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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.

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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.

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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.

reviewed 2026-06-26 · how reviews work

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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}
}
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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.

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Reference graph

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This paper was first reviewed by grok-4.3 on June 26, 2026.