Bulk-dependence bounds for the Dirichlet-to-Neumann operator

Establish that for every Riemannian manifold with boundary γ, the Dirichlet-to‑Neumann operator DN^{γ,M}_{m^2} satisfies Assumption (i)–(ii): in the splitting DN^{γ,M}_{m^2} = √(Δ^γ + m^2) + (DN^{γ,M}_{m^2})′, the remainder (DN^{γ,M}_{m^2})′ is a bounded operator; moreover, for any δ > 0 there exists C ≫ 1 such that for complex masses with Re(m^2) > C one has the norm bound ∥(DN^{γ,M}_{m^2})′∥ < δ·|m|. This is conjectured to hold for all Riemannian manifolds with boundary and would provide the technical input needed to derive the heat-kernel gluing formula purely from boundary and subdomain data.

Background

Section 3 introduces a technical assumption on the Dirichlet-to-Neumann operator that controls its dependence on bulk geometry. The operator is split as DN^{γ,M}_{m^2} = √(Δ^γ + m^2) + (DN^{γ,M}_{m^2})′, where the leading term depends only on boundary geometry and the subleading term encodes bulk effects. Assumption 3.1 requires (DN′) to be bounded and asymptotically small relative to |m| for large Re(m^2).

The paper proves the main gluing results for heat kernels under this assumption (e.g., when the metric is a product near γ), and provides examples where it holds (cylinders and product neighborhoods). The explicit conjecture asks for this assumption to hold for any Riemannian manifold with boundary, which would extend the gluing formula beyond the special cases and give strong control of bulk dependence in general.