  Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let
$\Delta_p$ and $D_p$ be the realizations of the real and complex Laplacians on
p forms with either Dirichlet or Neumann boundary conditions. We generalize
previous results in the closed setting to show that (M,g,J) is Kaehler if and
only if $Spec(\Delta_p)=Spec(2D_p)$ for p=0,1. We also give a characterization
of manifolds with constant sectional curvature or constant Ricci tensor (in the
real setting) and manifolds of constant holomorphic sectional curvature (in the
complex setting) in terms of spectral geometry.
