Global conformal boundary area rigidity without a closeness assumption

Prove that if a smooth metric $\bar g$ and its conformal deformation $e^{2f}\bar g$ are both area simple on a three-dimensional Riemannian ball, then equality of their boundary area spectra on the space of oriented round circles, $\mathcal A_{\bar g}=\mathcal A_{e^{2f}\bar g}$, implies $f\equiv0$ without assuming that the two metrics are close.

Background

The paper proves local conformal rigidity for sufficiently small C14C^{14} conformal factors and global rigidity when the conformal factor is real analytic. The stated conjecture removes the smallness or analyticity restriction while retaining area simplicity of both metrics.

The authors explain that the main missing ingredient is an analogue of Santaló’s formula for the foliation by minimal surfaces; such a volume-preserving flow structure is generally absent in the present setting.

References

Similarly, we expect that Theorem \ref{theorem:main} should hold without any closeness assumption on $\bar g$ and $e{2f} \bar g$, provided they both satisfy the assumptions of Theorem \ref{th.Foliated_Plateau_posta}.

— 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 “Conformal case and Santaló's formula,” Conjecture conjecture:a