The uniform Dirichlet-to-Neumann bound for Riemannian manifolds with boundary

Let $M$ be a Riemannian manifold with a boundary that need not be compact. The Dirichlet-to-Neumann operator and the associated mass-dependent quantities are as in Assumption

$$in the source. **Uniform Dirichlet-to-Neumann bound.** Assumption$$

holds for any Riemannian manifold $M$ with (not necessarily compact) boundary. The surrounding discussion notes that versions of this assumption are known under additional conditions, including when the mass has a large imaginary part, while the general case remains unresolved.