Injectivity of the surface Radon transform for every simple metric

Prove that, for every smooth $(\Phi,k)$-simple metric on a three-dimensional Riemannian ball, the surface Radon transform $R_0:C(X)\to C(\mathcal C)$, defined by integrating functions over the unique $(\Phi,k)$-disc spanning each oriented round circle, is injective.

Background

The paper proves that R0R_0 has finite-dimensional kernel for every (Φ,k)(\Phi,k)-simple metric and establishes injectivity on an open dense subset of smooth metrics containing every real analytic metric. The conjecture asks whether the genericity restriction can be removed.

Injectivity is important because it would yield stronger rigidity and inversion results for the associated surface integral geometry.

References

We expect the Radon transform to be always injective when the metric is $(\Phi,k)$-simple. However, we are only able to prove it on an open dense subset of smooth metrics containing every analytic metric so far. We formulate this as a conjecture.

— 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 “Injectivity of the Radon transform” (label ssection:boundary), Conjecture conj:r0

However, the case of general Riemannian manifolds is still open for now, and we intend to study it in a follow-up paper.

— 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 “Non-conformal case” (label ssection:non-conformal)