DEFORMATIVE MAGNETIC MARKED LENGTH SPECTRUM RIGIDITY

James Marshall Reber

correcthigh confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the stated rigidity by (i) constructing isometries via a Livšic-based argument for the metric variation and (ii) handling the magnetic intensity through a non-homogeneous Jacobi equation plus new magnetic Carleman estimates, forcing the intensity to be constant after gauge. The candidate solution correctly treats the metric part using Livšic/s-injectivity ideas, but its treatment of the magnetic intensity assumes, without a justified first-variation formula, that the degree-1 component identifies φ̇ with the Lie derivative Y(φ). The paper’s Section 5 shows this step is nontrivial and requires the Jacobi/Carleman machinery to conclude φ̇′=0, not φ̇=Y(φ). Hence the model’s proof is incomplete at exactly the crucial intensity step, while the paper’s argument is complete and correct.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper rigorously establishes deformative marked length spectrum rigidity for magnetic flows on surfaces of negative magnetic curvature. The metric component follows a classical scheme adapted to the magnetic setting, while the intensity component is resolved via a careful non-homogeneous Jacobi equation and magnetic Carleman estimates. The work is well-motivated, technically sound, and fills a gap that naive s-injectivity arguments do not cover. Minor editorial refinements would further clarify the exposition.