Closed magnetic geodesics on non-compact manifolds

Gong proves Theorem 1 via Benci–Giannoni penalization, obtains global minima of the penalized free-period functional, and then uses a curvature-at-infinity index-form estimate to prevent escape and recover a closed orbit; the steps are explicit (definition of S_k, penalization, first/second variations, and the key negativity estimate (58)–(60) in the proof of Theorem 1) . The candidate solution, while capturing the right variational object and the same Ricci–FΩ index-form mechanism, relies on an unproven characterization of cu(L) on noncompact manifolds to select a closed 1-form β with ∥θ−β∥∞<√(2k), omits the paper’s boundedness assumption ∥θ∥∞<∞, and contains a wrong claim T=ℓ(x)/√(2k). These gaps are substantive and not merely cosmetic.