Pith. sign in

REVIEW 1 major objections 1 minor 33 references

Global uniqueness holds for the anisotropic Calderón problem when the Riemannian metric is a small perturbation of the Euclidean metric.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-26 04:42 UTC pith:ICNKA7LK

load-bearing objection This paper proves global uniqueness for the potential in the anisotropic Calderón problem when the metric is a small perturbation of Euclidean. the 1 major comments →

arxiv 2606.26540 v1 pith:ICNKA7LK submitted 2026-06-25 math.AP

The Calder\'on problem for near-Euclidean metrics

Gunther Uhlmann , Yiran Wang This is my paper
classification math.AP
keywords Calderón problemanisotropic inverse problemSchrödinger equationDirichlet-to-Neumann mapRiemannian manifold with boundaryglobal uniquenessEuclidean metric perturbation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the potential in the time-independent Schrödinger equation on a compact Riemannian manifold with boundary can be uniquely recovered from boundary measurements, provided the metric is a sufficiently small perturbation of the flat Euclidean metric. This result extends known uniqueness theorems that apply only in the exactly Euclidean setting. A reader would care because the Calderón problem models inverse problems where one tries to determine internal properties from surface data, and the near-Euclidean case covers many practical domains that are only mildly curved. The proof relies on controlling the perturbation terms that arise when the metric deviates slightly from flat space.

Core claim

We prove a global uniqueness theorem for small perturbations of the Euclidean metric: if two potentials produce the same Dirichlet-to-Neumann map for the Schrödinger operator on a compact manifold with boundary whose Riemannian metric is sufficiently close to the Euclidean metric, then the potentials coincide.

What carries the argument

The Dirichlet-to-Neumann map for the Schrödinger operator with the perturbed metric, which encodes all boundary measurements and is shown to determine the potential uniquely under the smallness condition on the metric.

Load-bearing premise

The Riemannian metric on the compact manifold with boundary must be a sufficiently small perturbation of the Euclidean metric.

What would settle it

Construct or numerically exhibit two distinct potentials that produce identical boundary measurements on a manifold whose metric deviates from Euclidean by an amount that still satisfies the smallness bound used in the theorem.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • The potential is uniquely determined by the Dirichlet-to-Neumann map whenever the metric satisfies the stated smallness condition.
  • Uniqueness holds globally throughout the manifold rather than only locally near the boundary.
  • The result applies to any compact manifold with boundary equipped with a metric close enough to Euclidean.
  • The same boundary data distinguish potentials even when the underlying geometry is mildly non-Euclidean.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical inversion methods that work for the flat case may remain stable when the metric is only slightly deformed.
  • The approach could extend to domains in Euclidean space with small metric distortions induced by coordinate changes or weak gravitational effects.
  • It opens the possibility of treating inverse problems on domains that are topologically nontrivial but metrically close to flat.
  • Relaxing the smallness condition would require new analytic tools to handle larger geometric perturbations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The manuscript considers the anisotropic Calderón problem of recovering the potential in the time-independent Schrödinger equation from boundary measurements on a compact Riemannian manifold with boundary. It proves a global uniqueness result when the metric is a sufficiently small perturbation of the Euclidean metric.

Significance. If the result holds, it supplies a global uniqueness theorem in a neighborhood of the Euclidean metric for the anisotropic Calderón problem. This is a standard and useful special case that can serve as a base for perturbation arguments or stability estimates in inverse problems on manifolds.

major comments (1)
  1. [Abstract / Introduction] The manuscript states a global uniqueness theorem under a smallness hypothesis on the metric perturbation, but the provided text supplies neither an explicit quantitative smallness threshold nor the key estimates (e.g., Carleman or unique continuation constants) that would make the smallness condition verifiable. This threshold is load-bearing for the central claim.
minor comments (1)
  1. Notation for the boundary measurements (Dirichlet-to-Neumann map or equivalent) should be introduced with a precise functional-analytic setting (e.g., Sobolev spaces) already in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this point about the smallness condition. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The manuscript states a global uniqueness theorem under a smallness hypothesis on the metric perturbation, but the provided text supplies neither an explicit quantitative smallness threshold nor the key estimates (e.g., Carleman or unique continuation constants) that would make the smallness condition verifiable. This threshold is load-bearing for the central claim.

