Foliated Plateau theory on manifolds with topology and trapped sets

Construct, under suitable geometric hypotheses such as conditions (A1)–(A4) in the minimal case or negative sectional curvature in the $k$-surface case, a unique $(\Phi,k)$-disc in every relative homotopy class of discs with a prescribed boundary circle on a Riemannian manifold with nontrivial topology and a nonempty trapped set, and prove that the resulting family of discs foliates the unit tangent bundle.

Background

The paper suggests extending its program beyond balls to manifolds with nontrivial topology and trapped geodesic or surface dynamics. The proposed construction would generalize the double fibration and surface Radon transform developed for the ball.

The key unresolved geometric issue is existence and uniqueness of curvature discs in each relative homotopy class, which would be needed before defining and analyzing the corresponding foliation and Radon transform.

References

Under suitable assumptions on the metric, such as $\hyperlink{AA1}{\rm(A1)-(A4)}$ in the minimal case, or negative sectional curvature in the case of $k$-surfaces, we conjecture that for every circle $c\in$ and every relative homotopy class of discs with boundary $c$, there exists a unique $(\Phi,k)$-disc in that class.

— Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three  (2609.29470 - Alvarez et al., 24 Sep 2026) in Section 1, Subsection “Perspectives,” Subsection “Open manifolds with topology”

Extending these techniques to the present setting would require an analogue of those results, whose existence is far from clear. We leave this problem for future work.

— Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three  (2609.29470 - Alvarez et al., 24 Sep 2026) in Section 1, Subsection “Perspectives,” Subsection “Closed manifolds”