The anisotropic Calderón problem: rigidity near the Euclidean metric
Abstract
We prove rigidity near the Euclidean metric for the anisotropic Calderón problem on smooth bounded connected domains in dimensions three and higher. Any smooth Riemannian metric sufficiently close to the Euclidean metric in a suitable Hölder norm and having the same Dirichlet-to-Neumann map agrees with it up to a diffeomorphism fixing the boundary pointwise. The result applies to general smooth anisotropic perturbations, without analytic or quasianalytic regularity assumptions, a prescribed conformal class, or a transversal product structure, and requires no convexity of the boundary. The proof combines harmonic coordinates with a Carleman estimate for compactly supported correctors to obtain a quadratic Fourier estimate on a frequency range determined by the perturbation. A frequency decomposition then yields rigidity.
Related
- Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem
- Quasianalyticity and geometric rigidity in anisotropic Calderón's problem
- The Calderón problem for near-Euclidean metrics
- The conformal Poisson embedding method for the anisotropic Calderón problem with a potential
- Local rigidity of conformally Euclidean metrics for the anisotropic Helmholtz equation at fixed energy
BibTeX
@misc{lin2026anisotropic,
title = {{The anisotropic Calderón problem: rigidity near the Euclidean metric}},
author = {Yi-Hsuan Lin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17261},
eprint = {2609.17261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17261},
}
RIS
TY - UNPB
TI - The anisotropic Calderón problem: rigidity near the Euclidean metric
AU - Yi-Hsuan Lin
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17261
UR - https://arxiv.org/abs/2609.17261
AB - We prove rigidity near the Euclidean metric for the anisotropic Calderón problem on smooth bounded connected domains in dimensions three and higher. Any smooth Riemannian metric sufficiently close to the Euclidean metric in a suitable Hölder norm and having the same Dirichlet-to-Neumann map agrees with it up to a diffeomorphism fixing the boundary pointwise. The result applies to general smooth anisotropic perturbations, without analytic or quasianalytic regularity assumptions, a prescribed conformal class, or a transversal product structure, and requires no convexity of the boundary. The proof combines harmonic coordinates with a Carleman estimate for compactly supported correctors to obtain a quadratic Fourier estimate on a frequency range determined by the perturbation. A frequency decomposition then yields rigidity.
ER -