Validity of the Neumann-condition argument in Disconzi–Khuri

Verify or refute the passage following equation (5.6) in Proposition 5.1 of Disconzi–Khuri that infers zero normal derivatives for the individual harmonic components from the zero normal derivative of their sum under the Neumann condition.

Background

The paper discusses positive-scalar-curvature, minimal-boundary noncompactness constructions and compares them with the result of Disconzi–Khuri. The authors explicitly state that they could not verify a specific step in the cited proof: the transition from a vanishing normal derivative of a sum to vanishing normal derivatives of its individual harmonic components. Resolving this issue is necessary to establish the validity of that argument.

References

We have been unable to verify the argument for the Neumann condition in Proposition~5.1, specifically the passage following (5.6) from a zero normal derivative of the sum to zero normal derivatives of its individual harmonic components.

— On critical dimensions for compactness in the boundary Yamabe problem, I  (2609.39945 - Gong et al., 30 Sep 2026) in Remark following Corollary 1.2 in Section 1, Introduction