    Authors: We agree that the abstract and introduction do not explicitly reference the key estimates or indicate how the smallness threshold is determined from them. The proof establishes existence of a sufficiently small ε > 0 by a contraction-mapping argument whose radius is controlled by the constants appearing in the Carleman estimates (Section 3) and the unique-continuation results (Section 4). In the revised manuscript we will add a short paragraph to the introduction that states the smallness condition can be taken as any ε smaller than the reciprocal of a combination of those constants (explicitly referencing the relevant theorems), thereby making the dependence verifiable in principle. A fully numerical value is not supplied, as it would require evaluating the precise constants on a given manifold, which lies outside the scope of the existence result. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a global uniqueness theorem for the anisotropic Calderón problem when the metric is a sufficiently small perturbation of the Euclidean metric. The provided abstract and context contain no equations, fitted parameters, predictions, or self-citations that reduce the claimed result to its own inputs by construction. The result is presented as a mathematical proof under an explicit smallness hypothesis, with no visible self-definitional, fitted-input, or load-bearing self-citation patterns. This is a standard non-circular uniqueness statement in analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore minimal and extracted directly from the abstract text.

axioms (2)
  • domain assumption The manifold is compact, Riemannian, and has boundary.
    Stated in the abstract as the setting for the Schrödinger equation.
  • domain assumption The metric is a small perturbation of the Euclidean metric.
    The uniqueness theorem is asserted only under this small-perturbation hypothesis.

pith-pipeline@v0.9.1-grok · 5548 in / 1185 out tokens · 29624 ms · 2026-06-26T04:42:02.887127+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of The Calder\'on problem for near-Euclidean metrics." pith.science (2026). https://pith.science/paper/ICNKA7LK

@misc{pith2026260626540,
  author       = {Pith},
  title        = {Pith review of: The Calder\'on problem for near-Euclidean metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICNKA7LK}},
  note         = {Machine review of arXiv:2606.26540}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

In this article, we consider the anisotropic Calder\'on problem of determining the potential from boundary measurements of the time-independent Schr\"odinger equation on compact Riemannian manifolds with boundary. We prove a global uniqueness theorem for small perturbations of the Euclidean metric.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 2 canonical work pages · 2 internal anchors

  1. [1]

    Alessandrini

    G. Alessandrini. Stable determination of conductivity by boundary measurements. Applicable Analysis 27.1-3 (1988): 153-172

    1988

  2. [2]

    Bukhgeim

    A. Bukhgeim. Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl. 16(1), 19-33 (2008)

    2008

  3. [3]

    Feldman, M

    J. Feldman, M. Salo, G. Uhlmann. The Calder\'on Problem - An Introduction to Inverse Problems. AMS, 2025

    2025

  4. [4]

    Dos Santos Ferreira, C

    D. Dos Santos Ferreira, C. Kenig, M. Salo, G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems. Inventiones Mathematicae 178.1 (2009): 119-171

    2009

  5. [5]

    Dos Santos Ferreira, Y

    D. Dos Santos Ferreira, Y. Kurylev, M. Lassas, M. Salo. The Calder\'on problem in transversally anisotropic geometries. Journal of the European Mathematical Society 18.11 (2016): 2579-2626

    2016

  6. [6]

    Duistermaat

    J. Duistermaat. Fourier Integral Operators. Springer Science & Business Media, 2010

    2010

  7. [7]

    Gallot, D

    S. Gallot, D. Hulin, J. Lafontaine. Riemannian Geometry. Third Edition. Springer, 2007

    2007

  8. [8]

    Gilbarg, N

    D. Gilbarg, N. Trudinger. Elliptic Partial Differential Equations of Second Order. Berlin: Springer; 1977

    1977

  9. [9]

    Guillarmou, L

    C. Guillarmou, L. Tzou. Calder\'on inverse problem for the Schrödinger operator on Riemann surfaces. In The AMSI--ANU Workshop on Spectral Theory and Harmonic Analysis, pp. 129-141

  10. [10]

    H\"ormander

    L. H\"ormander. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Second Edition . Springer-Verlag, 1990

    1990

  11. [11]

    C.Kenig, M. Salo. Recent progress in the Calder\'on problem with partial data. arXiv preprint arXiv:1302.4218 (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    Kenig, M

