proof in the complete, noncompact setting would require substantially different techniques and suggests that purely local arguments are unlikely to establish the same rigidity.

References

The author does not know whether the hypothesis of compactness can be relaxed to completeness, as in this case the involved techniques would be totally different, but it seems unlikely that the same rigidity can be obtained from purely local considerations; the proof presented here makes essential use of compactness.

— Lagrangian Foliations on compact Kähler manifolds  (2609.01142 - Podestà, 1 Sep 2026) in Introduction, paragraph immediately preceding the Acknowledgments