The anisotropic Calderón conjecture

From papers

Let $(M,g_1)$ and $(M,g_2)$ be compact Riemannian manifolds with smooth boundary, with $\dim(M)\geq 3$ . For each metric, let $C_{g_i}$ denote the Cauchy data set of harmonic functions on $(M,g_i)$ . Anisotropic Calderón conjecture. If

C_{g_1}=C_{g_2},

then there is a diffeomorphism $\psi:M\to M$ such that $\psi|_{\partial M}=\operatorname{Id}$ and

g_2=\psi^*g_1.

This is the uniqueness assertion for the anisotropic Calderón problem: boundary Cauchy data should determine the Riemannian metric up to a boundary-fixing diffeomorphism in dimensions at least three. The general problem remains open, although uniqueness is known under additional geometric hypotheses, including the transversally anisotropic settings studied in this paper.

Progress summary

Partially solved

The full uniqueness conjecture remains open, despite a recent claimed proof and new results covering important special settings.

The conjecture asks whether complete boundary measurements determine a smooth metric in three or more dimensions, up to a diffeomorphism that fixes the boundary. A 2024 preprint claimed a proof for the general case, but later literature continues to treat the general smooth problem as unresolved.

Known results

Salo’s exposition records the established two-dimensional case and the boundary-fixing diffeomorphism obstruction.
Uniqueness is known for real-analytic manifolds and several fixed-conformal-class or admissible settings.
Transversally anisotropic geometries yield uniqueness under injectivity hypotheses for the relevant ray transform.
Counterexamples for disjoint boundary measurement sets do not apply to the full Cauchy-data conjecture.

August 2026 developments

Chen, Jiang, Liu, and Tao report complementary uniqueness results in quasianalytic and symmetric settings, including a partial-boundary consequence, while explicitly leaving general smooth metrics open. The 2024 preprint’s general proof claim remains unverified by the retrieved sources.

Current status (as of August 2026): The general smooth anisotropic Calderón conjecture remains open, with substantial uniqueness results only under additional geometric or regularity assumptions.

Sources

Sources & referencesView supporting material

Primary source

David Dos Santos Ferreira, Yaroslav Kurylev, Matti Lassas and Mikko Salo, “The Calderon problem in transversally anisotropic geometries”, arXiv:1305.1273 (2014).

Additional references

2 papers in this index state this conjecture (2010–2013). The statement above is taken from the most recent of them; the others are arXiv:1011.2507.

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