The anisotropic Calderón problem for Riemannian metrics at high fixed frequency

@misc{cekic2026anisotropic,
  title = {{The anisotropic Calderón problem for Riemannian metrics at high fixed frequency}},
  author = {Mihajlo Cekić and Katya Krupchyk and Suman Kumar Sahoo and Gunther Uhlmann},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01280},
  eprint = {2610.01280},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01280},
}

TY  - UNPB
TI  - The anisotropic Calderón problem for Riemannian metrics at high fixed frequency
AU  - Mihajlo Cekić
AU  - Katya Krupchyk
AU  - Suman Kumar Sahoo
AU  - Gunther Uhlmann
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01280
UR  - https://arxiv.org/abs/2610.01280
AB  - We study an inverse boundary value problem for the Helmholtz equation on a smooth compact non-trapping Riemannian manifold with strictly convex boundary. We prove that, given two such metrics, for sufficiently large but fixed frequency λ, equality of their Dirichlet-to-Neumann maps implies equality of their lens data, up to a smooth boundary-fixing diffeomorphism. This establishes a high-frequency bridge between the Calderón problem and lens rigidity. We also prove a quantitative version of this result uniformly valid in a suitable bounded set of Riemannian metrics. Crucially, we show that Gaussian beam type solutions concentrating on maximal geodesics, split near the boundary into outgoing/incoming parts, such that the phase of the outgoing part contains information on exit point, direction, and travel time. The recovery of lens data then proceeds by a careful stationary phase analysis and a boundary integral identity.
ER  -