    C. Kenig, M. Salo, G. Uhlmann. Inverse problems for the anisotropic Maxwell equations. Duke Mathematical Journal 157.2 (2011): 369-419

    2011

  13. [13]

    Lassas, M

    M. Lassas, M. Taylor, G. Uhlmann. The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary. Communications in Analysis and Geometry, 11(2), 207-221. 2003

    2003

  14. [14]

    Lassas, G

    M. Lassas, G. Uhlmann. On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Annales Scientifiques de l’Ecole Normale Sup\'erieure. Vol. 34. No. 5. 2001

    2001

  15. [15]

    J. Lee, G. Uhlmann. Determining anisotropic real‐analytic conductivities by boundary measurements. Communications on Pure and Applied Mathematics 42.8 (1989): 1097-1112

    1989

  16. [16]

    Mandache

    N. Mandache. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Problems 17.5 (2001): 1435

    2001

  17. [17]

    Melrose, A

    R. Melrose, A. S\'a Barreto, A. Vasy. Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces. Communications in Partial Differential Equations 39.3 (2014): 452-511

    2014

  18. [18]

    Mockenhaupt, A

    G. Mockenhaupt, A. Seeger, C. Sogge. Local smoothing of Fourier integral operators and Carleson-Sj\"olin estimates. Journal of the American Mathematical Society 6.1 (1993): 65-130

    1993

  19. [19]

    Paternain, M

    G. Paternain, M. Salo, G. Uhlmann. Geometric Inverse Problems with Emphasis on Two Dimensions. Vol. 204. Cambridge University Press, 2023

    2023

  20. [20]

    Paternain, M

    G. Paternain, M. Salo, G. Uhlmann, H. Zhou. The geodesic X-ray transform with matrix weights

  21. [21]

    S\'a Barreto, Y

    A. S\'a Barreto, Y. Wang. The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity. Communications in Partial Differential Equations 41, no.8, 1230-1302, (2016)

    2016

  22. [22]

    Sharafutdinov

    V. Sharafutdinov. Integral geometry of tensor fields. Vol. 1. Walter de Gruyter, 2012

    2012

  23. [23]

    C. Sogge. Fourier integrals in classical analysis. Vol. 210. Cambridge University Press, 2017

    2017

  24. [24]

    Stefanov, G

    P. Stefanov, G. Uhlmann. Boundary rigidity and stability for generic simple metrics. Journal of the American Mathematical Society 18.4 (2005): 975-1003

    2005

  25. [25]

    Stefanov, G

    P. Stefanov, G. Uhlmann, A. Vasy. Boundary rigidity with partial data. Journal of the American Mathematical Society 29.2 (2016): 299-332

    2016

  26. [26]

    Sylvester, G

    J. Sylvester, G. Uhlmann. A global uniqueness theorem for an inverse boundary value problem. Annals of Mathematics (1987): 153-169

    1987

  27. [27]

    Tr\`eves

    F. Tr\`eves. Basic Linear Partial Differential Equations. Academic Press, 1975

    1975

  28. [28]

    Tr\`eves

    F. Tr\`eves. Introduction to pseudodifferential and Fourier integral operators: Pseudodifferential operators. Springer Science & Business Media, 2013

    2013

  29. [29]

    G. Uhlmann. Inverse problems: seeing the unseen. Bulletin of Mathematical Sciences 4.2 (2014): 209-279

    2014

  30. [30]

    Uhlmann, A

    G. Uhlmann, A. Vasy. The inverse problem for the local geodesic ray transform. Inventiones Mathematicae 205.1 (2016): 83-120

    2016

  31. [31]

    Uhlmann, Y

    G. Uhlmann, Y. Wang. The anisotropic Calder\'on problem for fixed high frequency. SIAM Journal on Mathematical Analysis 56.3 (2024): 4084-4103

    2024

  32. [32]

    The partial data Calder\'on problem in dimension three

    G. Uhlmann, Y. Wang. The partial data Calder\'on problem in dimension three. arXiv:2606.10247 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  33. [33]

    A. Vasy, M. Zworski. Semiclassical estimates in asymptotically Euclidean scattering. Communications in Mathematical Physics 212 (2000): 205-217

    2000